Parametric Solitons Due to Cubic Nonlinearities

PARAMETRIC SOLITONS DUE TO CUBIC
NONLINEARITIES
A THESIS SUBMITTED FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
By
Ka7imir Kolossovski
M.Sc.
School of Mathematics and Statistics,
University College,
The University of New Smith Wales,
Anstralian Defence Force Academy.
Angnst 2001

I hereby declare that this snbmission is my own work and to
the best of my knowledge it contains no material previonsly
pnblished or written by another person, nor material which to
a snbstantial extent has been accepted for the award of any
other degree or diploma at UNSW or any other edncational
institntion, except where dne acknowledgement is made in the
thesis. Any contribntion made to the research by colleagnes,
with whom I have worked at UNSW or elsewhere, dming my
candidatme, is fnlly acknowledged.
I also declare that the intellectnal content of this thesis is the
prodnct of my own work except to the extent that assistance
from others in the project's design and conception or in style,
presentation and lingnistic expression is acknowledged.
Ka7imir Kolossovski

Abstract
The main snbject of this thesis is solitons dne to degenerate parametric fom-wave
mixing. Derivation ofthe governing eqnations is carried ont for both spatial solitons
(slab wavegnide) and temporal solitons (optical fibre). Higher-order effects that are
ignored in the standard paraxial approximation are discnssed and estimated.
Detailed analysis of conventional solitons is carried ont. This inclndes discovery
of varions solitons families, linear stability analysis of fnndamental and higher-
order solitons, development of theory describing nonlinear dynamics of higher-order
solitons. The major findings related to the stationary problem are bifmcation of
a two-freqnency soliton family from an asymptotic family of infinitely separated
one-freqnency solitons, jnmp bifmcation and violation of the bonnd state principle.
Linear stability analysis shows a rich variety of internal modes of the fnndamental
solitons and existence of a stability window for higher-order solitons. Theory for
nonlinear dynamics of higher-order solitons snccessfnlly predicts the position and
si7e of the stability window, and varions instability scenarios. Eqnivalence between
direct asymptotic approach and invariant based approach is demonstrated.
A general analytic approach for description of localised solntions that are in
resonance with linear waves (qnasi-solitons and embedded solitons) is given. This
inclndes norm:::tl form theory :::tnd :::tpproxim:::ttion of inter:::~cting p:::trticles. The m:::tin
resnlts are an expression for the amplitnde of the radiating tail of a qnasi-soliton,
and a two-fold criterion for existence of embedded solitons.
Inflnence of nonparaxiality on soliton stability is investigated. Stationary insta-
bility threshold is derived. The major resnlts are shift and decreasing of the si7e of
the stability window for higher-order solitons. The latter is the first demonstration
of the destabili7ing inflnence of nonparaxiality on higher-order solitons.
Analysis of different aspects of solitons is based on nniversal approaches and
methods. This inclndes Hamiltonian formalism, consideration of symmetry proper-
ties of the modeL development of asymptotic models, constmction of pertmbation
theory, application of general theorems etc. Thns, the resnlts obtained can be
extended beyond the particnlar model of degenerate fom-wave mixing.
All theoretical predictions are in good agreement with the resnlts of direct mi-
merical modelling.
111

Acknowledgements
The resnlts presented in this thesis were obtained in three-year research condncted
at the Anstralian Defence Force Academy Campns of UNSW. It is hard to express
all gratitnde to my snpervisors, Prof. Rowland Sammnt and Dr. Alexander Bmyak.
From the very start and np to now, I felt their wise gnidance, nnderstanding and
readiness to help. Thank yon, my Teachers!
In many respects, snccess of almost any research depends on working atmo-
sphere, relationship between colleagnes and encomagement. I am very gratefnl to
om relatively smalL bnt very friendly mathematical commnnity in ADFA for pro-
viding this. My special thanks are to Alexander (Sasha), who encomaged me in
tronblesome periods. His motto "If a problem does not give np straight away it has
something interesting to hide'· will accompany my fntme research. Many thanks to
Victoria Steblina whose help and snpport extended far beyond the scientific related
matters. T am thankfnl to my colleagne, Tsaac Towers, for interesting discnssions.
Also, his expertise in finding reqnired information helped me a lot.
I wonld like to thank Prof. Alan Champneys for his kind invitation to visit his
department at University of BristoL UK. His knowledge, experience and vivid inter-
est in different kinds of problems made om project "Mnltipnlse embedded solitons
:::ts bonnd st:::~tes of qn:::tsi-solitons " one of the most interesting p:::trt of my Ph.D.
Also, his hospitality made my stay in Bristol enjoyable and nnforgettable.
Nowadays theoretical analysis, especially in applied fields, is closely accompa-
nied by extensive nnmerical experiments. I deeply acknowledge ANU Optical Sci-
ences Centre for allowing me to nse its compnter resomces. I am gratefnl to Prof.
Ymi Kivshar (now Nonlinear Physics Gronp at ANU) for his interesting comse of
seminars thronghont all these years. I also appreciate interesting discnssions with
Andrey Snkhornkov.
Finally I gratefnlly acknowledge the financial snpport from Anstralian Gov-
ernment (International Postgradnate Research Scholarship) and ADFA (University
College Scholarship).
v

Publications
Refereed papers
1. K. Y. Kolossovski, A. V. Bmyak R A. Sammnt, "Quadrati~ snlitary 1nm;~s in
a ~mmt~rprnpagating quasi-phas~-mat~h~d ~nn.figuratinn", Optics Letters, 24,
835-837 (1999).
2. K. Y. Kolossovski, A. V. Bmyak V. V. Steblina, A. R Champneys, R A.
Sammnt, "High~r-nrd~r rwnlinmr mnd~s and bifurmtinn ph~rwm~na du~ tn
d~g~n~rat~ param~tri~ .fnur-u;m;~ mixing", Physical Review K 62, 4~09-4~17
(2000).
3. K. Y. Kolossovski, A. V. Bmyak R A. Sammnt, "Stability n.f high~r-nrd~r
rwnlinmr mnd~s du~ tn d~g~n~rat~ .fnur-1JJn1J~ mixing", Physics Letters A, 279,
355-360 (2001).
4. K. Y. Kolossovski, A. V. Bmy:::tk D. V. Skry:::tbin, :::~no R A. S:::~mmnt, "Nnn-
linmr dynami~s n.f high~r-nrd~r snlitnns n~ar th~ nsdllatnry instability thm~h
nld", Physical Review E, 64, 056612, 1-11 (2001).
Conference proceedings
1. K. Y. Kolossovski, A. V. Bmyak ano R A. Sammnt, "Quadrati~ bright snli-
tnns: ~mmt~rprnpagating s~h~m~ ", Technical Digest on Nonlinear Gnioeo
Wave Phenomena (Optical Society of America, Washington D.C., 1999), 58-60
(1999).
2. K. Y. Kolossovski, A. V. Bmyak ano R A. Sammnt, "Stability n.f quadrati~
snlitnns in a ~nunt~rprnpagating ~nn.figuratinn ", Proceeoings of 24th Ans-
trali:::tn Conference on Optical Fibre Technology, pg. 72 (1999).
3. K. Kolossovski, A. V. Bmy:::tk :::~no R A. Sammnt, "lnt~rnal mnd~s and in-
stabiliti~s n.f param~tri~ s~4f-trapping in K~rr planar 1nm;~guid~s ", Qnantnm
Electronics ano L:::tser Science Conference, QELS (San Francisco, California,
May 2000), OSA Technical Digest, 47-48.
4. A. V. Bmyak K. Y. Kolossovski, ano R A. Sammnt, "lnflu~n~~ n.f nnn-
paraxiality nn snlitnn stability ", Nonlinear Optics: M:::tteri:::tls, Fnnoamentals,
ano Applications (K:::tnai, Hawaii, USA, Angnst 2000), OSA Technical Digest,
350-352.
Vll

5. K. Kolossovski, A. V. Bmyak and R A. Sammnt, "Stability analysis nf snli-
tnns du~ tn param~tri~ 1nm;~-mixing in K~rr planar 1nm;~guid~s ", International
workshop on Nonlinear Gnided Waves, Institnte of Advanced Stndies, Ans-
tralian National University 2000.
6. K.Y. Kolossovski, A. V. Bmyak D. V. Skryabin, and R A. Sammnt, "High~r
nrd~r snlitnns: rwnlin~ar dynami~s n~nr th~ nsdllatnry instability thr~shnld ",
Nonlinear Gnided Waves and Their Applications (Clearwater, Florida, March
2001), OSA Technical Digest, 214-216.
7. K.Y. Kolossovski, A. V. Bmyak and R A. Sammnt, "Quasisnlitnns du~ tn
parnm~tri~ fnur-1JJn1J~ mixing", OECC/IOOC Incorporating ACOFT (Sydney,
.Jnly 2001), Conference Proceedings, 460-461.
Vlll

Contents
Declaration
Abstract
Acknowledgements
Pnblications
Chapter 1 Introdnction
1.1 From solitary waves to solitons
1.2 Family of solitons .....
1.3 Solitons or solitary waves?
1.4 Optical solitons .
1.5 Snbject of thesis .
1.6 Ontline ofthesis .
Chapter 2 Fnndamental eqnations
2.1 Degenerate fom-wave mixing in a bnlk xC3
) medinm
2.2
2.~
2.4
2.5
2.6
2.7
Degenerate fom-wave mixing in a x(3) slab wavegnide
Temporal solitons d11e to degenerate fom-wave mixing.
Conventional normalintion
Symmetries, invariants and other related issnes .
Higher-order corrections
Conclnsion ....... .
Chapter 3 General classification of stationary solntions
3.1 Nnmerical methods ......... .
3.2 Physical and nnmerical normalintion
3.3 Conventional solitons
3.4 Qnasi-solitons ...
3.5 Embedded solitons
3.6 Conclnsion .....
1X
111
v
V11
1
1
2
3
4
5
6
7
7
9
10
12
13
14
17
19
19
20
21
23
24
27

Chapter 4 Conventional solitons
4.1 Continnation method and bifmcation diagram
29
29
4.2 Bifmcation from one-freqnency family and asymptotic families 32
4.~ Violation of the "bmmd state'· principle ~6
4.4 '.Jnmp' bifmcation 38
4.5 Snmmary and discnssion 39
Chapter 5 Linear stability of conventional solitons
5.1 Preliminary discnssions ........... .
5.2 Stability threshold for fnndamental solitons .
5.3 Nnmerical methods ....... .
5.3.1 Beam propagation method
5.3.2 Fomier decomposition method .
.1.4 Stability res11lts . . . . . . . . . . . .
5.4.1 Stability of one-wave family .
5.4.2 Symmetric two-wave families .
5.4.3 Asymmetric two-wave family .
5.5 Discnssion .
5.6 Conclnsion .
Chapter 6 Nonlinear dynamics of stability
6.1 Asymptotic model: invariant based approach
6.2 Asymptotic model: direct approach
6.3 Nnmerical resnlts .....
6.3.1 Families of solitons
6.3.2 Calcnlation of the coefficients
6.3.3 Linear limit . . . . .
6.3.4 Instability scenarios .
6.4 Conclnding rem:::trks .....
Chapter 7 Qnasi-solitons and embedded solitons
7.1 Reversible two degree of freedom Hamiltonian systems
7.2 Asymptotic analysis
43
43
44
48
48
49
.12
53
54
56
56
59
61
62
66
70
70
72
73
74
80
R1
82
87
7.2.1 Exact solntion . 87
7.2.2 Qnasi-solitons . 88
7.2.3 Embedded soliton as a bmmd state of two qnasi-solitons 90
7.3 Nnmerical resnlts .....
7.4 Conclnsion and discnssion
Chapter 8 Nonparaxial solitons
R.1 Stationary instability threshold for nonparaxial solitons
X
93
96
99
100

8.2 Nnmerical resnlts ..... .
8.2.1 Stationary solntions .
8.2.2 Stability window
R.~ Conclnsion ....... .
Chapter 9 Snmmary and discnssion of open qnestions
9.1 Conventional solitons ........ .
9.2 Qnasi-solitons and embedded solitons
9.3 Nonparaxial solitons ..
9.4 Beginning of the Boom?
Appendix A Third order terms for the invariant-based approach
References
Xl
103
104
105
10R
111
111
112
113
113
115
116

