RossiDissertation_TwoSided

San Diego State University and
Claremont Graduate University
Dissertation
Non-Conservative Variational
Approximation for Nonlinear Schrödinger
Equations and its Applications
Author:
Julia M. Rossi
Advisor:
Dr. Ricardo Carretero
A dissertation submitted to the faculties of
San Diego State University and Claremont Graduate University
in partial fulﬁllment of the requirements for the degree of
Doctor of Philosophy in Computational Science
June 2016

c Copyright 2016 by
Julia Michelle Rossi
All rights reserved.

Approval of the Dissertation Committee
The dissertation has been duly read, reviewed, and critiqued by the Committee listed
below, which hereby approves the manuscript of Julia M. Rossi as fulﬁlling the scope
and quality requirements for meriting the degree of Doctor of Philosophy.
Dr. Ricardo Carretero, Chair
Department of Mathematics & Statistics, Computational Science Research Center
San Diego State University
Dr. Christopher Curtis
Department of Mathematics & Statistics, Computational Science Research Center
San Diego State University
Dr. Ali Nadim
Institute of Mathematical Sciences
Dr. Marina Chugunova
Institute of Mathematical Sciences
Dr. Michael W.J. Bromley
School of Mathematics and Physics
The University of Queensland
Approval Date

SAN DIEGO STATE UNIVERSITY
CLAREMONT GRADUATE UNIVERSITY
Abstract
Doctor of Philosophy in Computational Science
Non-Conservative Variational Approximation for Nonlinear Schrödinger
Equations and its Applications
by Julia M. Rossi
June 2016
Recently, Galley [Phys. Rev. Lett. 110, 174301 (2013)] proposed an initial value prob-
lem formulation of Hamilton’s principle applied to non-conservative systems. Here,
we explore this formulation for complex partial differential equations of the nonlinear
Schrödinger (NLS) type, using the non-conservative variational approximation (NCVA)
outlined by Galley. We compare the formalism of the NCVA to two variational tech-
niques used in dissipative systems; namely, the perturbed variational approximation and
a generalization of the so-called Kantorovitch method. We showcase the relevance of
the NCVA method by exploring test case examples within the NLS setting including
combinations of linear and density dependent loss and gain. We also present an example
applied to exciton-polariton condensates that intrinsically feature loss and a spatially
dependent gain term. We also study a variant of the NLS used in optical systems called
the Lugiato-Lefever (LL) model applied to (i) spontaneous temporal symmetry break-
ing instability in a coherently-driven optical Kerr resonator observed experimentally by
Xu and Coen in Opt. Lett. 39, 3492 (2014) and (ii) temporal tweezing of cavity soli-
tons in a passive loop of optical fiber pumped by a continuous-wave laser beam observed
experimentally by Jang, Erkintalo, Coen, and Murdoch in Nat. Commun. 6, 7370 (2015).
For application (i) we perform a detailed stability analysis and analyze the temporal bi-
furcation structure of stationary symmetric configurations and the emerging asymmetric
states as a function of the pump power. For intermediate pump powers a pitchfork loop
is responsible for the destabilization of symmetric states towards stationary asymmetric
ones while at large pump powers we find the emergence of periodic asymmetric solutions
via a Hopf bifurcation. For application (ii) we study the existence and dynamics of cavity
solitons through phase-modulation of the holding beam. We find parametric regions for
the manipulation of cavity solitons by a tweezer in the LL model. For both applications
we also explore the ability of the NCVA method at capturing the evolution of solitary
waves.

“I think it’s very important to have a feedback loop, where you’re constantly thinking about
what you’ve done and how you could be doing it better."
Elon Musk

Acknowledgements
This dissertation is not solely mine, a great many people are responsible for its production
and I owe my gratitude to all of them.
First and foremost, I would like to express my deepest gratitude to my advisor Prof. Ri-
cardo Carretero for his continuous support of my research, and for patience, motiva-
tion, enthusiasm, and immense knowledge. I am very appreciative of his willingness to
accept me into his research group as a third year graduate student who found herself
without an advisor and little to no experience in nonlinear dynamical systems. Without
his support, this dissertation would not be possible.
Besides my advisor, I would like to thank the rest of my dissertation committee: Prof. Ali
Nadim, Dr. Michael Bromley, Dr. Marina Chugunova, and Dr. Christopher Cur-
tis, for their time, service, and interest in my research. Dr. Michael Bromley is owed
an extra debt of gratitude for participating on my committee despite being on Australia
time, and also for being my advisor for my master’s thesis. He introduced me to the
variational approximation in his classes, which has become central to my dissertation.
Thank you for guiding and advising me through all my graduate research.
I would like to acknowledge Dr. Panos Kevrekidis for his collaboration and key in-
sights on this research. I also would also like to thank Dr. Mariana Haragus for her
collaboration.
I am also grateful to the ARCS Foundation for giving me financial support over the
years and choosing me to be part of the prestigious group of scholars.
There are special people who offered me friendship and moral support through this entire
process. Thank you to Josh Staker who was the first cool grad student I met at SDSU.
I have to especially thank Dr. Eduardo Sánchez, without whom I would have never
survived trips to CGU or who managed to get Trefethen’s autograph for me. He is
incredibly intelligent and I am very grateful for his encouragement and advice. I wish
to thank Brad Dutkiewicz and his wife Heather Ruderian; I’ve never known two
people who have lived so much. The response to running another mile, going surfing, or
drinking another beer is always, “Why not?”, and thankfully we’ve never had a reply to
that question. Next, I am very thankful to Baptiste Buchler and Michelle O’Connor
for their friendship and for letting me be in the band. You rock! It has been a long
journey from that day at Cafe 976, mucho mahalo to Matt Burgess and my aloha sista
Maggie Burgess. Gila Cohen is a compassionate soul who I know is sending me love
and encouragement all the time. Becca Underdown, thanks for all the positive vibes
#YouAreEnough. I want to thank the community of Shore Colony, it no longer exists
xi

at 6767 Neptune Place, but it was home. I am also thankful for the Trinidad’s – Greg
and Denay – who were instant friends and have shared so many epic moments.
I am eternally grateful to my Floridian family Steve Deeb and Zea Deeb for all their
love, support and encouragement. Thank you to Jeff Speaks, Elyse Speaks, Amelia
Speaks, Violet Speaks, Ryan Deeb, and Sean Deeb for being my family. Also a
special thank you to my big brother Sean Rossi and his tribe–Jessica Rossi, Inanna
Rossi, Giancarlo Rossi, and Giada Rossi.
None of this would be possible without my mom Cheri Rossi and my dad Carl Rossi.
I owe my parents everything, they are a constant source of love, support, strength, and
guidance. They have taught me to persevere and to be strong, both of which were
needed to complete this dissertation. Last but not least, I would like to express my
immense gratitude to my heart, Robby Deeb. He is my best friend, co-conspirator and
accomplice. He was selfless during the making of this dissertation and has been by my
side every step of the way to get to this moment. Thank you for making the coffee when
I needed it, which was always.

Contents
Abstract vii
Acknowledgements xi
Contents xiii
List of Figures xv
List of Tables xvii
1 Introduction 1
1.1 A Brief Introduction to NLS and Solitons . . . . . . . . . . . . . . . . . . 1
1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Non-Conservative Variational Approximation 9
2.1 Non-conservative Variational Approximation
Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Derivation of Non-Conservative Variational
Method for Nonlinear Shrödinger Equation . . . . . . . . . . . . . . . . . 17
2.2.1 A Brief Example for Constructing R . . . . . . . . . . . . . . . . . 19
2.2.2 NCVA Recovery of NLS Equation . . . . . . . . . . . . . . . . . . 20
2.3 Other Non-Conservative Methods . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Perturbed Variational Approach (PVA) Formalism . . . . . . . . . 22
2.3.2 Modiﬁed Kantorovitch Method Formalism . . . . . . . . . . . . . . 25
2.3.3 Equivalence Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Applications of NCVA to the Nonlinear Schrödinger Equation 29
3.1 Nonlinear Schrödinger Equation with Linear Loss . . . . . . . . . . . . . . 29
3.1.1 Non-conservative Variational Approximation . . . . . . . . . . . . . 30
3.1.2 Numerical Results: NLS with Linear Loss . . . . . . . . . . . . . . 34
3.2 Nonlinear Schrödinger Equation with Density Dependent Loss . . . . . . . 39
3.2.2 Numerical Results: NLS with Density Dependent Loss . . . . . . . 41
3.3 Exciton-Polariton Condensate - The Nonlinear Schrödinger Equation with
Linear Gain and Density Dependent Loss . . . . . . . . . . . . . . . . . . 43
xiii

Contents xiv
3.3.2 Numerical Results: Exciton-Polariton Condensate . . . . . . . . . . 46
4 Spontaneous Symmetry Breaking of the Lugiato-Lefever Equation 51
4.1 Spontaneous Time-Reversal Symmetry Breaking in Passive Kerr Resonator 53
4.1.1 The Full Lugiato-Lefever Model: Equilibria, Stability and Bifur-
cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.2 Numerical Convergence of the Stability Spectrum . . . . . . . . . . 60
4.1.3 Bifurcation Analysis Using the NCVA Approach . . . . . . . . . . 67
4.1.4 Local Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . 77
4.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Temporal Soliton Tweezing of the Lugiato-Lefever Equation 91
5.1 Temporal Tweezing of Cavity Solitons . . . . . . . . . . . . . . . . . . . . 92
5.1.1 Theory of Temporal Tweezing . . . . . . . . . . . . . . . . . . . . . 92
5.1.2 Nondimensionalization of LL Model . . . . . . . . . . . . . . . . . 94
5.1.3 Power-Balance Constraint . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 Tweezabiltiy of Cavity Solitons . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 NCVA of Tweezed Cavity Solitons . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Numerical Results for the Temporal Tweezing . . . . . . . . . . . . . . . . 102
5.4.1 Tweezer with Narrow Width . . . . . . . . . . . . . . . . . . . . . . 107
5.4.2 Tweezer with Natural Width . . . . . . . . . . . . . . . . . . . . . 114
5.4.3 Tweezer with Wide Width . . . . . . . . . . . . . . . . . . . . . . . 122
5.4.4 Demonstration of Temporal Tweezing . . . . . . . . . . . . . . . . 128
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6 Conclusions 133
6.1 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A NCVA Maple Worksheet for Temporal Tweezing 139
B NCVA System of Equations for Temporal Tweezing of Cavity Soliton 145
C NCVA Non-Conservative Integrals 163
D Additional Temporal Tweezers 171
Bibliography 175

List of Figures
3.1 NLS with Linear Loss, = 0.01 . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 NLS with Linear Loss, = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 NLS with Linear Loss, = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 NLS with Density Dependent Loss, = 0.1 . . . . . . . . . . . . . . . . . . 42
3.5 Exciton-Polariton Above Equilibrium . . . . . . . . . . . . . . . . . . . . . 48
3.6 Exciton-Polariton Below Equilibrium . . . . . . . . . . . . . . . . . . . . . 49
4.1 LL Equation Linearization Spectrum for the Symmetric and Asymmetric
Steady State Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Density Evolution of Unstable Symmetric States at X = 8 and 16 and
Snapshots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Comparison of Frequency Spectrum for Domain Length L=10 . . . . . . . 65
4.4 Comparison of Frequency Spectrum for Domain Length L = 50 . . . . . . 66
4.5 SSB Bifurcation Diagram Comparison for LL Steady States and NCVA
4-Parameter Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 Fluid Velocity of LL Equation and NCVA Solutions . . . . . . . . . . . . . 72
4.7 NCVA ODE Linearization Spectrum . . . . . . . . . . . . . . . . . . . . . 75
4.8 SSB Bifurcation Diagram Comparison for LL Steady States and NCVA
6-Parameter Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.9 LL Model and Center Manifold Approach Steady State Comparison near
Pitchfork Bifurcation Points . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.10 LL Model and Center Manifold Reduction Pitchfork Bifurcation Orbits . . 87
5.1 Temporal Proﬁles of Fundamental States . . . . . . . . . . . . . . . . . . . 103
5.2 Tweezers of Narrow, Natural, and Wide Widths . . . . . . . . . . . . . . . 106
5.3 Power Ratios Inside and Outside Tweezer with Narrow Width . . . . . . . 108
5.4 Tweezer with Narrow Width Power Comparison . . . . . . . . . . . . . . . 109
5.5 Comparison of Power Ratio Inside Narrow Tweezer for LL Model and NCVA110
5.6 Dynamic Evolution of Narrow Tweezer with Tweezed CS . . . . . . . . . . 111
5.7 Dynamic Evolution of Narrow Tweezer with no-CS . . . . . . . . . . . . . 113
5.8 Power Ratios Inside and Outside Tweezer with Natural Width . . . . . . . 115
5.9 Tweezer with Natural Width Power Comparison . . . . . . . . . . . . . . . 116
5.10 Comparison of Power Ratios of Natural Tweezer for LL Model and NCVA 117
5.11 Dynamic Evolution of Natural Tweezer with Tweezed CS . . . . . . . . . . 118
5.12 Dynamic Evolution of Natural Tweezer with no-CS . . . . . . . . . . . . . 119
5.13 Dynamic Evolution of Natural Tweezer with Non-Tweezed CS . . . . . . . 120
5.14 Dynamic Evolution of Natural Tweezer with LL Tweezed-CS and NCVA
No-CS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
xv

List of Figures xvi
5.15 Power Ratios Inside and Outside Tweezer with Wide Width . . . . . . . . 123
5.16 Tweezer with Wide Width Power Comparison . . . . . . . . . . . . . . . . 124
5.17 Comparison of Power Ratio Inside Wide Tweezer for LL Model and NCVA 125
5.18 Dynamic Evolution of Wide Tweezer with Tweezed CS . . . . . . . . . . . 126
5.19 Dynamic Evolution of Wide Tweezer with no-CS . . . . . . . . . . . . . . 127
5.20 Dynamic Evolution of Wide Tweezer with Non-Tweezed CS . . . . . . . . 128
5.21 Dynamic Evolution of Wide Tweezer with Artiﬁcial Tweezing . . . . . . . 129
5.22 Temporal Tweezing of a CS . . . . . . . . . . . . . . . . . . . . . . . . . . 130
D.1 Power Ratios Inside and Outside Tweezer with σφ = 0.5 and hφ = 2 . . . . 172
D.2 Power Ratios Inside and Outside Tweezer with σφ = 1 and hφ = 2 . . . . . 173
D.3 Power Ratios Inside and Outside Tweezer with σφ = 2 and hφ = 2 . . . . . 174

List of Tables
4.1 Comparison of the Spectral Radius for Dirichlet, Neumann, and Periodic
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Comparison of Unstable Eigenvalue for Dirichlet, Neumann, and Periodic
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
xvii

For my family
Mom, Dad, and Robby
xix

Chapter 1
Introduction
The nonlinear Schrödinger (NLS) equation is a dispersive nonlinear partial differential
equation (PDE) describing a wide range of physical nonlinear systems. The earliest
applications of the NLS were introduced by Ginzburg, Landau, and Pitaevskii in the fields
of superconductivity [1, 2] and superfluidity [3]. However, the wider physical importance
of the NLS equation was made evident by Chiao et. al. [4] and Talanov [5] in studying
self-focusing phenomenon. The equation and its variants are of principal interest to
applications from optical physics [6], atomic physics [7] and other areas of mathematical
physics [8], not only in their conservative, but also in dissipative variants of the model [9].
In what follows, we give you a brief review of the NLS equation and its soliton solutions.
1.1 A Brief Introduction to NLS and Solitons
The NLS is the lowest order (i.e. normal form) nonlinear wave partial differential equa-
tion (PDE) describing envelope waves in nonlinear media. The one-dimensional NLS
equation in nondimensonal form is usually cast as
i∂tψ +
1
2
∂xxψ + g|ψ|2
ψ = 0, (1.1)
1

Chapter 1. Introduction 2
where ψ is the complex field and g is the nonlinearity. The NLS has two forms de-
pending on the sign of the nonlinearity: an attractive/focusing NLS for g = +1 and
a repulsive/defocusing NLS for g = −1. The NLS admits soliton solutions which are
solitary, localized wavepackets traveling without distortion due to the interplay of non-
linearity (|ψ|2ψ) and dispersion (∂xxψ). The focusing NLS allows for bright soliton
solutions characterized by spatial attenuation towards infinity, while the defocusing NLS
allows for dark soliton solutions with a nontrivial background intensity (i.e. the soliton
does not vanish at infinity).
To find these solitons, we look for solutions using an ansatz of the form:
ψ(x, t) = A(x, t) exp(iφ(x, t)), (1.2)
where A(x, t) describes the envelope wave and φ(x, t) is the carrier wave. We substitute
Eq. (1.2) into Eq. (1.1) and separate into the real and imaginary parts to obtain the
system of equations
At + Axφx +
1
2
Aφxx = 0, (1.3)
−Aφt +
1
2
Axx −
1
2
Aφ2
x + gA3
= 0. (1.4)
We use a linear phase φ = b(x − ct) + φ0, which satisfies Eq. (1.4), then the amplitude
Eq. (1.3) is integrated and becomes
Axx − bf + 2gA3
= 0. (1.5)
Depending on the sign of g, Eq. (1.3) can be integrated with the appropriate boundary
conditions to obtain the soliton solution. In the case of defocusing NLS (g = −1) the