CHAPTRR 1
Introduction
There never was in the world two opinions alike, no more than two hairs
or two grains; the most nniversal qnality is diversity.
Michel de Montaigne, French essayist
Of the Resemblance of Children to their Fathers
1.1 From solitary waves to solitons
To a physicist, the history of solitons started abont a centmy and a half ago from an
accidental bnt fatefnl encmmter in a barge canal. In 1834 a Scottish scientist and en-
gineer, John Scott Rnssell (1808-1882), was astonished by the sight of a bell-shaped
w:we th:::~t dep:::trted from the front of :::tn :::tbmptly stopped bo:::tt (Rnssell 1844).
The wave started its own long trip and travelled for abont two miles withont no-
ticeable change in the shape and velocity. This inspired Rnssell to carry ont an
immense nnmber of experiments trying to investigate and nnderstand this phe-
nomenon. Among the smprising properties of the Great Solitary Waves (he also
called them "the Wave of Translation") were dependence of their velocity on the
amplitnde, decay of a solitary wave with large enongh amplitnde into smaller waves
and preservation of the form and velocities of two colliding solitary waves.
Being essentially nonlinear objects, solitary waves were for a long time ontside
the main direction of modern science. The excitement cansed by great progress
in the nnderstanding and application of snch linear phenomena as sonnd waves,
electricity and magnetism postponed recognition of solitary waves for many years.
The pessimistic attitnde of leading scientists towards solitary waves did not change
even after de Bonssinesq (1871), Rayleigh (1876) and Saint-Venant (1885) confirmed
the possibility of solitary waves by establishing a mathematical basis for the shallow
water problem.
To a mathematician, the starting point in developing a mathematical theory of
solitons was a work by Korteweg and his stndent de Vries where they constmcted
a simple model for the shallow water problem and demonstrated the possibility
of solitary wave generation from a long periodic wave. They derived what we
now call the KdV eqnation (Korteweg and de Vries 1895). Despite this progress,
the research of solitary waves was still not an active topic. After some period of
1

relative inactivity the solitary waves of Rnssell were fnlly recogni7ed as a valid
area of research. The most striking property of solitary waves, their particle-like
behaviom, was revealed by Zabnsky and Kmskal (1965) in nnmerical experiments.
They renamed the "solitary wave" a "solitron". Rnt the latter term happened to be
the name of a registered firm so it was transformed into 'soliton'. After this work
the mathematical theory of solitons has been developing by the collective efforts of
scientists from many cmmtries. A new branch of mathematical physics has been
fonnded.
1.2 Family of solitons
While solitons were on the w:::ty to recognition, p:::trticle-like objects were discovered
in many different branches of science, e.g. biology, hydrodynamics, cosmology. In
1868 Bernstein fonnd that the time dependence of electric potential in a nerve had
the form of a bell-shaped pnlse. Investigating Enler's eqnations describing dynamics
of an idealliqnid, in 1858 Helmholt7 discovered that vortices and vortex rings behave
like interacting particles. Tn his work on a similar topic, Kelvin nncovered particle-
like behaviom of conpled vortices (1869). In 1848 vortices were fonnd in miter
space when Parsons, an Irish astronomer, nsed the biggest telescope at that time
to observe spiral stmctmes in many nebnlae.
Since then the family of solitons has immensely increased. For example, solitons
were fmmd in solid state physics (Frenkel-Kontorova solitons of dislocations, mag-
netic solitons of domain walls), in oceanology (tsnnami, envelope solitons), in cos-
mology (vortex-like objects in the atmospheres of the planets and stars). Examples
of the "modern" newcomers are qnanti7ed vortices in snperflnids and snpercondnc-
tors, Josephson's solitons in a layer ofthin insnlator connecting two snpercondncting
materials, solitons in particle physics (magnetic monopoles, skyrmions), vortices,
classical and qnantnm solitons in nonlinear atom optics (solitons in Bose-Einstein
condensation), bright and dark solitons, rings, propellers, light bnllets (solitons in
nonlinear optics). For fmther details see, e.g. (Remoissenet 1996).
As we have seen, the family of solitons inclndes a great variety of members. They
differ by age, place of residence, dimensionality, characteristic scale etc. Classifica-
tion of these species is directly related to the mathematical place of the solitons'
birth, i.e. to the basic types of differential eqnations which allow soliton-like soln-
tions. B:::tsed on this :::tppro:::tch, we c:::tn distingnish the three m:::tin represent:::~tives:
• solitons of the KdV eqnation Ut + Uxxx + UUx = 0,
• solitons of the sine-Gordon eqn:::ttion Utt- Uxx +sin u = 0,
• solitons of the Nonlinear Schrodinger eqnation iut + Uxx + lul2
u= 0,
where u = u(t, x) is a fnnction describing the soliton profile and the actnal meaning
of x and t differs from problem to problem. Knowledge of these members can
2

dramatically help in the nnderstanding of other relatives. One of the key properties
of the KdV solitons is that their length is inversely proportional to the sqnare root
of amplitnde, whereas in the case of nonlinear Schrodinger (NLS)-like solitons it is
inversely proportional to the amplitnde. Sine-Gordon solitons are the head of the
vast branch of topological solitons.
1.3 Solitons or solitary waves?
Strictly speaking, the term 'soliton' refers to a localised solntion of an int~grabl~
model. Dne to their origin, ideal solitons are completely stable. Existence of in-
finitely many conserved qnantities of integrable models makes the interaction of
solitons f:::tiry trivi:::tl. For ex:::tmple, the only resnlt of collision of two solitons is :::t
shift in the low of motion, i.e. in the dependence of the centre of soliton position
on time.
The nnmber of physical phenomena that can be described in terms of integrable
models forms only a small part of the whole diversity of real complex processes.
The goal of a physicist is development of a model that adeqnately describes the
major properties of a phenomenon while neglecting the minor details. Some times,
this becomes possible by adopting exactly integrable or nearly integrable models.
Bnt generally, an adeqnate description of an actnal process leads to formnlation of
an essentially nrm-int~grnbl~ model. In this context, the non-integrable models are
of special importance on the way to the nltimate aim of physics- nnderstanding of
Natme.
Stationary localised solntions to a non-integrable model are called 'solitary
waves'. The difference between solitons and solitary wave is principal for a math-
ematician as it indirectly refers the class of methods nsed for analytical treatment
of the nnderlying model. For a physicist, this distinction is not so cmcial since
it corresponds to models with different levels of approximation. Thns, in modern
literatme on physical snbjects, the term 'soliton' often refers to stationary localised
solntions of both integrable and non-integrable models. Keeping in mind the above
mentioned difference, fmther on we nse the term 'soliton' in a physicaL rather than
in a strict mathematical sense.
Solitons of non-integrable models have a few distinct properties. All solitons
of some particnlar model can be combined into families which are described by
snlitrm param~t~rs. In some region of these p:::tr:::tmeters, solitons might h:::tve int~rnal
modes and exhibit nontrivial instability indnced dynamics, e.g. decay, persistent
oscillations, snake-like instability. Interaction of these solitons is a rich snbject
itself which inclndes, e.g. soliton spiraling, emission of radiation dming collision,
formation of bonnd states and many other phenomena. All these properties make
the stndy of solitons arising in non-integrable models extremely interesting.
3

1.4 Optical solitons
One of the vast branches descending from solitons of the Nonlinear Schrodinger
eqnation inclndes npt1ml snliinns. Among this gronp one can distingnish two main
types. Temporal solitons correspond to stationary wave packets localised in time
(pnlses) whereas spatial solitons are signals localised in space (self-gnided beams).
In the c:::tse of these solitons, the n:::~tm:::tl bro:::tdening of pnlses dne to dispersion or
diffraction is compensated for by focnsing properties of nonlinearity.
Existence of temporal solitons in an optical fibre was predicted in 1973 when
Hasegawa and Tappert demonstrated that propagation of a pnlse throngh a weakly
nonlinear optical fibre was described by the NLS eqnation (Hasegawa and Tappert
19n). Tn 19RO this prediction was snccessfnlly confirmed (Mollenaner Pi nl. 19RO).
The main reason for the interest of the scientific commnnity in optical solitons
following this discovery relates to the possibility of making a highly effective com-
mnnication system and in developing of all-optical signal processing.
The concept of a spatial soliton as a self-gnided beam was snggested by Askar'yan
(1962) in the context of :::t medinm with Kerr nonline:::trity. Experiment:::~] observ:::ttion
of spatial solitons in sodinm vapor (Bjorkholm and Ashkin 1974) and liqnid carbon
disnlphide-CS2 (Barthelemy Pi nl. 1985) were the main marks of the 'Kerr' period
of nonlinear optics (Aitchison Pi nl. 1990, Kang Pi nl. 1996, Bartnch Pi nl. 1997).
Dming this time the mechanisms responsible for the Kerr soliton formation (focns-
ing of a beam d11e to increase of the local refractive index) and drawbacks of the
Kerr media (weakness of nonlinear properties) were fnlly nnderstood. Snbseqnent
interest focnssed largely on the search for solitons in other types of materials.
One gronp of solitons that have received a great deal of attention over the
past five years are the qnadratic solitons predicted by Karam7in and Snkhomkov
(1976). This type of soliton dramatically differs from those mentioned above. The
beam-trapping mechanism is dne to parametric wave-mixing, leading to energy
exchange between the fnndamental and the second harmonic. Starting from the
first observations (Tormellas Pi nl. 1995, Schiek Pi nl. 1996, Stegeman Pi nl. 1996),
qnadratic solitons are now the snbject of very many experiments (Lope7-Lago Pi nl.
2001, Conderc Pi nl. 2001, Carrasco Pi nl. 2001). Dne to the fast electronic response
of materials (np to femtoseconds), qnadratic solitons revive the hope for bnilding
of nltra-fast switching and beam manipnlation schemes, Fig. LL
The diversity of optical solitons does not end with those mentioned above, see
e.g. reviews (Zakharov and Wabnit7 1999, Stegeman Pi nl. 2000). In this section
we covered only those types of solitons that are the most relevant to the snbject of
the thesis.
4

(a) (b)
Figme 1.1: M<mip11lation of light by light. (a) M11tlJal deflection of q11adratic:
solitons, (b) f11sion of two q11adratic: solitons. !From Bmyak and Steblina (1999)1.
1.5 Snhject of thesis
In this thesis we present the detailed analysis of solitons d11e to degenerate para-
metric: fom-wave mixing. The c:orresponding non-integrable model inc:h1des terms
featming both nonparametric: interaction (self- and c:ross-phase mod11lation) and
parametric: mixing (third harmonic: generation). This ab1mdanc:e of different nonlin-
ear terms promises the model to be ric:h and extremely interesting. Indeed, rec:ently
parametric: wave mixing in Kerr media has attracted signific:ant theoretic:al atten-
tion (Saltiel Pi nl. 1997, Samm11t Pi nl. 1998, Bmyak Steblina and Samm11t 1999).
On the other hand, the theoretic:al analysis of S11c:h a model is timely d11e to rec:ent
experimental advanc:es, e.g., a novel sc:heme for q11asi-phase matc:hed third harmonic:
generation has been s11ggested (Williams Pi nl. 1998, Sc:hneider Pi nl. 2001).
Thf' main SlJbjPct of thf' thPsis is analysis of highf'r-ordf'r solitons. In prPvimJs
works devoted to spatial solitons d11e to third harmonic: generation in planar waveg-
llides, only families of f1mdamental self-trapped beams were mm:idered in detail.
Problems Sllc:h as types, stability and dynamic:s of higher-order families have been
left entirely 1minvestigated. However, higher-order modes have rec:ently mme into
the foc:11s of interest of modern nonlinear sc:ienc:e. Stable m11ltih11mp sc:alar and vec:-
tor solitons d11e to nonparametric: interactions are known in some c:ases !for c:m1pled
NLS type eq11ations with satmable nonlinearity (Ostrovskaya Pi nl. 1999), the gen-
erali7ed Korteweg-de-Vries eq1mtion (Bmyak and Champneys 1997), nonlinear op-
tic:al c:m1pler model (Malomed 1995)1. Also a dynamic:ally stable soliton spiraling in
b11lk satmable media was disc:overed (Bmyak Kivshar, Shih and Segev 1999). The
first indic:ation of possible stability of higher-order solitons d11e to proc:esses involv-
ing parametric: interaction was fmmd for gap solitons of waveg11ides with q11adratic:
nonlinearity (Pesc:hel Pi nl. 1997, Mak Pi nl. 1998), where solitons were fmmd to be
m1meric:ally rob11st.
5

In addition to the extensive analysis of the solitons mentioned above, we also
consider other types. These inclnde stationary solntions that are in resonance with
linear waves (qnasi-solitons and embedded solitons) and nonparaxial solitons.
The long history of soliton science has res11lted in the appearance of a commonly
established 'schednle' for almost any trip into the world of solitons for a new model.
These gniding marks inclnde attending to the following topics: search for the sta-
tionary localised solntions of the model. stability analysis of the solitons and their
dynamics, interaction between solitons. Dne to time limitations, this thesis will
cover the first three steps from the above mentioned list. Please, take yom seats -
om trip is abont to start!
1.6 Ontline of thesis
The main line of the thesis is solitons dne to degenerate fom-wave mixing. To anal-
yse different aspects of these solitons we nse snch nniversal approaches and methods
as Hamiltonian formalism, consideration of symmetry properties of the model. de-
velopment of asymptotic models, constr11ction of pertmbation theory if a small
parameter is presented, application of general theorems etc. Some obtained resnlts
can be extended beyond the particnlar model of degenerate fom-wave mixing. They
can be readily applied or generali7ed, if reqnired, to other physical models. Nowa-
days, there is a close relation between theoretical analysis and nnmerical modelling
of a non-integrable system. As a resnlt, almost all chapters consists of two parts,
analytics and nnmerics.
The ontline of the thesis is as follows. We start from the derivation of the model
for degenerate parametric fom-wave mixing in a slab wavegnide (spatial solitons)
and in an optical fibre (temporal solitons). After the standard system of parabolic
eqnations is obtained we discnss the higher order phenomena snch as vectorial
effects, conpling between the transverse and longitndinal components of the field
and nonparaxiality (chapter 2). General classification of solitons dne to the paraxial
model is given in chapter~- Detailed consideration of conventional solitons inclnding
stability analysis and nonlinear dynamics is yielded in chapters 4,5 and 6. Analytical
treatment of qnasi-solitons and embedded solitons based on the normal form theory
and particle-like approximation is done in chapter 7. Neglecting all higher-order
effects except nonparaxiality, we investigate inflnence of small nonparaxial terms on
st:::tbility of solitons in ch:::tpter 8. Ch:::tpter 9 is devoted to conclnsion :::tnd discnssion
of fntme prospects.
6