solution has non-zero boundary conditions and a hyperbolic tangent-type profile describes
the wave packet envelope, which is known as a dark soliton. However, our main interest
is the focusing NLS (g = +1) for which the elliptic Eq. (1.3) is easily integrated by
assuming zero boundary conditions to obtain
ψ(x, t) =
√
b sech
√
b(x − ct − x0) exp i(cx +
b − c2
2c
t + φ0 , (1.6)
which is a four-parameter bright soliton solution. For the purposes of this dissertation,
we are most interested in bright solitons, which in its simplest form has a sech-type
profile describing the wave packet envelope with a spatial and time-dependent phase.
Understanding the fundamental nature and solutions of the conservative NLS is necessary
in order to begin developing the concepts in this thesis which are concerned with non-
conservative (dissipative) PDEs of the NLS-type.
1.2 Overview
The focus of this dissertation is to explore variational approaches to study nonlinear
waves including dissipative pulse propagation [10]. Applications of this technique in-
clude, but are not limited to, PT -symmetric variants in nonlinear optics [11], excitations
of Bose-Einstein condensates [12] and charged polymers [13]. Our variational approach
is based on using well-educated ansatze in the Lagrangian of complex, infinitely di-
mensional, problems cast in the form of dissipative variants of the NLS equation. By
choosing an ansatz with time dependent parameters such as center position, width, am-
plitude, phase, etc., the original problem can be reduced in complexity to a few degrees
of freedom. The variational approximation (VA) method projects the high-dimensional
(or infinite-dimensional) dynamics to a low-dimensional system on the dynamics of the

time-dependent parameters to describe the qualitative and quantitative behavior of the
original dynamical, complex system. Classically, the variational method relies on the
existence of a Lagrangian or Hamiltonian structure from which the Euler-Lagrange equa-
tions can be derived. This prerequisite limits the application of the variational approach
to conservative systems. It is this limitation that we want to overcome by extending the
VA to non-conservative (non-Hamiltonian) systems.
The recent publication by Galley [14] offers a new perspective to the classical mechanical
formulations. He asserts that Hamilton’s principle has a pitfall in that it is formulated as
a boundary value problem in time but used to derive equations of motion that are solved
with initial data. By treating the extremization problem as an initial value problem,
a variational calculus can be applied to non-conservative systems. Although Galley’s
proposal was originally cast for classical mechanics systems, i.e. systems described by
ordinary differential equations (ODEs), it paved the way for the application to disper-
sive complex nonlinear PDEs. In this dissertation, we extend Galley’s approach towards
a non-conservative variational approximation (NCVA) for general complex PDEs of the
NLS type. This extension is initially derived in Chapter 2 for the focusing NLS in order
to simplify basic test cases on soliton propagation, although the same procedure can be
applied to the defocusing NLS. There are at least two other variational methods that
have been applied to dissipative NLS equations: the perturbed variational approxima-
tion (PVA) and a generalization of the Kantorovitch method in a recent publication by
Cerda [10]. In the following chapter, we briefly summarize the formalism of these two
methods from literature and prove that they are equivalent to the NCVA in the case of
the NLS equation.

The application of the NCVA relies on obtaining a useful Lagrangian for the non-
conservative system. The NCVA produces a system of equations depending on the num-
ber of ansatz parameters and effectively reduces the original PDE model to a system of
ODEs. To show the relevance and validity of the NCVA, we explore three dynamical
system examples in Chapter 3. Two are dissipative NLS systems, one with linear loss
and the other with density dependent loss. The latter example deals with nonlinear
pulse propagation in the presence of two-photon absorption. The third example is a
non-Hamiltonian, non-conservative dynamical model for exciton-polariton condensates
which are bound electron-hole pairs (excitons) interacting with light (photons). Polari-
tons are important in solid-state Bose-Einstein condensates (BECs) due to their light
mass allowing for condensation temperatures on the order of tens of Kelvin; however,
a disadvantage is their short radiative lifetime of the order 1-10 ps so they have to be
continuously replenished from a reservoir of excitons. The external pumping from the
reservoir of excitons counterbalances the loss of polaritons due to the decay [15]. These
two effects yield a modified NLS model with linear gain (exciton pumping) and density
dependent loss (polariton decay). To validate the NCVA, we compare the NCVA ODEs
for the functional parameters of the ansatz to full numerical solutions of the original
PDE.
The main topics of interest elaborated in the dissertation are non-conservative PDEs of
the NLS type in nonlinear optics, specifically on the existence of spontaneous symmetry
breaking (SSB) and temporal tweezing in these systems. After developing the NCVA
methodology in Chapter 2 and showcasing its application in Chapter 3, we begin the
extension of the NCVA approach to a variant of the NLS equation: the mean-field
Lugiato-Lefever (LL) model by studying symmetry breaking instability in a coherently-
driven optical Kerr resonator in Chapter 4. SSB is the basis for many phase transitions

and accounts for effects including ferromagnetism, superconductivity, and convection
cells. SSB occurs in nonlinear Hamiltonian systems such as open systems in the case of a
synchronously-pumped passive optical resonator filled with a Kerr nonlinear material as
experimentally studied in Ref. [16]. In addition to the NCVA, we also perform a detailed
stability analysis of the LL model and analyze the temporal bifurcation structure of
stationary symmetric configurations and the emerging asymmetric states as a function
of the pump power. We also use local bifurcation theory in order to analyze the most
unstable eigenmode of the system.
In Chapter 5 we investigate temporal tweezing of cavity solitons in a passive loop of
optical fiber pumped by a continuous-wave laser beam which is described by a modified
LL model. The optical trapping and manipulation of the temporal position of light pulses
is highly desirable as it has immediate implications for optical information processing
which has recently been realized experimentally [17]. Information is treated as a sequence
of pulses that can be stored and reconfigured by trapping ultrashort pulses of light and
dynamically moving them around in time. In the experiment, temporal cavity solitons
(CSs) exist as picosecond pulses of light that recirculate in a loop of optical fibre and are
exposed to temporal controls in the form of a gigahertz phase modulation. It has been
shown, both theoretically and experimentally, that the CSs are attracted and trapped
to phase maxima, suppressing all soliton interactions. These trapped CSs can then be
manipulated in time, either forward or backward, which is known as temporal tweezing.
We study the existence and dynamics of temporally tweezed CSs. The key phenomena
reported herein are parametric intervals for the existence of tweezed CSs, dissipative CSs,
and non-trapped CSs. We also apply the NCVA to identify regions of temporal tweezing,
and compare to the full numerical solutions of the original PDE.

In summary, the dissertation is organized as follows. In Chapter 2 we present the for-
malism of the NCVA and its application to the NLS in Chapter 3. Chapter 4 is a
comprehensive analysis of SSB for the LL equation using a NCVA and local bifurcation
analysis. Finally, Chapter 5 identiﬁes parametric regions for temporal tweezing using
both a modiﬁed LL and NCVA approach, and Chapter 6 concludes our work, including
suggestions for future studies.

Chapter 2
Non-Conservative Variational
Approximation
A commonly used approximation method is known as the variational method. This
method is widely used in quantum chemistry, especially Hartree-Fock and variational
quantum Monte Carlo theories lacking exact solutions [18–20]. Variational methods are
also useful to describe nonlinear wave dynamics in nonlinear optics and atomic physics [6,
21–23]. In these methods a well-informed ansatz is substituted into an original partial
differential equation (PDE) model which reduces an infinite dimensional system to a few
degrees of freedom. Variational approximation (VA) methods rely on a conservative,
closed system with a Lagrangian or Hamiltonian formulation from which one derives
Euler-Lagrange equations for the approximate dynamics of the system projected into
the solution space of the ansatz.
The VA method projects the infinite-dimensional dynamics of the original PDE to a
small, finite-dimensional, dynamical system for the parameters of the ansatz space. The
intrinsic drawbacks of using an ansatz subspace is that it must contain enough degrees of
freedom to describe the dynamical properties of the system and requires prior knowledge
of these dynamics. Therefore, when the ansatz ceases to describe the full PDE dynamics,
9

Chapter 2. Non-Conservative Variational Approximation 10
the projection can lead to invalid results [24], a feature which is naturally expected (given
the large reduction in the number of degrees of freedom) when the full PDE dynamics
ceases to be well-described by the selected ansatz. Nonetheless, there have been some
efforts to control the corrections of the VA to increase the accuracy of the results [25].
Due to the limitations of the application of the VA method to conservative systems,
there are several well-known continuations for non-conservative systems such as linear
perturbed VA and Kantorovitch method. Another perspective to the classical mechanical
formulation was offered by Galley [14, 26] by recognizing that the Hamiltonian-Lagrange
formulation is a boundary value problem in time used to derive equations of motion
solved with initial data and confined to conserved systems. Instead, Galley proposes
treating the extremization as an initial value problem in order to apply the variational
calculus to non-conservative systems, specifically systems described by ODEs.
In Sec. 2.1 we extend Galley’s [14] initial value formulation to complex nonlinear PDEs.
In Sec. 2.2 we focus on the extension of NCVA method for NLS-type equations. The
two well studied methods currently used to derive initial value problems from the non-
conservative NLS are briefly outlined; the perturbed variational approximation (PVA) in
Sec. 2.3.1 and the modified Kantorovitch method [10] (KVA) in Sec. 2.3.2. In Sec. 2.3.3,
we prove that the three methods (PVA, KVA, and NCVA) are equivalent. After estab-
lishing the theoretical foundation of the NCVA method, we present in Chapter 3 results
for three bright soliton test cases: NLS with linear loss, NLS with density dependent loss
and NLS with linear gain and density dependent loss (exciton-polariton condensate).

2.1 Non-conservative Variational Approximation
Formalism
Hamilton’s principle relies on a Lagrangian formulation of a system to derive equations
of motion for conservative systems. The derivation of Lagrange’s equations considers the
entire evolution of the system between times ti and tf and small virtual variations of this
motion from the actual motion, known as an “integral principle”. The integral Hamilton’s
principle describes the motion of a monogenic system i.e. a physical system for which all
forces (except the force constraint) are derivable from a generalized scalar potential [27].
Hamilton’s principle for monogenic systems states: “The motion of a system from time
ti to time tf is such that the line integral (called the action of the action integral)
S =
tf
ti
L dt, (2.1)
where L = T − V has a stationary value for the actual path of the motion” [27]. L is
the Lagrangian density, T is kinetic energy and V is the potential energy of the system.
Therefore, from all the possible paths from the position at ti to the position at tf , the
system point will travel along that path for which the integral Eq. (2.1) is stationary.
Hamilton’s principle is summarized by saying that the motion is such that the variation
of the line integral S for ﬁxed ti and tf is zero:
δS = δ
tf
ti
L(q1, ..., qn, ˙q1, ..., ˙qn, t)dt = 0. (2.2)
Lagrange equations follow from Hamilton’s principle, which are formed as a bound-
ary value problem in time with initial data. However, we are interested in studying
the dynamics of non-conservative systems. For simple dissipative forces, one can use

Rayleigh’s dissipation function. The following section explains the Lagrangian formula-
tion for generic non-conservative systems.
Extending the variational approximation for non-conservative systems in classical me-
chanics described in Galley [14], we apply the technique to complex PDEs. The foun-
dation of the derivation of the non-conservative variational approximation is based on
using Hamilton’s principle of stationary action compatible as an initial value problem —
as opposed to a boundary value in time — derived to solve equations of motion used in
conservative systems. In the papers by Galley [14] and Kevrekidis [11], the authors treat,
respectively, dissipative systems in the form of ODEs and real PDEs. We are interested
in extending the initial value problem formulations of Hamilton’s principle to complex
PDEs, i.e. the NLS equation.
In the recent publication Galley [14] illustrated that the time-symmetric and conservative
dynamics is due to the boundary value form of the action extremization problem. Instead,
he proposed the extremization problem to be considered as an initial value problem
for two sets of variables, q1 and q2, then one could apply variational calculus for non-
conservative systems.
One can introduce two sets of variables q1 and q2 such that q1 gives the correct force
provided q2 = q1 after the variation. Let q ≡ {qi}N
i=1 and ˙q ≡ { ˙qi}N
i=1 be a set of N
generalized coordinates and velocities. Double both sets of quantities, q → (q1, q2) and
˙q → ( ˙q1, ˙q2) and parametrize both coordinate paths:
q1,2(t, ) = q1,2(t, 0) + η1,2(t), (2.3)
where q1,2(t, 0) are the coordinates of two stationary paths ( 1) and η1,2(t) are arbi-
trary virtual displacements. The following equality conditions are required for varying

the action:
η1,2(ti) = 0, (2.4)
q1(tf , ) = q2(tf , ), (2.5)
˙q1(tf , ) = ˙q2(tf , ). (2.6)
Therefore, the equality condition does not fix either value at the final time. After all
variations are performed, both paths are set equal and identified with the physical one,
q(t), the so-called physical limit.
The action functional of q1 and q2 is defined as the total line integral of the Lagrangian
along both paths plus the line integral of a functional R depending on both paths {qa}2
a=1:
S[qa] ≡
tf
ti
dt L(q1, ˙q1) +
ti
tf
dt L(q2, ˙q2) +
tf
ti
dt R(qa, ˙qa, t), (2.7)
=
tf
ti
dt[L(q1, ˙q1) − L(q2, ˙q2) + R(qa, ˙qa, t)]. (2.8)
The above action defines a new Lagrangian:
Λ(qa, ˙qa) ≡ L(q1, ˙q1) − L(q2, ˙q2) + R(qa, ˙qa, t). (2.9)
If R is written as the difference of two potentials V (q1)−V (q2), then it may be absorbed
into the difference of the Lagrangians, leaving R zero. A nonzero R describes non-
conservative forces and couples the two paths together.
For convenience, following [14], we make a change of variables to q+ = (q1 + q2)/2 and
q− = q1 − q2 because q− → 0 and q+ → q in the physical limit. The conjugate momenta
are found as π± = ∂Λ/∂ ˙q and the paths are parametrized as q±(t, ) = q±(t, 0)+ η±(t).

The new action is stationary under these variations if (dS[q±]/d ) =0 = 0 for all η±:
tf
ti
dt η+ ·
∂Λ
∂q+
−
d
dt
∂Λ
∂ ˙q+ =0
+ η− ·
∂Λ
∂q−
−
d
dt
∂Λ
∂ ˙q− =0
+ η+(t) · π−(t) + η−(t) · π+(t)
tf
t=ti
= 0, (2.10)
where η+ · π− = N
i=1 η+iπ−i. From the equality condition, η−(tf ) = 0, π−(tf ) = 0 and
η±(ti) = 0, the boundary terms all vanish. Therefore, the action is stationary for any
η±(t) when the two variables q±(t) solve
dπ
dt
=
∂Λ
∂q±
. (2.11)
In the q1,2 coordinates instead of the ± variables, the action is found by solving dπ1,2/dt =
∂Λ/∂q1,2 with conjugate momenta π1,2 = (−1)1,2∂Λ/∂ ˙q1,2 as a function of q1,2 and ˙q1,2.
In the physical limit (PL), only the ∂Λ/∂q− = dπ+/dt equation survives, such that
d
dt
π(q, ˙q) =
∂Λ
∂q−
PL
=
∂L
∂q
+
∂R
∂q−
PL
, (2.12)
with conjugate momenta
π(q, ˙q) =
∂Λ
∂ ˙q− PL
=
∂L
∂ ˙q
+
∂R
∂ ˙q− PL
. (2.13)
When R = 0 and under the presence of conservative forces, the usual Euler-Lagrange
equations are recovered. A nonzero R is derived from non-conservative forces and modi-
ﬁes the trajectories of Eqs. (2.12) and (2.13). In our special case, we are concerned with
complex non-conservative forces. In the case of complex R, the action which deﬁnes a
new Lagrangian, Eq. (2.9), includes a line integral in which q1 and q2 paths are coupled

to each other. As we show in Section 2.2 below, the complex conjugate of the functional
terms in L are similarly necessary for solving the Euler-Lagrange equations with com-
plex PDEs, such as the NLS equation. In the physical limit, only the Euler-Lagrange
equation for the + variable survives. Therefore, expanding the action in powers of q−
the equations of motion follow the variational principle:
δS[q±]
δq−(t)
PL
= 0. (2.14)
Only terms in the new action that are perturbatively linear in q− contribute to physical
forces. In the following section, we formulate Hamilton’s principle with initial conditions
for systems described by complex PDEs.
2.1.1 An Illustrative Example
In order to understand Galley’s [14] new formulation, we consider a well-known second
order diﬀerential equation of motion for the harmonic oscillator with a linear damping
given by
¨x + 2β ˙x + w2
0x = 0, (2.15)
where w0 and β are, respectively, frequency and damping parameter. The conservative
harmonic oscillator (Eq. (2.15) with β = 0) is derived by forming a Lagrangian
L = T − V, (2.16)
for the mass on the end of a spring wth kinetic energy, T = m ˙x2/2 and potential energy,
V = kx2/2. Using the Lagrangian, we apply the Euler-Lagrange equations to ﬁnd the

equation of motion
¨x + w2
0x = 0, (2.17)
where w0 = k/m. This method works for the conservative system, but if we want
to add the linear damping term, i.e. 2β ˙x, we do not have a Lagrangian that can de-
scribe non-conservative forces. Using Galley’s approach, we consider the following new
Lagrangian, given in the ± variables:
Λ(x±, ˙x±) = ˙x− ˙x+ − w2
0x+x− + 2β ˙x+x−, (2.18)
where the first term is the kinetic energy, the second term is the potential energy and the
third term is R containing all non-conservative forces. The new Lagrangian Eq. (2.18)
is unique for terms linear in x− and its time derivatives, which do not contribute to
physical forces. With the new Lagrangian we can recast the Euler-Lagrange equations
using Eqs. (2.14), or (2.12) and (2.13), which result in the standard equation of motion
Eq. (2.15), at the physical limit, where x+ → x and x− → 0. The key point is that
these equations for dissipative motion are derived from the (new) Lagrangian Eq. (2.18),
and solved through a modified Euler-Lagrange formulation which results in equations of
motion.
With the new Lagrangian Eq. (2.18), we can use variational techniques with an ansatz,
and find the equations of motion for the variational parameters. In this example using
an ansatz of the form x = Aewt would recover the well-known solutions for underdamped
(w2
0 > β2), overdamped (w2
0 < β2), and critically damped (w2
0 = β2) systems.