CHAPTRR 2
Fundamental equations
The grand aim of all science is to cover the greatest nnmber of empirical
facts by logical dednction from the smallest nnmber of hypotheses or
axioms.
Albert Einstein
In this chapter we derive two kinds of basic eqnations which describe spatial
and temporal solitary waves resnlting from degenerate fom-wave mixing. The first
case corresponds to a resonant interaction between a wave of freqnency w and its
third harmonic taking place in a slab wavegnide. The second model deals with
pnlses propagating along a weakly nonlinear optical fiber. We stress only the most
important steps of the derivation. For a more detailed analysis see, e.g., excellent
works with comprehensive derivations of similar models (Menynk ~t al. 1994, Bang
1997, Etrich ~t al. 2000).
2.1 Degenerate four-wave mixing in a bulk x(3) medium
Let 11s consider the propagation of light throngh nonmagnetic media withont free
cmrents or charges. We start from Maxwell's eqnations (in Ganssian nnits)
~ 18H
7 X E = - - -
c at '
~ 18D
7 X H = - -
c at '
7 · H = 0, 7 · D = 0,
(2.1)
where E, H and D are the vectors of electrical field, magnetic field and electric
displacement, respectively. Eqnations (2.1) rednce to the expression
(2.2)
In the framework of the electric dipole approximation the response of the medinm
is assnmed to be local. In this case, for sufficiently weak fields, the displacement
vector can be presented in the simple form:
7

where x(l) is the first-order snsceptibility tensor of rank 2 and pNL is nonlinear
polarintion. Assnming that the nonlinearity is weak we can expand the fields in a
snm of nearly monochromatic waves. For convenience, we firstly extract the explicit
time dependence of the fields and present them in the following form
E(r, t) = E1(r)e-iwt + E3(r)e-3iwt +c.c.,
D(r, t) = D1(r)e-iwt + D3(r)e-3iwt + c.c.,
pNL(r, t) = p~VL(r)e-iwt + p~VL(r)e-3iwt + c.c.,
(2.4)
where c.c. stands for complex conjngate. Independence of the envelope of electric
field on time means that the gronp-velocity dispersion, and therefore the temporal
walk-off, do not enter the problem. Tn this notation, the relation between nonlinear
response of the medinm and the electrical field can be expressed as
P NL - ~ (3) ( - )E E E*j,a - ~ Xa,;3,,,8 Wj - wk +Wz - Wm k,;3 z,, m,8· (2.5)
;3,,,8
Here snbscripts a, {3, (, 6 refer to Cartesian components of the fields and the tensor
x(3) (w) of rank 4 is the Fomier transform ofthe the third-order snsceptibility (Shen
1984). Now, snbstitnting the field presentations (2.4) into Eqs. (2.2) and considering
a homogeneons meoinm we obtain
(2.6)
where scalar dielectric permittivity Ej 1+47r J0
00
x(l)(t) exp(iwJt) dt and wJ = jw.
We note that, Eq. (2.6) is the direct resnlt of Maxwell's eqnations for harmonic
fields and no qnantitative approximations have been made yet. The next step is
simplification of Rq. (2.6) by red11cing it to the scalar form.
Degenerate £om-wave mixing corresponds to interaction ofthe fnndamental wave
with its third harmonic snch that wave vectors of all fields are parallel. Withont loss
of generality, we assnme that electric field is linearly polari7ed in the x direction. We
present the fields (2.4) in the form of slowly varying complex envelopes propagating
along the z direction
pNL(r) = { pNL(r)eikjzJ X J )
(2.7)
where the nnit polari7ation vector {x is assnmeo to be real. In general. the ele-
ments x~3
~ 8 of the nonlinear snsceptibility are different. For a particnlar case of,tJ,{,
an isotropic cnbic crystal with non-resonant mechanism of nonlinearity, nonlinear
8

response of the medinm (2.5) takes the form
P['L = 3x(3)[IE1I2
E1 + 2IE312
E1 + E*2
(w)E3e-iilkz],
Pf!L = 3x(3)[1E312
E3 + 2IE1i2
E3+ ~E3 (w)eiilkz],
(2.8)
where xC3
) = x~~xx(w) = x~~xx(3w) and 6.k = 3k1 - k3 . Snbstitnting expres-
sions (2.7) and (2.R) into Rq. (2.6) and neglecting the second derivatives of the
envelope fnnctions E1 , E3 with respect to z (paraxial approximation) we obtain a
pair of conpled nonlinear eqnations
(2.9)
where the snbscript T refers to the transverse component of a vector, V} = []2 jox2
+
82
/8y2
, and the linear dispersion relation
(2.10)
has been invoked. To obtain Eq. (2.9) we have neglected the 7(lPNL) term which
contribntes to higher-order effects. Discnssion of the inflnence of this and other
higher-order terms is given in a snbseqnent section. Eqnations (2.9) constitnte
the fnndamental system describing degenerate fom-wave mixing in bnlk. We note
that plane-wave eigenmodes of Eqs. (2.9) were considered in detail by Podoshvedov
(1997).
2.2 Degenerate four-wave mixing in a x(3) slab waveguide
Let ns consider a slab wavegnide where all waves are tightly confined in one direc-
tion. Withont loss of generality we can assnme that a wavegnide is located in the
(x, z) plane. Derivation of the model for a slab wavegnide remains essentially the
same as in the case of interaction in bnlk bnt involves some minor changes. Namely,
the linear permittivity in Eq. (2.6) is now a fnnction of y. Each freqnency compo-
nent of the propagating beam is confined in the y direction by the linear refractive
index n(wj, y) = y'EJjj}. If the thickness of a slab gnide is mnch smaller than the
beam width it is possible to separate variables in EJ in Eq. (2.9)
(2.11)
9

Defining the permittivity of the cladding area as Ejo we obtain the following system
determining the transverse modes of the wavegnide
(2.12)
where the real eigenvalne TIJ is a constant of separation which is allowed to be
different for the two waves. Solving Eqs. (2.12) for the transverse modes one finds
eigenvalnes T/J and conseqnently the effect of the wavegnide on the linear dispersion
relations
(2.13)
Paraxial approximation (2.9) becomes
where coefficients qi are given by
(2.15)
where FJ are fnndamental modes of the wavegnide normali7ed to satisfy J~: F} dy = 1.
In this c:::tse, qJ :::tre of oroer nnity. Below we :::tssnme th:::~t this is :::tlw:::tys the c:::tse.
2.3 Temporal solitons due to degenerate four-wave mixing
Derivation ofthe paraxial model for temporal solitons is slightly different in compar-
ison to the spatial case, see e.g. (Hasegewa 1990, Akhmediev and Ankiewic7 1997).
The difference is that the wave envelopes (2.7) are now slow fnnctions of both spa-
tial coordinates and time. Inclnding the time dependence in the envelope fnnctions
affects the expression for electric displacement vector (2.~). Tnserting the corre-
sponding terms in expressions (2.4), (2.7) into relation (2.3), expanding EJ(r, t 1) in
a Taylor series armmd EJ(r, t) and applying the slowly varying envelope approxi-
mation we obtain the following relations
(2.16)
10

where an jowj an;awniW=Wj and nonlinear p0larinti0n iS given by relatiOnS (2.8).
Snbstitntion of Eand :5 into Eq. (2.2) and disregarding the 7 · (7 ·E) term leads
to the following system:
where the linear permittivity Ejo is related to the cladding area and
(2.18)
Assnming that the beam propagates along the z direction while it is tightly confined
in the transverse direction we can nse the method of separation of variables:
(2.19)
To find the transverse modes we shonld solve the following eigenvalne problem:
(2.20)
which allows one to determine the effect of the wavegnide on the linear dispersion
relation,
(2.21)
To simplify Eq. (2.17) we integrate ont the dependence in x andy and go into
a reference frame moving slowly with respect to the normali7ed gronp velocity at
the fnndamental freqnency, ki/k~. At last, assnming that the gronp velocities of
all waves are close to each other we arrive to the fnndamental system describing
propagation of temporal solitary waves in x(3) two-dimensional wavegnide:
2i 8A1 _ k~ 8
2
A1 12KX(
3
) ( lA 12A 2 lA 12A A*2A -i~kz) _ 0
k
:::J k :::J
2 + q1 1 1 + q2 3 1 + q3 1 3e - ,
1 uz 1 UT E1
2i 8A3 .k~ - k~ 8A3 k'£ 82
A3
---2z ----+
k3 [)z k3 OT k3 OT2
121TX(
3
) ( 2 2 1 3 i~kz)
+ q4IA3I A3+2q2IA1I A3+-q3A1e =0,
E3 3
(2.22)
where T t - k~ z is retarded time, kj okjIowlw=Wj are inverse gronp veloc-
ities of the waves. Coefficients qi are defined by expressions (2.15) where addi-
tional integration over x mnst be carried ont and the eigenmodes are normalised as
J_~: F} dx dy = 1. Again, we assnme that all qi = 1.
11

2.4 Conventional normali7:ation
Normalintion of the fnndamental eqnations to a dimensionless system is similar
in both spatial and temporal cases. To show this we start from the case of spatial
solitary waves (2.9). Introdncing the characteristic scales of the system snch as
beam width r0 and diffraction length zd = 2k1r6, and scaling amplitndes AJ
A - U ~ A - W ~ i/::,kz
1
- 6k1ro V~' 3
- 2k3ro V~ e '
(2.2~)
one obtains
(2.24)
where ~ ~kzd and a k3 /k1 = 3- ~/(2kir6). In the framework of paraxial
approximation a = 3. In dimensionless Eqs. (2.24), Z denotes the normali7ed
propagation distance, X is the transverse coordinate.
The temporal case is treated in a similar way. We normali7e Eqs. (2.22) mea-
sming retarded time in nnits of the pnlse dmation t0 and the propagation distance
in nnits of the dispersion length zd 2t6/lk{l:
(2.25)
The normali7ed eqnations take the form
au a2
u 1 1
i az + r ax2 + CgiUI2+ 2IWI2)U + 3U*
2
W = 0,
. aw . aw a2
w 3 [(
1 1
2 1 1
2) 1 3]za--z6-+s---a~W+- 9W +2U W+-U =0
az ax aX2 a 9 '
(2.26)
where parameters ~ ~kzd, a lk{!kii, 6 2t0 (k~ - k~)/lkii and coefficients
r -sign(k{), s -sign(ki). In eqnations (2.26), X denotes the normali7ed
retarded time.
In this thesis we almost exclnsively concentrate on the case of spatial solitons.
Temporal solitary waves are considered only for a particnlar case which corresponds
to the following conditions: (i) the fnndamental wave is in the regime of anomalons
dispersion (r = 1), (ii) the gronp velocity difference is negligible (5 = 0), (iii)
12

a is close to 3.0 snch that deviation of the factor in front of the sqnare brackets
in Eqs. (2.26) from nnity introdnces higher-order corrections and can be omitted.
With these assnmptions, both models for spatial (2.24) and temporal (2.26) solitons
can be combined into the following general system
(2.27)
with obvions meaning of the parameters and coefficients.
2.5 Symmetries, invariants and other related issues
System (2.27) can be presented as dynamical eq11ations of the Hamiltonian system
H{pi(Z),qi(Z)}, i = 1,2, with canonical variables (p1 ,q1) = (iU,U*) and (p2 ,q2) =
(iaW, W*) and Z as the evolntional parameter (time or propagation distance)
.CJU 6H .CJU* 6H
z - = - z - - = - -
CJZ 6U*' CJZ 6U'
. aw 5H . oW* 5H
(2.28)
za CJZ = 6W*' za CJZ =- 6W'
where the asterisk and 6 correspond to complex conjngate and the variational deriva-
tive respectively. The Hamiltonian can be readily calcnlated
1+oo{laul
2
lawl
2
1 4
9 4
H= - +- --lUI --IWI--00 ax ax 1s 2
- 2IUI2
IWI2
- ~ (WU*
3
+ W*U
3
) + a~IWI2 } dX
(2.29)
and is invariant with respect to change in Z. Invariance of Hamiltonian (2.29) nnder
some transformation corresponds to existence of some qnantity that is conserved
with respect to variation of Z. Hamiltonian (2.29) is symmetrical with respect to
tnmslation in the transverse direction
U(X, Z) ----+ U(X- X 0 , Z), W(X, Z) ----+ W(X- X 0 , Z), (2.~0)
and phase rotation
(2.31)
where constants X 0 and 'Po are parameters of the corresponding transformation.
This resnlts in conservation of momentnm
p = !._ l+oo {u* CJU- UCJU* +sa (w*CJW- WoW*)} dX
2 _00
ax ax ax ax
(2.32)
13