2.2 Derivation of Non-Conservative Variational
Method for Nonlinear Shrödinger Equation
The NCVA formalism is extended for the NLS equation. The one-dimensional (1D) NLS
equation in non-dimensional form is [28]
iut +
1
2
uxx + g|u|2
u = 0, (2.19)
where u(x, t) is the complex field and g is the nonlinearity coefficient [g = +1 (g = −1)
corresponding to attractive/focusing (repulsive/defocusing) nonlinearity]. This NLS is a
conservative Hamiltonian PDE with Lagrangian density [28–30] given by
L =
i
2
(u∗
ut − uu∗
t ) +
1
2
|ux|2
−
1
2
g|u|4
, (2.20)
where (·)∗ denotes complex conjugation. We will adopt the following notation for clarity:
densities are denoted with calligraphic symbols (cf. L), effective quantities integrated over
all x use standard symbols L =
∞
−∞ L dx. The corresponding Euler-Lagrange equation
for the conservative Lagrangian density is
d
dt
∂L
∂u∗
t
=
∂L
∂u∗
−
d
dx
∂L
∂u∗
x
. (2.21)
We verify that the Lagrangian density Eq. (2.20) indeed corresponds to the NLS Eq. (2.19)
by noticing that −1
2|u|4 = −1
2(u∗)2uu such that d(−1
2(u∗)2uu)/du∗ = −|u|2u, and the
partial of L with respect to u∗
x, comes only from the term 1
2|ux|2. Using these terms,

Eq. (2.21) becomes
i
2
ut = −
i
2
ut −
1
2
uxx − |u|2
u, (2.22)
i
2
ut +
i
2
ut = −
1
2
uxx − |u|2
u, (2.23)
iut +
1
2
uxx + |u|2
u = 0, (2.24)
and we recover the conservative focusing NLS Eq. (2.19).
We are interested in non-conservative terms (P) that may depend on the field u, its
derivatives, and/or its complex conjugate. The non-conservative NLS may be cast in the
following general form:
iut +
1
2
uxx + g|u|2
u = P. (2.25)
For the variational formulation of non-conservative systems [14] we define coordinates u1
and u2 and construct the total Lagrangian:
LT = L1 − L2 + R, (2.26)
where Li ≡ L(ui, ui,t, ui,x, ..., t) for i = 1, 2, correspond to the conservative Lagrangian
densities for u1 and u2 as defined by Eq. (2.20), and R contains the non-conservative
forces originating from the term P in Eq. (2.19). The non-conservative part of the total
Lagrangian (2.26) is related to the term P in Eq. (2.25) by
P =
∂R
∂u∗
− PL
. (2.27)

It follows by construction that
R = P u∗
− + const, (2.28)
where the constant of integration is with respect to u∗
−.
We deﬁne a change of variables u+ = (u1 + u2)/2 and u− = u1 − u2 strictly for con-
venience. In the physical limit (PL) u+ → u and u− → 0. Based on the equality
conditions in Sec. 2.1, η−(tf ) = π−(tf ) = η±(ti) = 0, the boundary terms all vanish.
The corresponding conjugate momenta are deﬁned as in Sec. 2.1 and the equation of
motion is
∂
∂t
δL
δu∗
t
=
δL
δu∗
+
δR
δu∗
− PL
, (2.29)
where δ denotes Fréchet derivatives. Therefore, the NCVA method recovers the Euler-
Lagrange equations for the conservative terms and lumps all the non-conservative terms
into [δR/δu∗
−]PL. The most crucial part of the NCVA method is constructing R in such
a way that at the physical limit, we recover the non-conservative forces [P in Eq. (2.25)]
2.2.1 A Brief Example for Constructing R
In the case of the NLS Eq. (2.25) where P is a dissipative non-conservative term i.e. P =
−iκ|u|2u [10]. The non-conservative force must be of the form
∂R
∂u∗
− PL
= (−iκ|u|2
u). (2.30)

A possible choice is to use R = −iκ|u+|2u+u∗
− + const in which the non-conservative
forces couple the two paths to each other:
R = −iκ|u+|2
u+u∗
− + iκ|u+|2
u+u−, (2.31)
satisfying the criteria in the physical limit in Eq. (2.30).
2.2.2 NCVA Recovery of NLS Equation
In order to showcase NCVA methodology and in particular the use of the u1/2 and u±
variables, let us solve the conservative NLS Eq. (2.26) where R = 0. We begin with
variables
u1 =
(2u+ + u−)
2
, (2.32)
u2 =
(2u+ − u−)
2
. (2.33)
Again, we can solve the total Lagrangian (L = L1 + L2, where R = 0) of Eq. (2.26) in
the u1 and u2 coordinate system and switch to the u+ and u− variables:
L =
i
2
(u1u∗
1,t − u1,tu∗
1 − u2u∗
2,t − u2,tu∗
2) +
1
2
(u1,xu∗
1.x − u2.xu∗
2.x − u2
1u∗2
1 + u2
2u∗2
2 ),
=
i
2
(2u+ + u−)
2
(2u∗
+,t + u∗
−,t)
2
−
(2u+,t + u−,t)
2
(2u∗
+ + u∗
−)
2
−
(2u+ − u−)
2
(2u∗
+,t − u∗
−,t)
2
+
(2u+,t − u−,t)
2
(2u∗
+ − u∗
−)
2
+
1
2
(2u+,x + u−,x)
2
(2u∗
+,x + u∗
−,x)
2
−
1
2
(2u+,x − u−,x)
2
(2u∗
+,x − u∗
−,x)
2
+
1
2
(2u+ − u−)
2
(2u∗
+ − u∗
−)
2
−
(2u+ + u−)
2
(2u∗
+ + u∗
−)
2
×
(2u+ − u−)
2
(2u∗
+ − u∗
−)
2
+
(2u+ + u−)
2
(2u∗
+ + u∗
−)
2
. (2.34)

The terms that survive are:
L =
i
2
u−u∗
+,t + u+u∗
−,t − u−,tu∗
+ − u∗
−u+,t +
1
2
u−,xu∗
+,x + u+,xu∗
−,x
+
1
4
− u−u∗
+ − u+u∗
− 4u+u∗
+ + u−u∗
− ,
=
i
2
u−u∗
+,t + u+u∗
−,t − u−,tu∗
+ − u∗
−u+,t +
1
2
u−,xu∗
+,x + u+,xu∗
−,x
− u+u∗
+u−u∗
+ −
1
4
u−u∗
−u−u∗
+ − u+u∗
−u+u∗
+ −
1
4
u+u∗
−u−u∗
−. (2.35)
Now take Eq. (2.35) at the physical limit (PL),
∂L
∂u∗
−,t PL
=
i
2
u+
PL
=
i
2
u. (2.36)
The Euler-Lagrange equation can then be evaluated in the physical limit:
d
dt
∂L
∂u∗
−,t
=
∂L
∂u∗
− PL
−
d
dx
∂L
∂u∗
−,x PL
, (2.37)
d
dt
i
2
u = −
i
2
u+,t − u+|u+|2
−
1
4
u−u∗
+u− −
1
4
u+|u−|2
PL
−
d
dx
1
2
u+,x
PL
.
(2.38)

The individual terms in Eq. (2.38) evaluated at the physical limit are:
d
dt
i
2
u =
i
2
ut,
−
i
2
u+,t
PL
= −
i
2
ut,
−
1
4
u−u∗
+u−
PL
= 0,
−
1
4
u+|u−|2
= 0,
−u+|u+|2
= −|u|2
u,
−
d
dx
1
2
u+,x
PL
= −
d
dx
1
2
u+,x
PL
= −
1
2
uxx.
Plugging in all the physical limits into Eq. (2.38) one gets:
i
2
ut = −
i
2
ut − |u|2
u −
1
2
uxx,
and we therefore arrive at the focusing NLS Eq. (2.19). Similar variational formulations
can be applied to other PDE (or ODE) systems.
2.3 Other Non-Conservative Methods
In this section, the methodologies of the standard perturbed variational approach [31]
and modiﬁed Kantorovitch methods [10, 32–34] are compared to the NCVA method.
2.3.1 Perturbed Variational Approach (PVA) Formalism
We start with the conservative focusing NLS Eq. (2.19) [g = +1] and the Lagrangian
density Eq. (2.20). For consistency of notation we will use calligraphic symbols (cf. L) to
denote densities while their eﬀective (integrated over all x) quantities we will use stan-
dard symbols. Namely L =
∞
−∞ L dx. In the perturbed variational method, Eq. (2.19)

becomes a non-conservative modified NLS equation with the addition of non-conservative
generalized force P = Q, where is a formal perturbation parameter
iut +
1
2
uxx + |u|2
u = Q(u, ux, ut, . . . , x, t). (2.39)
The Euler-Lagrange equation for the unperturbed ( = 0) NLS, Eq. (2.19), is given by:
∂ ¯L
∂p
−
d
dt
∂ ¯L
∂ ˙p
= 0, (2.40)
where ¯L(p) = ¯Ldx where ¯L ≡ L[¯u(x, t, p)] is the conservative Lagrangian Eq. (2.20)
evaluated on the chosen variational ansatz ¯u containing a vector of variational parameters
p and the over-dot denotes the derivative with respect to t. The effective Lagrangian ¯L
depends on ¯L, where we will use a bar over quantities that are evaluated at the variational
ansatz. To solve the Euler-Lagrange equation for the perturbed NLS, Eq. (2.39), we find
the remainder of
∂ ¯LT
∂p
−
d
dt
∂ ¯LT
∂ ˙p
= 0, (2.41)
which is nonzero for the total Lagrangian ¯LT = ¯L + ¯L with conservative terms ¯L of the
NLS Eq. (2.19) and non-conservative terms ¯L , i.e. Q(¯u, ¯ux, ¯ut, . . . , x, t). The first term
in the perturbed Euler-Lagrange equation, Eq. (2.41), for ansatz ¯u is
∂ ¯LT
∂p
=
∂
∂p
∞
−∞
¯LT dx,
=
∞
−∞
∂ ¯LT
∂¯u
∂¯u
∂p
+
∂ ¯LT
∂¯ut
∂¯ut
∂p
+
∂ ¯LT
∂¯ux
∂¯ux
∂p
+
∂ ¯LT
∂¯u∗
∂¯u∗
∂p
+
∂ ¯LT
∂¯u∗
t
∂¯u∗
t
∂p
+
∂ ¯LT
∂¯u∗
x
∂¯u∗
x
∂p
dx,
=
∞
−∞
∂ ¯LT
∂¯u
∂¯u
∂p
−
∂
∂x
∂ ¯LT
∂¯ux
∂¯u
∂p
+
∂ ¯LT
∂¯u∗
∂¯u∗
∂p
−
∂
∂x
∂ ¯LT
∂¯u∗
x
∂¯u∗
∂p
dx.

The second term of the perturbed Euler-Lagrange Eq. (2.41), yields
d
dt
∂ ¯LT
∂ ˙p
=
∞
−∞
∂
∂t
∂ ¯LT
∂¯ut
∂¯ut
∂ ˙p
+
∂ ¯LT
∂¯u∗
t
∂¯u∗
t
∂ ˙p
dx,
=
∞
−∞
∂
∂t
∂ ¯LT
∂¯ut
∂¯u
∂p
+
∂ ¯LT
∂¯ut
∂
∂t
∂¯u
∂p
+
∂
∂t
∂ ¯LT
∂¯u∗
t
∂¯u∗
∂p
+
∂ ¯LT
∂¯u∗
t
∂
∂t
∂¯u∗
∂p
dx,
=
∞
−∞
∂
∂t
∂ ¯LT
∂¯ut
∂¯u
∂p
+
∂
∂t
∂ ¯LT
∂¯u∗
t
∂¯u∗
∂p
dx
where the only term with ˙p is ut. Therefore, by combining the above two terms in
Eq. (2.41), we obtain
∂ ¯LT
∂p
−
d
dt
∂ ¯LT
∂ ˙p
=
∞
−∞
∂¯u
∂p
∂ ¯LT
∂¯u
−
∂
∂x
∂¯u
∂¯ux
−
∂
∂t
∂ ¯LT
∂¯ut
+
∂¯u∗
∂p
∂ ¯LT
∂¯u∗
−
∂
∂x
∂¯u∗
∂¯u∗
x
−
∂
∂t
∂ ¯LT
∂¯u∗
t
dx.
Using only the conservative term in the Lagrangian ¯LT , the solution to the unperturbed
NLS, Eq. (2.19), is
∂ ¯L
∂p
−
d
dt
∂ ¯L
∂ ˙p
= 0
=
∞
−∞
∂¯u
∂p
∂ ¯L
∂¯u
−
∂
∂x
∂¯u
∂¯ux
−
∂
∂t
∂ ¯L
∂¯ut
+
∂¯u∗
∂p
∂ ¯L
∂¯u∗
−
∂
∂x
∂¯u∗
∂¯u∗
x
−
∂
∂t
∂ ¯L
∂¯u∗
t
dx.
(2.42)

Now, using only the non-conservative term in the Lagrangian ¯LT , the solution to the
perturbed NLS Eq. (2.39) is
∂ ¯L
∂p
−
d
dt
∂ ¯L
∂ ˙p
=
∞
−∞
∂¯u
∂p
∂ ¯L
∂¯u
−
∂
∂x
∂¯u
∂¯ux
−
∂
∂t
∂ ¯L
∂¯ut
+
∂¯u∗
∂p
∂ ¯L
∂¯u∗
−
∂
∂x
∂¯u∗
∂¯u∗
x
−
∂
∂t
∂ ¯L
∂¯u∗
t
dx,
=
∞
−∞
∂¯u
∂p
¯Q∗
+
∂¯u∗
∂p
¯Q dx,
=
∞
−∞
∂¯u
∂p
¯Q∗
+ ¯Q
∂¯u∗
∂p
dx.
The perturbed variational approximation gives the following perturbed Euler-Lagrange
equation combining the ¯L and ¯L terms:
d
dt
∂ ¯L
∂ ˙p
−
∂ ¯L
∂p
=
∞
−∞
¯P∗ ∂¯u
∂p
+ ¯P
∂¯u∗
∂p
dx, (2.43)
where we substituted ¯P = ¯Q [35]. The right hand side is equivalent to the following
modiﬁed Kantorovitch [see Eq. (2.52) below] method such that
∞
−∞
¯P∗ ∂¯u
∂p
+ ¯P
∂¯u∗
∂p
dx ≡ 2Re ¯P
∂¯u∗
∂p
dx. (2.44)
2.3.2 Modiﬁed Kantorovitch Method Formalism
As developed in Cerda et al. [10], a variational technique is outlined to deal with nonlinear
pulse propagation. Ref. [10] uses a generalization of the Kantorovitch method for non-
conservative systems in the NLS equation. The total Lagrangian is the sum of the
conservative Lagrangian L and a non-conservative Lagrangian, L :
LT (u, u∗
, x, t, ux, ut, u∗
x, u∗
t , ..., etc) = L + L , (2.45)

where u(x, t) represents the soliton. In the method, the function u(x, t) must render the
Lagrangian integral stationary as expressed by Hamilton’s principle:
δ LT dxdt = δ (L + L )dxdt = 0, (2.46)
such that the Euler-Lagrange equations of the system are given by
δLT
δui
=
d
dt
∂L
∂(∂ui
∂t )
+
d
dx
∂L
∂(∂ui
∂x )
−
∂L
∂ui
= Pi. (2.47)
The non-conservative dynamics are taken into account through Pi:
Pi =
∂L
∂ui
−
d
dt
∂L
∂(∂ui
∂t )
−
d
dx
∂L
∂(∂ui
∂x )
, (2.48)
where the index i is either 1 or 2 with u1 = u and u2 = u∗. The approximate Euler-
Lagrangian equations for non-conservative systems uses a generalization of the Rayleigh-
Ritz method known as the Kantorovitch method assuming the extremum of the varia-
tional integral of the Lagrangian function is expressed as
u(x, t) = f(b1(t), b2(t), ..., bN (t), x), (2.49)
where f is an ansatz. Through the substitution of the ansatz f, ¯L = ¯Ldx, the Euler-
Lagrange equations for the generalized function parameters, p, are deﬁned as follows,
d
dt
∂ ¯L
∂ ˙p
−
∂ ¯L
∂p
= P
∂u
∂bi
dx. (2.50)

Since u and its complex conjugate u∗ are linearly independent and the Euler-Lagrangian
equations are related by
δL
δu∗
=
δL
δu
∗
= P, (2.51)
the modified Kantorovitch method [36] yields
d
dt
∂ ¯L
∂ ˙p
−
∂ ¯L
∂p
= 2Re P
∂u∗
∂p
dx. (2.52)
The Kantorovitch method has been successfully applied to bright soliton solutions for
the cubic-quintic Ginzburg-Landau equation [32] and to vortical solutions [33, 34].
2.3.3 Equivalence Proof
The following proof will illustrate the NCVA is equivalent to the PVA and modified KVA
method. Given a non-conservative NLS Eq. (2.25), where P is assumed complex, then
R is evaluated at the variational ansatz as
¯P =
∂ ¯R
∂¯u∗
− PL
. (2.53)
The formulations require that the variational parameters are real for the ansatz. There-
fore, we ensure real values for the parameters and the solution satisfies Eq. (2.53) such
that
¯R = ¯P(¯u±, ¯u∗
±, ¯u±,t, . . .) ¯u∗
− + c.c., (2.54)
where c.c. stands for complex conjugate. In order to be concise, we denote p for a single
variational parameter (i.e., an entry of p). All equations with the symbol p are a set of
coupled equations for each of the entires p in p.