and power
(2.33)
respectively. Another direct conseqnence of the above mentioned symmetries of the
H:::tmiltoni:m is existence of so-c:::tlled int~rnal p:::tr:::tmeters of system (2.27). We nse
this term extensively thronghont the thesis. Internal parameters natmally appear
when transformation parameters are taken in the form of linear fnnctions of time.
For example, velocity C arises when one nses translation (2.30) with X 0 = CZ and
nonlinear phase shift {3 resnlts from snbstitntion 'Po= {3Z into phase rotation (2.31).
The combined transformation has the form
U(X, Z) = U(X- CZ, Z)eif3Z,
W(X, Z) = W(X- CZ, Z)e3if3Z
and Eqs. (2.27) written with internal parameters become
.au a2
u . au (1
1 12 1
l2) 1 *2z- + - - zC- + - U + 2 W U + -U W- {3U = 0
az ax2 ax 9 3 '
where tildes are omitted for brevity.
2.6 Higher-order corrections
(2.34)
The parabolic form of scalar Eqs. (2.9) is basic in wave optics and is well-jnstified.
Neglected nonparaxial terms and 7(i'PNL) are negligible for sufficiently wide
beams. For narrow beams their inflnence becomes important. Moreover, the 1J~~
tnrial effects become significant too. The latter relates to the fact that initially
completely transverse field gradnally gains a small longitndinal constitnent which
becomes nonlinearly conpled to other components of the field. One of the strict
methods to obtain post-paraxial corrections is nsage of radiation modes as a basis
for presentation of the overall field, see e.g. (Crosignani ~tal. 1997a), (Crosignani
~t al. 1997b). Here we nse an order-of-magnitnde analysis which allows ns to eval-
nate the order of effects we have neglected dming the derivation of (2.9). For the
basic concept of the method see (Marcnse 1982) and for an example of application
see (Sien and Gno 1995). Let ns assnme that the diffraction/dispersion and non-
linear effects are balanced. This allows ns to regard all terms in Eqs. (2.9) to be of
the same order. Then the following scaling holds
(2.36)
14

where p = 1/(4kir6) = .A2
/(167r2
r6),). is the wave length of the first harmonic and
r0 is the characteristic beam width. Generally, p is very small. For the maximnm
focnsed beam ro rv ). and p rv 10-2
.
Now we retmn to Rq. (2.6). First, to gain q11alitative 11nderstanding of the
vectorial effects instead of Eq. (2.7) we consider the following field presentation
E(r, t) = El(r)ei(kF-wt) + E3(r)ei(k3z-3wt) +c.c.,
pNL(r, t) = pfL(r)ei(k1 z-wt) + p~L(r)e(k3 z-3wt) +c.c.,
(2.37)
where the z projection of the fields is mnch smaller then the x component. From
the last expression in Eqs. (2.1) we obtain
(2.38)
We ass11me that the f11ndamental relations (2.~6) are still valid for the fields (2.~7).
Then the last two terms in (2.38) are of 1/2 order in p smaller then the first two.
In the leading order we obtain
(2.39)
i.e. transverse components of the field become conpled to the small longitndinal
component. From Eq. (2.39) it follows th:::~t Ej,z/IEJ,TI rv p1
12
. Dne to nonline:::tr
conpling an inpnt beam linearly polari7ed in the x direction gains a negligible y
component, EJ,y/EJ,x rv p. Appearance of the small z component (2.39) leads
to change in the resnlting nonlinear response. As a resnlt, Pf:xL is now defined by
linear conpling dne to x~~xx, x~~zz, x~~xz components ofthe third-order snsceptibility
tensor. In addition to (2.5) the nonlinear polarintion has small terms proportional
to x(3
) Ei,xEJ,zEk,z· Using (2.39) we obtain
(2.40)
Terms dne to vectorial effects are 1 order in p smaller than other terms in Eq. (2.9).
Secondly, the term 7(i'PNL) in Eq. (2.6) introdnces into Eqs. (2.9) a correction
that can be easily compared to any of the terms in Eqs. (2.9)
(2.41)
The last, inflnence of non-paraxial effects can be estimated straightforwardly.
Comparison of terms of the form 1/k] []2 EJ,x;l3z2
with any of the terms in (2.9)
15

gives
~ []2 E;,x / (2_OEj,x) rv p.
kj oz kl oz
(2.42)
Analysis of all mentioned higher-order effects reqnires qnite elaborate work. We
constrain it by consideration only the nonparaxial terms (Akhmediev, Ankiewic7
and Soto-Crespo 1993), (Fibich 1996). Namely, we consider the following nonparax-
ial model of degenerate fom-wave mixing
1 CJ
2
E1 2i CJE1 1 2 121TX(
3
) (I 12 I 12 *2 -i~kz)
k2~ + -k ~ + k2V rEI + E1 E1 + 2 E3 E 1 + E 1 E3e = 0,
1 uz 1 uz 1 E1
1 82
E3 2i CJE3 1 2 12KX(3) ( 1 )
k2~ + -k ~ + k2 'VrE3 + IE31
2
E3 + 21E1I
2
E3 + -
3Efei~kz = 0,
3 uz 3 uz 3 E3
(2.4~)
where the snbscript x is omitted for brevity. Derivation of the non-paraxial model
in the case of a slab wavegnide and the corresponding normali7ed eqnation is similar
to the sections 2.2 and 2.4. Eqs. (2.43) for a slab wave gnide take the form
1 8Ai 2i 8A1 1 82
A1 12KX(
3
) ( 2 2 *2 -i~kz)
k
2~ + -k ~ + k2~ + q1IA1I A1 + 2q2IA3I A1 + q3A1 A3e = 0,
1 uz 1 uz 1 uX E1
1 oA§ 2i 8A3 1 82
A3 12KX(
3
) ( 2 2 1 3 i~kz)
k§ CJz2 + k
3
oz + k§ ox2 + E
3
q4IA3I A3 + 2q2IA1I A3 + 3q3A1e = 0,
(2.44)
whereas normalintion procedme leads to eqnations with velocity and nonlinear
shift
o2
U o2
U au au
Paz2 - 2pCaxaz + i(1 + 2pf3) az- iC(1 + 2pf3) ax+
+(1 + pC
2
) ~~ - {3(1 + pf3)U + (~IUI2
+ 2IWI
2
)U + ~U*2
W = o,
o2
W o2
W aw aw
P az2 - 2pCaxaz + 3i(1 + 2pf3) az - 3iC(1 + 2pf3) ax+
82
W 1
+(1 + pC
2
) ax2 - (3{3 + ~)(3 + 3pf3- p~)W + (9IWI
2
+ 2IUI
2
)W +
9u3 = o.
(2.45)
Comp::uison between Eqs. (2.35) ::md Eqs. (2.45) shows th:::~t introdnction of non-
paraxial terms breaks the Hamiltoni:::tn stmctme of the corresponding paraxial
model. The main conseqnences are in the fact that power and momentnm fnnc-
tionals (2.33) and (2.32) are not conserved any longer. Instead, they slowly change
according to the expressions
oQ l+oo ( 8
2
U 82
W)CJZ = -2p -oo Im U* oz2 + 3W* oz2 dX,
CJP _ l+oo (CJU* 8
2
U oW* 82
W)CJZ - 2p -oo Re oX oz2 + oX oz2 dX.
(2.46)
16

Deviation from the power and momentnm conservation law is small and proportional
top.
2.7 Conclusion
In this chapter we have shown that propagation of spatial solitons confined in one
transverse direction and soliton-like p11lses confined in both transverse dimensions is
described by similar normali7ed dynamical models. It has been demonstrated that
the model for the temporal case has a more general form with parameters varying
in a broader range. We have also estimated the next order effects which are beyond
the standard paraxial approximation.
17

CHAPTRR 3
General classification of stationary solutions
Data withont generalintion is jnst gossip.
Robert Pirsig, U.S. writer
We start this chapter with a short review of the nnmerical methods nsed to solve
ordinary differential eqnations. Then we state the problem of finding the solitons
dne to degenerate fom-wave mixing and carry ont classification of the stationary
solntions. Discnssion of solitons is accompanied by a description of the nnmerical
methods specifically nsed in each case.
3.1 Numerical methods
The t:::~sk of solving onlin:::~ry oifferenti:::t1eqn:::~tions t h:::~t :::~re reqnireo to s:::~tisfy bonno-
ary conditions at more then one valne of the independent variable is called a twn
pnint bmmdary 11alu~ prnbl~m. The most common case is when bonndary condi-
tions mnst be satisfied at the start and the endpoint of the interval of integration.
There are two distinct classes of nnmerical methods for solving this problem (Press
~t al. 1996).
In the shnnting method one chooses initial valnes for all independent variables.
These valnes mnst be in agreement with all bmmdary conditions at the starting
point. The difference between the nnmber of independent variables and the mim-
ber of bonndary conditions at this point corresponds to the set of free parameters.
Initi:::~lly the v:::tlnes for these p:::tr:::tmeters :::tre gnesseo. After integr:::~tion of ODEs ns-
ing an initial valne problem solver, for example fomth-order Rnnge-Kntta method,
one obtains a discrepancy in the bonndary conditions at the endpoint. Adjnsting
the free parameters in an appropriate way one can decrease discrepancy or elimi-
nate it completely. In many cases it is sufficient simply to scan trongh the space
of free parameters, while in others more caref11l approaches m11st be nsed. The
latter inclnde methods of root-finding which consider discrepancy as a fnnction of
the adjnstable parameters. The shooting method is strongly recommended when
there is no information available regarding the songht solntion except the bmmdary
conditions.
19

If one has a fairly good gness for the songht solntion then the r~laxatirm method
might be snggested. In this method the differential eqnations are replaced by finite-
difference eqnations on a mesh that covers the interval of interest. The trial solntion
might not satisfy the finite-difference version of the original ODEs and even the
reqnired bonndary conditions. One step of iteration consists of simnltaneonsly
adjnsting all the valnes on the mesh to bring them into closer agreement with the
eqnations and the bmmdary conditions. Relaxation works better then shooting for
solntions that are smooth and not highly oscillatory.
3.2 Physical and numerical normali7:ation
Dyn:::tmic:::tl eqn:::ttions for 7ero-velocity solitons of p:::tr:::txi:::tl :::tpproxim:::ttion h:::tve the
form
(3.1)
This model has two external parameters a,~, one internal parameter {3 and one
switch s. Physical presentation (3.1) has clear meaning for linear coefficients and
is convenient for analytical derivations. Nnmerical methods become specifically
important when one looks for solntions to a non-integrable system. For nnmerical
modelling it is more practical to rednce the nnmber of coefficients after the scaling
U = v7Ju, W = v7Jw, X= xjv7J, Z = z/{3. (3.2)
Snbstitntion of this scaling into Eqs. (3.1) yields
.au [)2u (11 12 I 12) 1 *2z- + - + - u + 2 w u + -u w - u = 0,
oz ox2 9 3
ow 82
w 1
ia--;::;- + s~ + (9lwl2
+ 2lul2
)w + -u3
- aw = 0,
uz ux 9
(3.3)
where a= a(3{3+~)/{3. There is one-to-one correspondence between Eqs. (3.1) and
Eqs. (3.3) in the whole range of parameters except the singnlar points {3 = 0, a-----+ oo
and a = 3a, {3 -----+ oo. Stationary solntions to Eqs. (3.3) are defined by
1 1
u" + (-u2
+ 2w2
)u + -u2
w - u = 0
9 3 '
1
s w" + (9w2
+ 2u2
)w + -u3
- aw = 0.
9
20
(3.4)

System of ODE's (3.4) can be formnlated in terms of a two-degree-of freedom dy-
namical system with Hamiltonian
where Pu = u', Pw = w' and x is considered as time. Any solntion to Eqs. (3.4)
is a trajectory in the phase space (u,pu, w,pw)· A bright soliton corresponds to
a solntion that is homoclinic to 7ero whose trajectory asymptotically starts and
finishes at (0, 0, 0, 0) for x -----+ ±oo. We mainly concentrate on bright solitons and
for brevity refer to them as 'so1itons'.
Classification of solitons is similar in both physical and nnmerical models which
allows ns to consider only Eqs. (3.4). In the asymptotic regions x -----+ ±oo the
amplitndes of the both harmonics are vanishing. Linearintion of Eqs. (3.4) and
snbstitntion u, w rv e,x gives a qnadmplet of eigenvalnes
). = ±1, ±y!S"a. (3.6)
3.3 Conventional solitons
Conventional solitons have exponentially decaying asymptotics, i.e. may exist only
for sa> 0 in spectmm (3.6). If this condition is satisfied, the origin of system (3.5)
has stable and nnstable manifolds which smoothly transform into each other. This
separatrix transformation corresponds to a soliton. The key property of the solitons
is that they form families that are continnons in a. Depending on the sign of s,
solitons continnonsly exist for :::~11 a > 0 or a < 0. Existence of solntion for :::~11
parameters in some region of parameter space responds to a codimension 7ero event.
The term "codimension 7ero" means that variation of a does not lead to transition
between the regions of soliton existence and non-existence.
Eqnations (3.4) have odd symmetry, that is, if [u(x), w(x)] is a solntion then so
is [-u(x), -w(x)]. Th11s all sol11tions m11st come in pairs, the second sol11tion being
simply a change in sign (a phase shift of 1r) of both harmonics. Also it is possible
to have solntions which are odd in both harmonics, or which are neither odd nor
even. The latter type of solntions we shall refer to as being 'asymmetric'.
Closed analytical expression for solitons is known only for a limited nnmber of
cases (Sammnt ~t aL 1998). For s = L the simplest solntions is the one-freqnency
soliton for the third harmonic
u(x) = 0,
yl2a
w(x) = -
3
-sech(yax). (3.7)
21