Given a set of real-valued parameters p of the ansatz defined in the ± coordinate space
such that p+ = (p1 + p2)/2 and p− = (p1 − p2), then we show that the NCVA method is
equivalent to the PVA and KVA:
¯P =
+∞
−∞
¯Pdx =
∞
−∞
∂ ¯R
∂¯u∗
− PL
dx, (2.55)
projected into the ansatz such that
¯P =
∞
−∞
∂
∂p∗
−
¯P¯u∗
− +
∂
∂p∗
−
¯P∗
¯u−
PL
dx,
=
∞
−∞
¯P
∂¯u∗
−
∂p∗
−
+ ¯u∗
−
∂ ¯P
∂p∗
−
+ ¯P∗
∂¯u−
∂p∗
−
+ ¯u−
∂ ¯P∗
∂p∗
− PL
dx,
=
∞
−∞
¯P∗ ∂¯u
∂p∗
+ ¯P
∂¯u∗
∂p
dx, (2.56)
since [¯u∗
−]PL = [¯u−]PL = 0.
The non-conservative integral in the Euler-Lagrange equation derived in Eq. (2.56) is
equivalent to the perturbed variational approximation in Eq. (2.43), which is equivalent
to the modified Kantorovitch method [10]:
∞
−∞
¯P∗ ∂¯u
∂p∗
+ ¯P
∂¯u∗
∂p
dx = 2Re
∞
−∞
¯P
∂¯u∗
∂p
dx. (2.57)
Therefore, the perturbed and modified Kantorovitch variational approximation methods
are equivalent to the NCVA for complex partial differential equations derived from Hamil-
ton’s principle as an initial value problem with two sets of variables u1 and u2. The next
chapter will present examples of the non-conservative variational approximation applied
to dissipative dynamical systems.

Chapter 3
Applications of NCVA to the
Nonlinear Schrödinger Equation
In the following sections, three dynamical systems using the NLS are described in order to
illustrate the application of the NCVA. The numerical results are found through compar-
ison of the ODE dynamics with the direct forward integration of the NLS complex-valued
PDE. The first two dynamical systems for the non-conservative variational approxima-
tion comparison are the NLS with linear loss [Sec. 3.1] and with density dependent loss
[Sec. 3.2]. In Sec. 3.3 we present the third dynamical system for an exciton-polariton
condensate defined by the NLS with linear gain and density dependent loss.
3.1 NLS Equation with Linear Loss
For the first dynamical system example we use the focusing (g = +1) NLS equation with
a linear loss term of strength :
iut +
1
2
uxx + |u|2
u = −i u. (3.1)
29

Chapter 3. Applications of NCVA to the NLS 30
In a system without linear loss ( = 0), the NLS (3.1) has a well-known, bright, solitary
wave solution [31, 37] of the form
u(x, t) = η sech[η(x − vt)] exp[i(kx − wt)], (3.2)
where η is the amplitude and inverse spatial width of the soliton, k is the soliton
wavenumber, w is the soliton frequency and v ≡ ∂w/∂k = k is the soliton velocity.
The Lagrangian corresponding to the conservative problem ( = 0) is given by:
L =
i
2
u
∂u∗
∂t
− u∗ ∂u
∂t
+
1
2
∂u
∂x
2
−
1
2
|u|4
. (3.3)
The proposed bright soliton ansatz, based on the exact solution for the loss-less case, is
uA(x, t; p) = a sech[w(x − ξ)] exp[i(b(x − ξ)2
+ c(x − ξ) + φ)], (3.4)
where the vector of time-dependent parameters corresponds to p = (a, w, ξ, c, b, φ) with
arbitrary height a, inverse width w, center position ξ, speed c, chirp b, and phase φ.
3.1.1 Non-conservative Variational Approximation
In the NCVA framework, the ¯u1 and ¯u2 ansätze are deﬁned as in Eq. (3.4)
¯u1 = uA(x, t; p1), (3.5)
¯u2 = uA(x, t; p2), (3.6)
where the solutions have corresponding parameters p1 = (a1, w1, ξ1, c1, b1, φ1) and p2 =
(a2, w2, ξ2, c2, b2, φ2), respectively. According to the non-conservative variational method

the Lagrangian is LT = L1 − L2 + R where
¯L1 =
i
2
¯u1 ¯u∗
1,t − ¯u∗
1 ¯u1,t +
1
2
|¯u1,x|2
−
1
2
|¯u1|4
,
¯L2 =
i
2
¯u2 ¯u∗
2,t − ¯u∗
2 ¯u2,t +
1
2
|¯u2,x|2
−
1
2
|¯u2|4
,
¯R =
i
2
(¯u1 ¯u∗
1 − ¯u2 ¯u∗
2 + ¯u2 ¯u∗
1 − ¯u1¯u∗
2) −
i
2
(¯u1 ¯u∗
1 − ¯u2 ¯u∗
2 + ¯u∗
2¯u1 − ¯u∗
1¯u2),
= i (¯u2¯u∗
1 − ¯u1¯u∗
2).
Note, it is very important to properly construct ¯R for the soliton dynamics. Plugging
ansätze into ¯L1 and ¯L2 results in the following fully expanded terms for i = 1, and 2:
¯Li =a2
i sech2
(wi(x − ξi)) ˙ci(x − ξi) − ci
˙ξi + ˙bi(x − ξi)2
− 2bi(x − ξi) ˙ξi + ˙φi
+
1
2
a2
i w2
i sech2
(wi(x − ξi))tanh2
(wi(x − ξi))
+
1
2
a2
i (ci + 2bi(x − ξi))2
sech2
(wi(x − ξi)) −
1
2
a4
i sech4
(wi(x − ξi)). (3.7)
Next, we find the effective Lagrangian ¯L =
∞
−∞
¯LT dx =
∞
−∞
¯L1dx −
∞
−∞
¯L2dx +
∞
−∞
¯Rdx, for which ¯L1 and ¯L2 recover the same equations of motion as the ‘conser-
vative’ variational approximation.
After integration and simplification, the two conservative terms ¯L1 and ¯L2 of the effective
Lagrangian ¯L = ¯L1 − ¯L2 + ¯R are given by the following with i = 1, and 2:
¯Li =
∞
−∞
¯Lidx = 2
a2
i
˙φi
wi
+
a2
i c2
i
wi
− 2
a2
i ci
˙ξi
wi
−
2
3
a4
i
wi
+
1
3
a2
i wi +
π2
3
a2
i b2
i
w3
i
+
π2
6
a2
i
˙bi
w3
i
. (3.8)

For the non-conservative loss term, we take derivatives with respect to p− at the physical
limit (PL) then integrate:
¯R =
∞
−∞
¯Rdx = i
∞
−∞
∂
∂p−
(¯u2 ¯u∗
1 − ¯u1 ¯u∗
2)
PL
dx. (3.9)
The total effective Lagrangian is given by:
¯L = ¯L1 − ¯L2 + i
∞
−∞
∂
∂p−
(¯u2¯u∗
1 − ¯u1¯u∗
2)
PL
dx, (3.10)
where ¯L1 and ¯L2 are given by Eq. (3.8).
For all the parameters we make the following ± coordinate substitutions into the expres-
sion for the effective Lagrangian:
p1 =
(2p+ + p−)
2
, p2 =
(2p+ − p−)
2
, (3.11)
˙p1 =
(2 ˙p+ + ˙p−)
2
, ˙p2 =
(2 ˙p+ − ˙p−)
2
, (3.12)
with p1 ∈ {a1, b1, c1, d1, ω1, ξ1} and p2 ∈ {a2, b2, c2, d2, ω2, ξ2}. For brevity, we express
the effective Lagrangian in 1,2 coordinates. Below we shocase two terms in the effective
Lagrangian expression in order to illustrate the cumbersome expansion into ± coordi-
nates:
2
a2
1
˙φ1
w1
− 2
a2
2
˙φ2
w2
=2
(2a+ + a−)2
4
(2 ˙φ+ + ˙φ−)
2
2
(2w+ + w−)
−
(2a+ − a−)2
4
(2 ˙φ+ − ˙φ−)
2
2
(2w+ − w−)
. (3.13)
The ± coordinates lend to a more straightforward implementation of the physical limit
where the (+) variables are the physical variables and (−) variables are zero. Both the
1,2 coordinates and ± coordinates give equivalent final results; therefore, the choice of

coordinate system is arbitrary and selected purely for convenience. From the ¯L1 and ¯L2
parts we recover the standard soliton evolution equations, i.e. variational approximation
for the Hamiltonian, conservative, NLS equation with the following equations of motion
(ODEs):



˙a = −ab,
˙b = 2
π2 w4 − 2
π2 a2w2 − 2b2,
˙c = 0,
˙ξ = c,
˙w = −2bw,
˙φ = 5
6a2 − 1
3w2 + 1
2c2.
(3.14)
From the non-conservative term ¯R, we expand in the ± coordinate systems and ﬁnd the
integrals:
∞
−∞
∂ ¯R
∂a− PL
dx = 0,
∞
−∞
∂ ¯R
∂b− PL
dx = −
π2 a2
3w3
,
∞
−∞
∂ ¯R
∂c− PL
dx = 0,
∞
−∞
∂ ¯R
∂ξ− PL
dx =
4 a2c
w
,
∞
−∞
∂ ¯R
∂w− PL
dx = 0,
∞
−∞
∂ ¯R
∂φ− PL
dx = −
4 a2
w
.