5
4
(a)
3
~ 2~
;:5
1
0
-1
-10 0
X
s = 1
a=l
10
3
(b)
-3~~~~~~~==~~~~
-5 0
u,w
5
Figme ~.1: Soliton (~.R). (a) Profile, (b) phase plot. Thin and thick lines correspond
to the fnndamental wave and its third harmonic, respectively.
Exact form of tme two-freqnency soliton is known only for a = 1 when the soliton
is a self-similar one and is expressible as
u(x) =a sech x, w(x) = b u(x), (3.8)
where the p::u:::tmeter b is the re:::tl root of the cnbic eqn:::ttion 63b3
- 3b2
+ 17b+ 1 = 0,
and a2
= 18/(18b2
+ 3b + 1) (Fig. 3.1).
Approximate solntion is known for the so-called cascading limit, when a» L
3v'2u(x) = h + O(a-1
),
COS X
6v'2w(x) = 3
+ O(a-2
).
a cosh x
(3.9)
Other soliton families can be obtained with the help of nnmerical methods. Nn-
merical problem of finding solitons for any a > 0 can be formnlated in a simple way
exploiting the fact that Hamiltonian (3.5) is 7ero for all x. Indeed, it is a conserved
qnantity of ODEs (~.4) and is 7ero for x---+ ±oo. Fmthermore. symmetrical solitons
have 7ero derivatives at the point of symmetry x0 . The 7ero level of Hamiltonian
at x0 defines a closed cmve C(u, w) that has the form in polar coordinates
C(r, e) := {r =
18(a + [1- a] cos2 e)
o::::; e::::; 21r, }
46 cos4 e+ 4 cos3 esine- 126 cos2 e+ 81'
(3.10)
where u = r cos e, w = r sin e. Withont loss of generality, let llS consider Xo = 0.
To obtain a soliton for each a we integrate ODEs (3.4) from x = 0 nsing bonndary
conditions u = u(r, e),Pu = 0, w = w(r, e),Pw = 0. The res11lting solntion is soliton
if for large x it satisfies one of the Canchy conditions u' +u = 0, w' + fow = 0. In
this formnlation the problem has only one free parameter, e, with the finite domain
22

of definition. The simple version of shooting that scans throngh the region of the
free parameter is efficient to obtain solitons with high accmacy.
Utili7ing the fact that the conventional solitons form continnons families the
soliton profile known at some a0 can be nsed as a good gness for the neighboming
a's. Using the relaxation method we can slightly change a for each step and track
the soliton family far beyond a0 . The drawback of this approach is that it works
slowly for the points where derivatives oujoa, owjoa are large enongh or fails
completely for the tmning points. In snch cases the special rJmtinuatirm methods
mnst be nsed (Seydel 1994). We will discnss them in detail in the next chapters.
3.4 Quasi-solitons
The case sa < 0 corresponds to the saddle-centre origin of the £om-dimensional
phase space. For a » L the solntions are characteri7ed by the tmly locali7ed
fnndamental wave and the third harmonic being in resonance with linear waves,
see e.g. (Boyd 1998). Generally in this region one finds steady almost locali7ed
stmctmes, quasi-snlitnns or hnmnrlinirs tn pPrindir nrbits, which take the forms of
a soliton core with a non-decaying radiating tail. The key property of the qnasi-
solitons is that the amplitnde of radiating tails is an exponentially small fnnction
of the resonant freqnency w = M, i.e. rv e-w. An intrigning property of these
solntions is that for any fixed system parameter, a, there exist infinitely many qnasi-
solitons differing by the phase shift between the core centre and the oscillatory tail.
Dne to appearance of phase as an additional parameter controlling the stmctme of
these solntions, the phenomenon of qnasi-soliton is a codimension-minns-one event
in the parameter space.
For model (3.4) there is no qnasi-soliton known in the closed form. The qnasi-
soliton form in the limit of large lal can be asymptotically approximated by the
expression
3v'2
u(x) rv coshx'
6v'2w(x) rv 3 + Bsin( Mx +¢),
a cosh x
(3.11)
where the amplitnde of oscillatory tails B = B(a, ¢) depends on the phase ¢ of
the oscillatory tail with respect to the soliton core of w-component (Kolossovski
Pi al. 2001).
The fact that for each a there are infinitely many qnasi-solitons with different
phases makes shooting inconvenient as a tool for finding solntions. A relaxation
method can be nsed instead where, as agness, we nse (3.11) with B = 0. We look
for qnasi-soliton in the interval X E [0, L] for large L (L rv 30). The bonndary
conditions at the ends are u'(O) = 0, w'(O) = 0 and u(L) + u'(L) = 0, w(L) = 0.
The last condition fixes phase ¢between the core and the tail. Varying the length
23

(a) (b)
B B o.1
0.0 ~~--~--~~~--~----~ 0.0 L---~=="=~~~==~......J
0.0 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8
¢, 1t ¢, 1t
Figme 3.2: The amplitnde of oscillatory tails versns the phase deference between
tail oscillations and soliton core at s = L a= -20.5066.
of the interval from L to L+flL where flL = 1rjw we cover all possible valnes that
phase can take. The amplitnde-phase dependence is presented in Fig. (3.2) whereas
two examples of qnasi-soliton profiles are presented in Fig. (3.3).
3.5 Embedded solitons
In the saddle-centre region sa < 0 there can be conditions when the small-amplitnde
radiation is snppressed altogether either for a single qnasi-soliton or for two (or
more) qnasi-solitons combined in a stationary radiationless bonnd state. Represen-
tatives of the former class, radiationless solitons in resonance with linear waves, are
called ~mb~dd~d solitons. See for example the review (Champneys ~tal. 2001). Ex-
amples of embedded solitons are known in models arising from nonlinear optics, snch
as second-harmonic generation (SHG) (Yang ~tal. 1999, Bmyak and Kivshar 1995),
three wave interaction (Champneys and Malomed 2000) and a model for Bragg-
grating solitons (Champneys ~t al. 1998, Champneys and Malomed 1999). Also,
examples have been fmmd in nonlinear Schrodinger eqnations with higher-order
pertmbation terms (Bmyak 1995, Fnjioka and Espinos7 1997, Kivshar ~t al. 1998)
and in higher-order and conpled Korteweg de Vries eqnations that arise in theory
of water waves (Grimshaw and Cook 1996, Champneys and Groves 1997). This
type of soliton req11ires f11lfilment of special conditions and is a more rare event
then qnasi-solitons or conventional solitons. Generally they are isolated solntions,
that is they are of codimension one in the region of physical parameter space where
there is resonance with linear waves (Champneys and Harterich 2000). To obtain a
continnons family of embedded solitons the parameter space mnst have dimension
eqnal to or higher than 2. As a conseqnence, embedded solitons of model (3.4) are
isolated and exist for a discrete set of a.
24

5
4 (a)
3
~ 2~
~
1
0
-1
-10
5
(c)4
3
~ 2~
;;5
1
0
-1
-10
0 10
X
0 10
X
(b)
-3 ~k=~~~~~~=k~~
-5
3
-3
-5
0
u,w
0
U,W
5
5
Figme 3.3: Ex:::tmples of qn:::tsi-soliton profiles (a, b) :::~no the corresponoing ph:::tse
plots (c,d) taken at points E,F in Fig. 3.2, (b). Thin cmves oenote u, thick cmves
stano for w.
The cooimension of the embeooeo solitons is higher than that of conventional
ano qnasi-solitons. It makes the finoing of embeooeo solitons a more challenging
task. Another tronblesome point is that there is no closeo form or asymptotic
expression known for embeooeo solitons. To locate snch solntions one has to rely
on the nnmerics, e.g., nse the shooting methoo. Formnl:::~tion of the bmmo:::try
conoitions reqnires knowleoge of the asymptotic behaviom of the solntion at x -----+
±oo. This information can be oeonceo from the consioeration of an embeooeo
soliton as a raoiationless snperposition of qnasi-solitons. More oetaileo oiscnssion
ofthis approach will be given in chapter 7. From approximation (3.11) we have the
following bmmoary conoitions for the limit of large a
u'(x) ± u(x) = 0, w'(x) ± 3w(x) = 0 for x-----+ ±oo. (3.12)
25

5
0
5
0
-5
0
(a)
(c)
s = 1
a=- 4.087
10 20
X
s =- 1
a= 49.483
10
X
4
(b)
0
-4
30 -5
4
(d)
0
-4
20 -5
0
u,w
0
u,w
5
5
Figme 3.4: Examples of embedded soliton profiles and the corresponding phase
plots taken at s = L (a,b) and s = -1 (c,d). Thin cmves denote u, thick cmves
stand for w.
The mim1s sign corresponds to the direction of the 1mstable manifold whereas the
plns sign defines the retmn of the trajectory along the stable manifold. We shoot
nsing the following initial conditions
u=A, u'=A
'
A3
W=-
9a'
A3
w'=-
3a'
(3.13)
where small amplitnde A lies in the range 10-3 - 10-4
. Initial conditions (3.13)
satisfy the bonndary conditions (3.12) for the nnstable manifold and have one free
parameter, a. To locate a0 for which the embedded soliton exists we follow the
following strategy. For some a 1 we integrate Eqs. (3.4) over large enongh interval.
At the endpoint we estimate discrepancy between the resnlt and bonndary condi-
tions (3.12). Smallness of the discrepancy means that a 1 is close to the songht a 0 .
Repeating everything for some a 2 which still mnst be close enongh to a0 we obtain
26

second valne for discrepancy. Using linear interpolation from this set of data we
dednce the valne of &0 that rednces the discrepancy to 7ero. This method is a one-
dimensional realintion of more general Newton-Rapson method for root finding.
Provided the trial valne a 1 is close en011gh to a0 the method converges after 4-6
iterations leading to discrepancy with the desired bonndary conditions as small as
10-10
. Examples of embedded solitons are presented in Fig. (3.4).
3.6 Conclusion
The overview resnlts presented in this chapter have introdnced the object ofthe the-
sis, solitons dne to degenerate fom-wave mixing. Temporally leaving the qnestions
of soliton dyn:::tmics :::~side, we h:::tve shown :::tn extremely rich v:::triety of st:::~tion:::try
solntions. The problem of soliton existence in different regions of the parame-
ter space has been tonched only slightly. Despite this we have demonstrated the
difficnlties associated with the location and nnmerical investigation of solitons of
different types.
27

CHAPTRR 4
Conventional solitons
Beware of the man who won't be bothered with details.
William Feather, Sr.
In this chapter we extend the known resnlt regarding conventional solitons presented
in section (3.3). We consider mainly the solitons of symmetric type, bnt also present
some resnlts for the asymmetric case.
4.1 Continuation method and bifurcation diagram
In this section we analy7e in fnll detail the stmctme and bifmcation phenomena of
conventional solitons of Eqs. (3.4) for s = 1. Withont loss of generality we assnme
that the centre of symmetry of solitons is at x = 0. Then it is snfficient to seek a
solntion in the interval 0 ::::; x ::::; oo.
To classify solitons existing for different parameters a one can nse any fnnctional
which measmes the norm of the solntion. To characteri7e conventional solitons
it is common to nse the valne of normali7ed total power (2.33). If the form of
solntion is known at some point ai, (ui, wi), (was obtained by shooting techniq11e
for example) it can be nsed as a trial fnnction in the relaxation method to find
solntion (ui+1, wi+I) for a neighboring valne ai+l· The open qnestion is the choice of
ai+1 for which the next solitons shonld be songht. A simple approach of monotonic
increasing or decreasing the parameter is not always appropriate as a family of
solitons might have loops in the Q- a plane. The task of branch tracing (or path
following) can be snccessfnlly solved with the help of a rJmtinuatirm procedme.
There is a great nnmber of different continnation methods developed by now.
They differ in the complexity of the implemented algorithms, level of antomatin-
tion, speed of calcnlations, amonnt of compnter resomces needed etc. Despite this
variety they necessarily have the following components
• parameterintion strategy
• predictor
• corrector
• step length control.
29

Q
Q.
J
Q. 1J-
~U.,W.
J J
U. 1' W. 1J- J-
a. 1
a.
J- J
a
Figme 4.1: Arclength parameterintion.
Parameterintion can be thonght of as a measme along a branch which allows ns to
distingnish between different families and also to specify qnantitatively the position
of a solntion [u(x; a), w(x; a)]. The most obvions choice of the control parameter as
a might be not very nsefnl when dealing with the tmning points. We nsed arclength
parameterintion instead
where control parameter s is a monotonically increasing qnantity. Initial valne s0 =
0 corresponds to the starting point on the branch where solntion [u(x; a 0 ), w(x; a 0 )]
is assnmed to be known. Arclength "coordinate'" of the ith solntion is approximately
si = L ~sj, where ~sJ = V(QJ- QJ_1)
2
+ (aJ- aj_1)
2
, (4.2)
j=1
see Fig. 4.1 .
The predictor provides an initial gness for the corrector iterations. As a corrector
we nse code to solve ODEs (3.4) by the relaxation method. A simple predictor
might prodnce too rongh a gness that can resnlt in drastically increased nnmber
of iterations reqnired to find the tme solntion or in losing the branch completely.
On other h::md, too sophistic:::tted :::t predictor yielding :::t good :::tpproxim:::ttion c:::tn
greatly slowdown the work of the code. The optimal choice varies for different
30

8.2 8.6 9.0
100 12.2
Sy
11.8
75
11.4
Q 50
20 30
a
Figme 4.2: Bifmcation diagram for symmetric solitons of Eqs. (3.3), s = 1. Dotted
cmves emerging at 7ero correspond to integer mnltiples of the primary one-wave
solntion S1 . Formally they represent mnlti-soliton states consisting of a concate-
nation of infinitely separated single solitons. Points at which branches of two-wave
solitons terminate by 'bifmcating' from one of these m11lti-solitons are depicted by
filled circles and all occm for a = 9.
problems. Tn om case, extrapolation 11sing c11bic polynomial tmned ont to be the
most efficient. Known valnes of qnantity fJ (! = uj, wJ or aJ) at fom neighboring
points sj, j = 0, 1, 2, 3, can be nsed for extrapolation off for point s nsing the
Lagrange formnla
!=
(4.3)
When predicting fnnctions (u, w) extrapolation (4.3) shonld be nsed for each x.
The problem of step si7e control closely relates to the sharpness of the branches.
For branches with snfficiently small derivatives dQ/ds, da/ds the constant step si7e
~s = 0.1 is efficient. The failme of the corrector to converge indicates the presence
of a tmning point with sharp folding or other singnlar point. In snch cases the step
si7e shonld be variable for more detailed investigation of this region. We nse the
adjnstment ~s ----t K ~s, where K is a constant. Starting from ~s E [10-4
, 10-2
] we
gradnally increase the step si7e, 1.0 < K < 1.5, if the corrector snccessfnlly finds
31

the tme solntion. In the opposite case we take K = 0.5 and repeat the corrector
step.
Combination of shooting and continnation methods allows ns to obtain the bi-
fmcation diagram presented in Fig. (4.2). The meaning of different elements of this
figme will be explained in the snbseqnent sections.
4.2 Bifurcation from one-frequency family and asymptotic families
As mentioned in the previons chapter, the first class of locali7ed waves of system
(3.4) consists of one-freqnency soliton families for the third harmonic w, which in
the cases= 1 exist for all a > 0. It represents scalar Kerr solitons described by the
st::md::ud cnbic (1+1)D NLS eqn:::ttion which follows from the second of Eqs. (3.4)
at u = 0:
(4.4)
It can be readily solved exactly giving the well-known nniqne single soliton solntion:
y'2a
w0 (x) = -
3
- sech(yax), Q = 4ya. (4.5)
To locate the point where small component u(x) bifmcates from the state (4.5)
we nse methods of standard bifmcation analysis, e.g. as in Refs. (Akhmediev and
Bmy:::tk 1994, Akhmediev :::tnd Bmy:::tk 1995, Pelinovsky :::tnd Kivsh:::tr 2000). We
consider solntion
(4.6)
where Eisa small parameter. Snbstitntion of (4.6) into Eqs. (3.4) and linearintion
with respect to E leads to the standard problem of existence of locali7ed states in
the sech2
-like potential
" 4a 2(va )u1 +
9
sech ax u1 = -Eu1 , (4.7)
where E = -1. The locali7ed state exists only for E = -a/9 and has the form
(4.8)
where A is a normalintion constant. We conclnde that the two-freqnency soliton
bifmcates from the one-freqnency family (4.5) at a = 9 and has two branches cor-
responding to the positive and negative fnndamental component. The approximate
form for the bifmcated branches in the vicinity of a= 9 is given by expressions (4.6)
and (4.8) with A= ±Ia- 91.
32

4 4
0: = 25 (b) 0: = 8.3
3 3
;:3 2 ;:3 2
i i
1 1
0 0
-1 -1
0 1 2 3 4 5 0 1 2 3 4 5
X X
4 4
(c) 0: = 9 (d) 0: = 9.5
3 3
;:3 2
~
2
i
1_
i
1 1
0 0
-1 -1
0 1 2 3 4 5 0 1 2 3 4 5
X X
2 2
1
0: = 7.1
1
0: = 7.3
0 0
;:3 ;:3
i -1
i -1
-2 -2
-3 -3
-4 -4
0 2 4 6 8 10 0 2 4 6 8 10
X X
Figme 4.3: Examples of two-wave and one-wave solitons. Thin line denotes the
f11ndamental wave, thick line shows the third harmonic. Labelling of all examples
corresponds to the labelling of the open circles in Fig. 4.2.
33

2 2
(u) ex = 10 (v) ex = 10
1 1
~ ~
j j
0 0
-1 -1
0 2 4 6 8 10 0 2 4 6 8 10
X X
2 2
ex = 10 ex = 10
1 1
~ ~
j i
0 0
0 2 4 6 8 10 0 2 4 6 8 10
X X
Figme 4.4: Examples of (l+l)D two-wave solitons, which are not directly linked to
the two-wave solitons of the cascading limit. Labeling is as for Fig. 4.3.
Strictly speaking there are no other one-wave locali7ed solntions. However, it
will be helpfnl in what follows to consider formal mnlti-soliton states consisting of
a different nnmber of infinitely separated single solitons (4.5), families of which we
denote by S1 (single soliton), S2 (two solitons), S3 (three solitons), etc. We are
mainly interested in families with an odd nnmber of separated solitons: S2i+l, i =
1, 2, 3, ..., bnt we also investigate 'bifmcations' from S2 . Note that, fori > L Si in
fact denotes more than a single one-wave family, becanse each single pnlse that is
glned together can be either positive or negative.
The resnlts of nnmerical continnation of limiting solntion (3.9), Fig. 4.3(a),
shows that the branch traces a convolnted path in the (Q, a)-plane, involving fom
'bifmcations' from one-wave soliton families (from the families sl, s3, s5, and s7)
all taking place at a= 9.
Let ns try to motivate what is happening at each of the 'bifmcations' from SJ;
for which at first sight it seems remarkable that each one occms precisely at a= 9.
34

Standard bifmcation analysis allowed ns to find the position of the single bifmcation
point from the one-wave soliton family S1 at a = 9.0, filled circle C in Fig. 4.2.
The corresponding soliton profile is shown in Fig. 4.3(c). This is a transcritical
bifmcation with one branch emerging to the left of the bifmcation point and one to
the right. This stmctme is confirmed by the inset to Fig. 4.2 which shows that the
branch emerging to the left nndergoes a fold (at point B), so that on a larger scale
both branches appear to bifmcate to the right. Examples of solitons for points B
and D are presented in Figs. 4.3(b, d) respectively.
The 'local' bifmcation from S1 canses a topological change in the £om-dimensional
phase space so that a global event mnst also happen at this parameter valne. This
global event is the possibility of glning together several copies of the S1 back to
back and forming a new branch of solitons with several large peaks that bifmcate
from a= 9. Phenomenologically this is similar to what happens in the second har-
monic generation case when the parameter eqnivalent to a passes throngh 1 (Yew
~t al. 1999, Yew ~t al. 2000). A key observation here is that in order to get a
symmetric (even) solntion, only an odd nnmber of copies of the S1 may be taken to
form solitons in this way. As a convenient short-hand for this global bifmcation of
mnlti-peaked solntions at a= 9, we have refereed to it as a local 'bifmcation' from
s2i+l where i = 1, 2, 3 0 0 . , althongh this is strictly a misnomer.
Representative examples of soliton profiles for a few nearby points of the branch
are presented in Fig. 4.3(g, h). Nnmerical continnation beyond point G of Fig. 4.2
shows th:::~t the two-w:::tve soliton br:::tnch :::tppro:::tches a = 9.0 from the left, where
it bifmcates from the s3 asymptotic one-wave family that has alternating phase
between each single-soliton component. However, we find that this is only one of
a total of fom symmetric two-wave solitons that come ont of S3. There are 8 in
total if yon inclnde the change of sign of both u and w. The second bifmcates to
the left from the same (alternating phase) s3 family and differs only in that the
first harmonic has the opposite sign. A representative of this branch, corresponding
to point H in Fig. 4.2 is shown in Fig. 4.3(h). The two other branches exist for
a > 9 and bifmcate from the S3 family where all peaks are in phase (positive), and
representatives are shown in Fig. 4.4(u, v). With the increase of a (cascading limit)
these complex mnlti-hnmped solitons keep their general stmctme intact, bnt be-
come more locali7ed. These two branches are not shown in the bifmcation diagram
(Fig. 4.2) bnt their Q(a) cmves lie very close to each other and to the S3 cmve to
the right of the bifmcation point.
A similar bifmcation pictme is observed at a = 9.0 for bifmcations from S5 and
S7 one-wave families. However, becanse of the increase in the nnmber of possible
one-wave mnlti-soliton families themselves, the nnmber of corresponding bifmcated
two-component branches also increases. For the even solitons we have the following
35

30
Q 20
10
11.8
0 ~~~~~~~~~~~~~~~~~~~~~~
5 10 15 20 25 30
a
Figme 4.5: Bifmcation diagram from the first three one-component families Si, i =
1, 2, 3. Asymmetric f:::tmily S2 is shown hy :::t thick line.
form11la to calc11late the n11mher of two-wave s11h-families bifmcating from one-wave
si f:::tmily: Ni = 2(i+l)/2
(dollble that if we COllnt the opposite signs of u and v). For
example, there are 16 branches that bifmcate from s7 branches which have Q = 84
at a= 9. Note that in the bifmcation diagram of Fig. 4.2, in order to red11ce, only
branches directly linked to the cascading limit two-wave family are shown. Close to
hifmc:::ttion points, the third h:::trmonic components of the depicted hr:::tnches h:::tve
neighboming h11mps of alternating sign and first harmonic components have all
h11mps of the same sign. Note that these branches all bifmcate to the left of a= 9.
For the branches which bifmcate to the right not all third harmonic neighboming
h11mps alternate in sign.
4.3 Violation of the "bound state'' principle
None of the m11lti-h11mp soliton branches bifmcating to the left of a = 9 can be
viewed as bo11nd st:::~tes of single partial solitons. Indeed, single one-h11mp solitons of
Eqs. (3.4) always have u and w components in-phase (of the same sign) for a< 9.0,
whereas some of the individ11al h11mps of m11lti-h11mp stmctmes bifmcating to the
left from Si (i > 1) families have u and w components of different signs. To
ill11str:::~te this point we show in Fig. 4.5 :::tn enl:::trged hifmc:::ttion di:::tgr:::tm in the
vicinity of a = 9 covering the first three families, Si, i = 1, 2, 3. Some of the
36

2 2
(Bt) A a= 8.6 (B2) a:= 8.6
1
Of------
-1 -1
-2~~~~~~~~~~~~ -2~~~~~~~~~~~~
-10 -5 0 5 10 -10 -5 0 5 10
X X
2 0.5 2 0.5
(B3) a:= 8.6 a:= 8.6
;;:=.----i 0.0 ;;l
-1
-2 ~~~~~~~~~~~~- 0.5 -2 "--"---~~~~~~~-"----'-~~__.__j -0.5
-10 -5 0 5 10 -10 -5 0 5 10
X X
Figme 4.6: Examples of the two-wave solitons close to bifmcation point at a = 9.
Weak component u(x) is enlarged in two bottom plots. Labelling of the profiles is
in agreement with Fig. 4..1.
corresponding ex:::tmples of soliton profiles plotted :::~t a = 8.6 :::tre given in Fig. 4.6. As
they approach a = 9.0 the separation between individnal hnmps ('partial solitons')
increases and the state begins to approach a concatenation of single solitons with
slightly overlapping tails. However, some of these partial solitons have ont-of-phase
u and w components and hence mnrwt ~x1.st on their own (i.e. withont being in
s11perposition with other 'partial' solitons).
Figme 4.5 shows something even more striking- that there is also a 'bifmcation'
from the S2 family. However, the solitary waves that bifmcate from there are not
symmetric bnt in fact are asymm~tri~ solitons, see Fig. 4.7. Also at least one of
these asymmetric solntions is born in a symmetry-breaking (pitchfork) bifmcation
from one of the symmetric soliton br:::tnches (:::~t the point Oas, see Fig. 4.5). Thns
there is a branch of asymmetric solitons which connects symmetric solitons with a
branch of asymptotic antisymmetric solitons (the S2 family). We conjectme that
there are similar asymmetric solitons that 'bifmcate' from SJ at a = 9 for all even j.
37

2 2
(S2 A 35
) (S2 B35
) a= 12
0 0
;3 -1 ;3 -1
::s· -J
-2 -2
-3 -3
-4 -4
-to -5 0 5 10 -10 -5 0 5 10
X X
2 2
(S2 c35
) a=7.7 (S2 0
35
) a= 12
0 0
;3 -1 ;3 -1
::s· -J
-2 -2
-3 -3
-4 -4
-10 -5 0 5 10 -10 -5 0 5 10
X X
Figme 4.7: Examples of asymmetric solntions bifmcated from the family S2 . La-
belling of the profiles is in agreement with Fig. 4.5.
4.4 'Jump~ bifurcation
The majority of resnlts obtained for (l+l)D solitons finds its similar connterpart in
the (1+2)D case, see (Bmyak Steblina and Sammnt 1999, Kolossovski ~tal. 2000).
The main difference is that in the (l+l)D case we have fmmd no examples of two-
wave solitons that smvive down to a = 0 where they might form a connection
with branches of qnasi-solitons existing for a < 0. Instead, a representative branch
coming from S7 bends abmptly, point R in Fig. 4.8. After this fold a increases nntil
it reaches Tat a :::::::; 3.65, where another nonlocal bifmcation occms. In this process,
the third harmonic gradnally forms a core with weakly separated wings. At T, the
latter become completely separated one-wave solitons [see Fig. 4.9(t)1. The solntion
at the point T can thns be viewed as a direct snm of two well-separated one-wave
solitons and the soliton at point N. Beyond T we were nnable to find any similar
solntions. This non-trivial "jnmp" bifmcation is indicated by the vertical arrow in
Fig. 4.8.
38

65
60
55
Q
50
45
40
0 2 4 6 8 10
a
Figme 4.8: Expanded portion of Fig. 4.2 in the range 0 ::::; a ::::; 10, 40 ::::; P ::::; 65.
4.5 Summary and discussion
As we have seen, the bifmcation diagram of the conventional solitons is qnite com-
plex. For convenience of the reader, in this section we snmmari7e the resnlts ob-
tained so far to stress the main featmes. We have fmmd the following
• the non-local bifmcation of mnlti-hnmped two-freqnency solntions which are
a conseqnence of the local bifmcation from the one-hnmped one-freqnency
soliton at a = 9,
• the so-called .Jump bifmcation,
• that some of the mnlti-hnmped states cannot be viewed as bonnd states of
several distinct one-hnmped states.
The first of these is particnlarly intrigning since not only are symmetric mnlti-
hnmped states formed in this way, bnt also asymmetric ones. The second novel
bifmcation, the jnmp, appears related to, bnt not the same as, the so-called nrbit-
flip bifmcation (Sandstede ~tal. 1997). The conclnsion that some of the discovered
mnlti-hnmped states cannot be viewed as bmmd states of several distinct one-
hnmped states has very significant physical implications. It demonstrates that a
conventional approach to the constmction of mnlti-hnmp solitons, see e.g. (Klander
~t al. 1993, Bmyak 1995, Calvo and Akylas 1997b), gives only one possibility and
that the parametric wave mixing may provide another, less straightforward way to
39

4
(l) (){ = 7.7
3
~ 2
~
1
a
-1
a 2 4 6 8
X
2
1 (n) (){ = 3.6
a
~ -1
~ -2
-3
-4
-5
a 5 1a 15
X
5
4 (){ = a.2t
3
~ 2
~ 1
a
-1
-2
a 5 1a 15
X
2
1
a
~ -1
~ -2
-3
-4
-5
1a a
2
1
a
~ -1
~ -2
-3
-4
-5
2a a
5
4
3
~ 2
~ 1
a
-1
-2
2a a
5
(a)
5
5
(){ = 7.4
1a 15
X
(){ = t
1a 15
X
(){ = 3.6
1a
X
15
2a
2a
2a
Figme 4.9: Examples oftwo-wave and one-wave solitons. Labelling is as for Fig. 4.3.
40

create stationary higher-order modes. This may find application in many fields of
physics where parametric interactions take place.
Stability of the discovered soliton families will be analy7ed in the next section.
Altho11gh llSllally higher-order soliton families are s11bject to one of several types
of instability, some exceptions are known, see e.g. (Ostrovskaya ~t al. 1999). The
promise of detecting stable m11lti-h11mp solitons is rml indeed beca11se at least some
ofthem cannot be viewed as bo11nd states oftwo or more single (one-h11mp) solitons.
For s11ch bo11nd state solitons of NLS-type system of eq11ations, there is practically
no hope of stability as shown in (Bmyak and Steblina 1999).
41

CHAPTRR 5
Linear stability of conventional solitons
Don't fear change-embrace it.
Anthony .J. D'Angelo
5.1 Preliminary discussions
Solitons can be formed when excitation applied to a medi11m is strong eno11gh to
provided sufficient nonlinear response. The importance of solitary waves in real ex-
periments is established on the fact that snch locali7ed modes can be the attractors
for many inpnt signals, they can be a natmal resnlt of evolntion of e.g. Ganssian,
beams dming propagation along the medinm. The possibility to attract initial in-
pnts of some form is closely reJ:::tted to st:::~bility of solitons. Physic:::tlly, st:::~bility
means that a weak pertmbation applied to soliton stays small dming its fmther
propagation. Thns, an important investigation that mnst be carried ont before the
design of actnal devices can be made is stability analysis of these stationary gnided
waves.
A problem of soliton stability has a long history which started sim11ltaneo11sly
with discovery of snch solntions (Zakharov 1968, Vakhitov and Kolokolov 1973).
The nsnal approach to the problem of stability consists in consideration of weakly
pertmbed solitons followed by linearintion of the eqnations of motion aronnd the
nnpertmbed solntion. Absence of growing eigenmodes in the lineari7ed model in-
dicates the linear stability of the corresponding soliton. A variety of methods
nsing this direct approach are known. Among them are the asymptotic stabil-
ity theory (Zakharov and Rnbenchik 1973, Pelinovsky p_t nl. 1995), method of
adiabatically varying soliton parameter (Newell 1985, Bmyak ~t nL 1996, Peli-
novsky ~t nL 1996, De Rossi ~t nL 1998, Kanp 1990, Lakoba and Kanp 1997),
method of Evans fnnctions (Pego and Weinstein 1992, Pego ~t nL 1995). A geo-
metrical approach involving consideration of conserved qnantities along paramet-
ric cmves was considered in (Akhmediev 1982, Knsmartsev 1989, Mitchell and
Snyder 1993, Akhmediev ~t nL 1999).
However, application of the mentioned methods is not always sufficient dne to
the fact that local linear stability does not imply global nonlinear stability of the
original eqnations. A rigorons method of treating stability was developed at the
43

end of the 19th centmy by the Rnssian mathematician A. M. Lyapnnov, for e.g.
see (Lyapnnov 1935, Mawhin 1994). The main idea of the direct Lyapnnov method
is to choose some special fnnction whose properties allow one to dednce the character
of the evolntion of the system (Makhankov pf nl. 19~M). Several modifications
of this method applied to stability of solitons inclnde the method of fnnctional
estimates (K117netsov pf nl. 1986), the energy method (Arnold 1965), the method
of Shatah and Stranss (Shatah and Stranss 1985), the Benjamin method (Benjamin
1972).
Depending on the type of applied pertmbation one can distingnish a few kinds
of stability:
• stability with respect to pertmbation of the stationary solntion which has the
same dimension as the soliton (lrmgitudinnl stability),
• stability with respect to pertmbation of the stationary solntion which has
larger dimension than the soliton (trnns1JPrsP stability),
• stability with respect to pertmbation of the form of the eqnations (strudurnl
stability).
In this thesis we examine only lrmgitudinnl stability of solitons. This chapter
covers linear stability which is the first step in the analysis of global stability. More-
over, we constrain om attention to consideration of only spPdrnl stability which has
the following definition.
8nlitnn snlutinn u n.f snmP nnnlinmr Pquntinn i Uz = F[u] is stnblP 1.f thP sppr.f.rum
n.f fhP linmrizPd Pqunt1nn i~z = i[u]~ F'[u]~ dnPs nnt hm;p PigPrwnluPs with
pn.sithP rPnl pnrts, i. P. RP).. ::::; 0 f).. E a(i), whPrP a(i) stnnds .fnr thP spPdrum n.f
npPrntnr L.
The difference between spectral and linear stability is that the latter allows
algebraically grown modes to exist. However, om extensive nnmerical simnlation
does not reveal existence of snch modes in the problem of degenerate fom wave
mixing.
5.2 Stability threshold for fundamental solitons
In this section we derive the stability threshold for solitons which have no 7ero
crossings in their transverse profile and monotonically decay to 7ero at asymptotic
regions x-----+ ±oo. Assnme that the two-parameter family {Us(x; {3, C), Ws(x; {3, C)}
of stationary solntions to Eqs. (2.35) is known. For stability analysis we consider a
weakly pertmbed soliton
U(X, Z) = Us(X) +c[U1 (X)ei-Z + U~(X)e-i-*Z],
W(X, Z) = Ws(X) +c[W1 (X)ei-Z + W{(X)e-i-*Z],
44
(5.1)

where E « 1 and the asterisk stands for complex conjngate. Snbstitntion of
ansat7 (5.1) into Eqs. (2.35) and linearintion aronnd the nnpertmbed soliton the
yields linear eigenvalne problem
L
The self-adjoint operator L is given by
Lw·
d2 ·c d L*
dX2 +Z dX + 1U
L1w•
L1w
and has the following components
L1w
Liw·
d2 . d
dX2 - wC dX + L2w
L~w·
L1w• = 2UsWs, L2w = -a(3p + ~) + 18IWsl2
+ 2IUsl2
, L2w• = gw_;.
(5.2)
(5.4)
In the vicinity of stability threshold it is natmal to assnme that growth rate is
sm:::tlL I.AI « 1. This :::tllows ns to seek the solntions of Eqs. (5.2) in the form of :::tn
asymptotic series in ).
00 00
(5.5)
n=O n=O
S11bstit11ting series (.1..1) into Rqs. (.1.2) we obtain a c011pled system of infinitely
many eqnations for approximations of different orders. Zero-order approximation
can be readily obtained nsing symmetries (2.30) and (2.31). By infinitesimal varia-
tion of X 0 and 'Po in these invariant transformations it can be shown that the two
locali7ed solntions
uco) iUs uco) dUs/dX
1 1
uco) -iu; uco) dU;jdX2 and
2 (5.6)wco) 3iWs wco) dWs/dX1 1
w,(O)
2 -3iW8
* w,(O)
2 dWs*/dX
1 d f A • A ( (0) (0) (0) (O))T A 1 . f bare nentra mo es 0 L, I.e. L u1 'u2 'w1 'w2 = 0. na YSIS 0 pertnr a-
tions in the form of series (5.5) is eqnivalent to consideration of the specific class of
45

pertmbations which, in the leading order, can be presented as a linear snperposition
ofthe nentral modes (Skryabin 2000). Generally speaking, snch an approach allows
one to obtain only a snfficient condition for instability. Other models might exhibit
different mechanisms of losing stability which can be detected nnmerically (Skryabin
and Firth 1998, Tran ~t al. 1992). We note, that nnmerical simnlations of the dy-
namics of fnndamental solitons dne to degenerate fom-wave mixing confirmed the
validity of asymptotic series (5.5).
The stmctme of the pertmbation to the first order in). is defined by the following
inhomogeneons problem
u(l) iUs dUs/dX
1
u(l) iU; -dU;/dX
L 2
= A1 +A2 (5.7)w(l) 3iaWs adWs/dX1
w,(1)
2 3iaWs* -adWs*/dX
where A1,2 are real constants. Eqnations (5.7) have nontrivial locali7ed solntions
only if the solvability condition is satisfied (Gorshkov and Ostrovsky 1981), i.e. if
the right-hand-side is orthogonal to all solntions of the homogeneons problem with
the adjoint operator. Dne to self-adjoint type of operator L, the nentral modes
of it coincide with (5.6). Solvability condition for Eqs. (5.7) is always satisfied.
Moreover, the exact solntion to Eqs. (5.7) has the form
u(l)
1 OUs/8{3 8Us/oC
u(1)
au;1of3 au;;ac2
= A1 +A2 (5.8)w(1)
8Ws/8{3 8Ws/8C1
w,(1)
2 aws*1of3 aws*;ac
The second order approximation can be fonnd from the following inhomogeneons
problem
uC2)
1 oUs/8{3 8Us/oC
uC2)
-au;1of3 -au;;ac
L 2
= A1 +A2 (5.9)wC2)
aoWs/8{3 a8Ws/8C1
w,(2)
2 -aoWs*I8{3 -aoWs*1ac
46

Demanding orthogonality between the right-hand-side of Eqs.(5.9) and nentral
modes (5.6) is eqnivalent to the following two conditions
(5.10)
where Q and Pare power and momentnm calcnlated for stationary solntion {Us, Ws}
nsing expressions (2.33) and (2.32). The existence of nontrivial solntion (A1 , A2)
indicates the onset of an exponentially growing mode of the operator L and th11s
leads to the stability threshold
8Q 8P 8Q 8P
8{3 8C - 8C 8{3 = o. (5.11)
Instability threshold (5.11) is the two-parameter version of the well-known pioneer-
ing resnlt of st:::tbility of one-p:::tr:::tmeter solitons (V:::tkhitov :::tnd Kolokolov 1973).
In tmn, condition (5.11) is a particnlar example of a more general expression for
instability threshold of mnlti-parameter solitons (Skryabin 2000, Pelinovsky and
Kivshar 2000).
Instability condition (5.11) can be simplified in the case of motionless solitons. It
is easy to verify that power, Q, is an even fnnction of velocity, Q({3, C) = Q({3, -C).
This implies 8QI8GIC=O = 0 and the threshold condition becomes
8Q 8P
8{3 8C = o. (5.12)
Factorintion (5.12) points ont that two different mechanisms of instability onset,
bifmcation from phase and translational nentral modes (5.6), are independent. In
the case a= 3 all moving reference frames are physically eq11ivalent to the still one
dne to Galilean invariance of Eqs. (2.35),
U(X, Z) = Us(X- CZ) ei(CX/2-C2Z/4)'
W(X, Z) = Ws(X- CZ) e3i(CX/2-C2Z/4)_
(5.13)
Gange transformation (5.13) enables constmction of moving solitons nsing the
motionless ones. Moreover, nsing Galilean invariance it is possible to show that
8P/8Gic=o = Q/2 > 0 and thns the instability onset dne to destabilintion of the
anti-symmetric nentral mode is absent. We assnme that for a rv 3 gange transfor-
mation (.1.1 ~)remains approximately valid and conclnde that the stability threshold
47

has the following simple form
oQ = o
8{3 0
(.1.14)
Condition (5.14) defines the instability threshold of the fnndamental solitons. In
more general context, this condition corresponds to valnes f3vK for which the cmve
,
2
= ,2
({3) changes its sign. Moreover, the generic form of this cmve in the vicinity
of PVK is defined by .A2
({3) rv ({3 - PVK) (Pelinovsky ~t al. 1995, Pelinovsky ~t nl.
1996). Derivation of this relation together with detailed definition of the constant
of proportionality reqnires more elaborate analysis and is done in the chapter 6.
From now on we consider mainly the 7ero-velocity solitons. For nnmerical simll-
lations it is preferable to nse scaled Eqs. (3.4). Recalcnlation of the threshold (5.14)
for this model gives
5.3 Numerical methods
oQ 1
(3a-a)-+-Q=O.
oa 2
(5.15)
Nnmerical simnlation becomes specifically important when analy7ing stability prop-
erties of solntions to non-integrable PDEs. In this section we describe the most
important nnmerical methods nsed to obtain information abont the spectr11m of
the pertmbed solitons.
5. 3.1 B~am prnpagatirm m~thnd
The direct and the easiest way to obtain preliminary resnlts regarding stability of a
soliton is to model its dynamics nsing PDEs (3.1) with a weakly pertmbed soliton
taken as the initial condition. The most common means in this case is to nse bmm
prnpagatinn m~thnd (BPM). Note, that PDEs like (3.1) model propagation of a beam
nnder interdependent action of diffraction/dispersion and nonlinearity. The physical
idea of RPM is to split the complete PDR-problem into two consecntive steps which
independently take into acconnt diffraction/dispersion and nonlinearity (Agrawal
1995). The general form of the eqnations modelling propagation of a beam throngh
a nonlinear media has the form
ou A A
i oz = (D + N)u. (5.16)
Here operator D inclndes all terms having derivatives with respect to the transverse
coordinate and acconnts for diffraction/dispersion or absorption in linear media.
Operator N describes nonlinear effects. Integrating Eqs. (5.16) over a small step
~zone formally obtains
(5.17)
48

Approximation of the exponential term in expression (5.17) can be done nsing the
Baker-Hansdorff formnla for non-commnting operators aand b
(5.18)
This gives expression
(5.19)
The symbolic expression involving the exponent with nonlinear operator N is eqniv-
alent to integration of Eqs. (5.16) with diffraction/dispersion terms switched off.
For this step we 11se fomth-order Rnnge-Kntta method with ~z = 0.005. The sym-
bolic expression with the exponential operator efJt:,z can be made into an algebraic
operation by direct and backward Fomier transformations
(5.20)
where F stands for Fomier transformation, w is Fomier freqnency, argnment iw is
the replacement of the differential operator ojox. To implement this step we nse
fast Fomier transformation (FFT) with 2k points in the transverse profile where
k E [12, 14]. For more detailed description of BPM and discnssion of other nnmerical
methods of solving NLS-like eqnations we refer to the review (Taha and Ablowit7
1984).
Stable propagation of a weakly pertmbed soliton does not necessarily imply its
stability. It might tmn ont the the form ofthe applied pertmbation is orthogonal to
the actnalnnstable mode and th11s can not excite instability. Th11s after propagation
of a soliton with a few kinds of applied pertmbation one can make only preliminary
conclnsions regarding its stability. More reliable nnmerical analysis of stability
shonld be based on sw~dral methods. These methods form the object of the next
section.
5. :1.2 Frmri~r d~rJJmpnsitirm m~thnd
In comparison to BPM the more accmate method of stability analysis of stationary
solntions to Eqs. (3.1) consists in nnmerical solntion of eigenproblem (5.2). An
important property of 7ero-velocity solitons is that they are real solntions. This
simplifies the stmctme of the operator i which becomes real. In spite of all the
simplifications, eigenvalne problem (5.2) is still not appropriate for the direct mi-
merical analysis dne to presence of differential terms. One way to transform the
operator problem into a standard matrix form is to approximate all derivatives by
finite differences. In this case one has to find eigenvalnes of 4N x 4N matrix, where
49

1.0
0.5
k=2 k=3
- 0 0 5 L_____j_----'---'------'-----'----'------'-----'---'----.J'-----'-----'---'----'-___.L__-'------'-----'----'-----'
-L/2 0
X
L/2
Figme 5.1: Soliton (dashed line) and the first three harmonics of the complete set
{<h} nsed in Fomier decomposition method.
N is the nnmber of mesh points in the transverse profile. The finite difference ap-
proach is efficient for problems with strongly locali7ed solitons of a simple shape
when a relatively small nnmber of mesh points (N rv 500) is snfficient to obtain a
highly accmate resnlt. To obtain accmate approximation of derivatives in the case
of a weakly locali7ed solntion with complex profile N mnst be greatly increased
(N rv 104
). This resnlts in inefficient memory nsage and drastically increases the
nmtime of the code. The reqnirement of accmate approximation of the derivative
terms and the demand to keep the si7e of matrix relatively small can be satisfied
nsing a Fomier decomposition method (FDM).
Assnme that the form of soliton {Us(X; {3), Ws(X; {3)} is known for the segment
X E [-L, L]. In FDM we expand the pertmbation in Fomier series
N
U2(X) = L Ck+N<h,
k=l
N
W1(X) = Lck+2N<Pk,
k=l
50
k=l
N
W2(X) = L ck+3N<Pk,
k=l
(5.21)

where symmetrical fnnctions {<Pk(X)} form a complete orthonormal set on the
interval X E [-L, L] and are given by
(.1.22)
see Fig.5.1. Snbstitnting series (5.21) into Eqs. (5.2) and nsing the orthogonality
condition
{
1,
i5k,m =
0,
n=m,
(5.23)
we obtain an eigenvalne problem in the form
M (5.24)
where the elements of the 4N x 4N matrix M are
Ml+N,k = -Mz,k+N, Ml+N,k+N = -Mz,k,
Ml+N,k+2N = - Ml,k+3N' Ml+N,k+3N = - Ml,k+2N'
1 1
Ml+2N,k = - Mz,k+2N, Ml+2N,k+N = - Mz,k+3N,
a a
1 11£/2
Ml+2N,k+2N = -- f?i5z,k +- <l>zL2w<Pk dX,
a a -L/2
11£/2
Ml+2N,k+3N =- <l>zL2w•<l>k dX,
a -L/2
1 1
Ml+3N,k = -- Ml,k+3N' Ml+3N,k+N = -- Ml,k+2N'
a a
(5.25)
Ml+3N,k+2N = - Ml+2N,k+3N' Ml+3N,k+3N = - Ml+2N,k+2N.
Knowing eigenvalnes ). and the corresponding eigenvectors {ck} we can calcnlate
the eigenmodes of the actnal pertmbation nsing expressions (5.21). In the practical
implementation of the algorithm described above we nse the following parameters:
interval length L rv 40, nnmber of the mesh points in the transverse profile M rv
8 X 103
, nnmber of harmonics N rv 150.
51

In the vicinity of the cnt-off freqnency (when ). is close to the bonndary of
continnons spectmm) the eigenmodes become weakly locali7ed. Similarly, station-
ary solntion U 8 , W 8 becomes weakly locali7ed when IaI« 1. In snch instances we
11se the modified Fomier decomposition method (Hewlett and Lad011cer 199.1). Tn
this method, we apply the tangential transformation to the transverse coordinate,
X = tan 1rr, to map an infinite domain X E [-oo, +oo] into the finite interval
r E [-1/2, 1/2]. This transformation allows for highly accmate calcnlation of the
spectmm and the eigenmodes.
5.4 Stability results
In this section we extend the resnlts of a stability analysis of the fnndamental
solitons (Sammnt ~t al. 1998) and extensively analy7e higher-order solitary waves.
The standard explanation of the absence of stable higher-order parametric families
is based on pertmbative soliton interaction theories (Gorshkov and Ostrovsky 1981.
Kanp 1990, Kanp 1991) and interpretation of higher-order solitons as soliton bmmd
states (Rmyak and Champneys 1997, Rmyak and Akhmediev 199.1, Rmyak 1996).
Bnt it has been shown in the previons chapter that higher-order nonlinear modes
dne to degenerate parametric fom wave mixing in principle cannot be considered
as bmmd states of single solitons. A violation of the bonnd state principle gives the
possibility for the corresponding mnlti-hnmped solitons to be stable in some region
of parameters.
Stability of higher-order optical solitons is an intrigning qnestion. The first indi-
cation of possible stability of higher-order solitons dne to processes involving para-
metric interaction is related to gap solitons of wavegnides with qnadratic nonlinear-
ity (Peschel ~tal. 1997, Mak ~tal. 1998). Also stable solitons dne to non-parametric
interaction are known (Malomed 1995, Bmyak and Champneys 1997, Ostrovskaya
~t al. 1999, Bmyak Kivshar, Shih and Segev 1999).
We perform linear stability analysis of Eqs. (3.3), s = 1. which is convenient for
nnmerical analysis. Om interest is focnsed on the part of the complete bifmcation
diagram (4.2) which displays two-wave families of solitons connecting the family of
fnndamental two-wave solitons of the cascading limit (large a) with the first fom
bifmcation points Oi, i = 1, 2, 3, 4, Fig. 5.2. Most ofthe soliton families presented in
Fig. 5.2 consist of solitons which are symmetric with respect to their centers. The
only exception is the branch of asymmetrically shaped solitons which bifmcates
from a symmetric family at point 0 2 and goes throngh the tmning point T3 to the
other bifmcation (from asymptotic antisymmetric two-soliton family) at the point
03.
To analy7e the linear stability of soliton families of Fig. 5.2 we investigate the
spectral properties of weakly pertmbed solitons, i.e. nnmerically solve eigenvalne
52

40
30
Q
20
10
10 20 30 40 50
a
Figme 5.2: Power venms a dependence for 7ero-velocity soliton of Eqs. (3.3). Filled
dots Oi correspond to the bifmcation points, open dots Ti denote the tmning points
of different soliton branches. Thin solid cmve S1 corresponds to primary (fnnda-
mental) one-wave soliton family. Dotted cmves S2 (S3) denote asymptotic families
which consist of two (three) infinitely sep::u:::~ted one-w:::tve solitons.
problem (5.2), C = 0. Pmely real valnes of ). lying in the intervals (.Ac, oo) and
(-Ac, -oo), where Ac = min(1, a), correspond to the continnons spectrnm of non-
locali7ed eigenmodes. Presence of a nomero imaginary part of ). implies soliton
instability.
As was already mentioned, in the vicinity of a = 3 the "translational" nentral
mode (dus/dx, dws/dx) does not lead to any nontrivial instability threshold dne to
Galilean invariance of Rqs. (~-~). The "phase rotational" ne11tral mode (ius, 3iws)
can lead to the onset of instability at the points defined by the Vakhitov-Kolokolov
criterion (VK) (Vakhitov and Kolokolov 1973, K117netsov ~t al. 1986, Pelinovsky
~tal. 1995, Pelinovsky ~tal. 1996). In om case VK has the renormali7ed form (5.15)
and determines the points where symmetric eigenmodes with pmely real eigenval-
nes transform into ones with pmely imaginary eigenvalnes. For symmetric soliton
families, nnmerically intensive eigenvalne analysis can be streamlined by separating
symmetric and antisymmetric eigenmode calcnlations.
5.,4.1 Stability nf nn~-1nm;~ family
Before considering two-wave soliton families we briefly discnss the spectmm of the
one-wave family S1 . The discrete spectmm of symmetric eigenmodes for this branch
53

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