The following equations are the modified Euler-Lagrange equations:
2
3
aw − 4
ac ˙ξ
w
+ 4
a ˙φ
w
+ 2
ac2
w
−
8
3
a3
w
+
π2
3
a˙b
w3
+
2π2
3
ab2
w3
= 0,
2π2
3
a2b
w3
−
π2
3
a˙a
w3
+
π2
2
a2 ˙w
w4
=
π2
3
a2
w3
,
a2
6w3
(−12w2 ˙ξ + 12w2
c) = 0,
4
ac˙a
w
− 2
a2c ˙w
w2
+ 2
a2 ˙c
w
= −4
a2c
w
,
a2
6w3
(8w3
− 24wc ˙ξ + 24w ˙φ + 12wc2
− 8a2
w)
−
a2
2w4
(2w4
− 12w2
c ˙ξ + 12w2 ˙φ + 6w2
c2
− 4a2
w2
+ π2 ˙b + 2π2
b2
) = 0,
−4
a˙a
w
+ 2
a2 ˙w
w2
= 4
a2
w
. (3.15)
The NCVA for the NLS with linear loss yield the following equations of motion by
simultaneously solving the modified Euler-Lagrange Eq. (3.15):



˙a = −a − ab,
˙b = 2
π2 w4 − 2
π2 a2w2 − 2b2,
˙c = 0,
˙ξ = c,
˙w = −2bw,
˙φ = 5
6a2 − 1
3w2 + 1
2c2,
(3.16)
corresponding to the same dynamics of the conservative case (3.14) and only differing
for the evolution of the amplitude with the added loss term − a.
3.1.2 Numerical Results: NLS with Linear Loss
Figures 3.1, 3.2, and 3.3 are numerical comparison between direct numerical integration
of the NLS with linear loss and the NCVA for = 0.01, 0.1, and 1, respectively. For
the equations of motion in the NCVA, Eq. (3.16), we used Matlab’s ode45 variable step

Runge-Kutta method to solve the ODEs numerically. For the modified NLS Eq. (3.1),
the PDE is numerically integrated for the focusing soliton using second-order central
differencing in space with periodic boundary conditions and fourth-order Runge Kutta
in time using the same initial ansatz as the ODEs. The top two panels in the figures
depict the spatial density profiles |u|2 at the initial time (t = 0) and at a time t = 1/
for the PDE and ODE solutions. The evolution of the NCVA ansatz parameters a, b, w,
φ, c, and ξ are plotted as functions of time. In order to compare the full NLS numerics
to the NCVA evolution, the numerical NLS solutions are projected onto the variational
ansatz uA at discrete time intervals using least-squares fitting (Matlab’s lsqcurvefit).
The time evolution of the projected parameters are compared (blue dots) in the bottom
six rows of panels in the figures. The initial conditions are a(0) = 1, b(0) = 0, w(0) = 1,
φ(0) = 0, c(0) = 0.1, and ξ(0) = −5 which gives an initial ansatz
u(x, t = 0) = sech(x + 5) exp[i(0.5(x + 5))].
For the full NLS with linear loss ( = 0.01) integration, the spatial domain is x ∈ [−50, 50]
with spatial step size dx = 0.05 and dt = 0.001 for temporal domain t ∈ [0, 100] in
Fig. 3.1. In the PDE integration for linear loss given = 0.1, the parameters are the
same except the temporal domain t ∈ [0, 10] (see Fig. 3.2) since the soliton solution
dissipates quickly. For = 1, the temporal domain is t ∈ [0, 2]. From the figures,
we observe the NCVA system of ODEs approximates very well the true numeric PDE
solution. Also, the NCVA system reflects the main dynamical features of the soliton
solution, mainly the decrease of amplitude ˙a = −2 a + ab, increase of the width w, and
constant speed (˙c = 0). For large dissipation = 1 in Fig. 3.3 we find a high fidelity of
the NCVA results, as expected in the case of linear dissipation.

0
0.5
1
|u|2
−10 −8 −6 −4 −2 0 2 4 6
0
0.5
x
|u|2
0.5
1
a
0
5
10
b
0
0.5
1
w
−0.1
0
0.1
φ
0
0.2
c
0 20 40 60 80 100
−5
0
5
t
ξ
Figure 3.1: Evolution of an NLS bright soliton under the presence of linear loss of
strength = 0.01. A bright soliton, as described by Eq. (3.4), is used as an initial
condition with the parameters: a(0) = w(0) = 1, c(0) = 0.1, ξ(0) = −5, and b(0) =
φ(0) = 0. The plots compare the NCVA approximations of Eq. (3.16) (red lines) with
the numerical NLS evolution of Eq. (3.1) (blue dots). The top subpanel depicts the
density |u|2
at the initial time (t = 0). The second subpanel depicts the density after
the system is evolved for a total time of t = 1/ . The bottom six subpanels detail
the evolution of the NCVA ansatz parameters a, b, c, ξ, w, and φ (red lines). For the
NLS evolution, the parameters are extracted by projecting the current solution into the
NCVA ansatz using least squares ﬁtting (blue dots).

0
0.5
1
|u|2
−10 −8 −6 −4 −2 0 2 4 6
0
0.5
x
|u|2
0.5
1
a
0
0.5
1
b
0.5
1
w
−0.1
0
0.1
0.2
φ
0
0.2
c
0 2 4 6 8 10
−5
−4.5
−4
t
ξ
strength = 0.1. The NCVA results are obtained from Eq. (3.16) (red lines) while the
full numerical solution is obtained from Eq. (3.1) (blue dots). Same initial conditions
and layout of panels as in the previous ﬁgure.

0
0.5
1
|u|2
−10 −8 −6 −4 −2 0 2 4 6
0
0.2
x
|u|2
0.5
1
a
−0.2
0
0.2
b
0.8
1
w
−0.1
0
0.1
0.2
φ
0
0.2
c
0 0.5 1 1.5 2
−5
−4.9
−4.8
t
ξ
strength = 1. The NCVA results are obtained from Eq. (3.16) (red lines) while the
and layout of panels as in previous ﬁgures. The system is evolved for a total time of
t = 2/ .

3.2 NLS Equation with Density Dependent Loss
In the second dynamical system example, we use the attractive NLS equation with a
density dependent (nonlinear) loss term of strength :
iut +
1
2
uxx + |u|2
u = −i |u|2
u. (3.17)
The Lagrangian corresponding to the conservative problem ( = 0) is the same as
Eq. (3.3). We again use the bright soliton ansatz Eq. (3.4) with a vector of time-
dependent ansatz parameters given by p = (a, w, ξ, c, b, φ).
In the NCVA framework, the ¯u1 and ¯u2 ansätze are deﬁned as in Eqs. (3.5) and (3.6).
According to the non-conservative variational method the Lagrangian is Lt = L1−L2+R
where
L1 =
i
2
¯u1¯u∗
1,t − ¯u∗
1 ¯u1,t +
1
2
|¯u1,x|2
−
1
2
|¯u1|4
, (3.18)
L2 =
i
2
¯u2¯u∗
2,t − ¯u∗
2 ¯u2,t +
1
2
|¯u2,x|2
−
1
2
|¯u2|4
, (3.19)
P¯u∗
− = i ¯u+ ¯u∗
+ ¯u+¯u∗
− = i
(¯u1 + ¯u2)
2
(¯u∗
1 + ¯u∗
2)
2
(¯u1 + ¯u2)
2
(¯u1 − ¯u2)∗
, (3.20)
R = i (¯u+ ¯u∗
+ ¯u+ ¯u∗
− − ¯u+¯u∗
+ ¯u− ¯u∗
+). (3.21)
Plugging the ansätze into L1 and L2 and integrating gives ¯L1 and ¯L2 of the same form
as Eq. (3.8). For the non-conservative terms, we take derivatives with respect to p− at

the physical limit (PL) and then integrate:
¯R =
∞
−∞
¯Rdx
=
i
4
∂
∂p−
|u1|2
(u2u∗
1 − u∗
2u1) + |u2|2
(u2u∗
1 − u∗
2u1) + u2u2u∗
1u∗
1 − u1u1u∗
2u∗
2
PL
dx.
(3.22)
Therefore, the total effective Lagrangian, ¯L = ¯L1 − ¯L2 + ¯R is given by:
¯L =2
a2
1
˙φ1
w1
+
a2
1c2
1
w1
− 2
a2
1c1
˙ξ1
w1
−
2
3
a4
1
w1
+
1
3
a2
1w1 +
π2
3
a2
1b2
1
w3
1
+
π2
6
a2
1
˙b1
w3
1
− 2
a2
2
˙φ2
w2
−
a2
2c2
2
w2
+ 2
a2
2c2
˙ξ2
w2
+
2
3
a4
2
w2
−
1
3
a2
2w2 −
π2
3
a2
2b2
2
w3
2
−
π2
6
a2
2
˙b2
w3
2
+
i
4
∂
∂p−
|u1|2
(u2u∗
1 − u∗
2u1) + |u2|2
(u2u∗
1 − u∗
2u1) + u2u2u∗
1u∗
1 − u1u1u∗
2u∗
2
PL
dx.
(3.23)
For all the parameters, we substitute the ± coordinates into the expression for the total
effective Lagrangian ¯L. From the ¯L1 and ¯L2 conservative terms we recover the standard
soliton evolution equations [see Eq. (3.14)], so we just need to obtain the non-conservative
ones. From the non-conservative term ¯R, we expand in the ± coordinate systems and
find the integrals:
∞
−∞
∂ ¯R
∂a− PL
dx = 0,
∞
−∞
∂ ¯R
∂b− PL
dx = −
2π2
9
a4
w3
+
4
3
a4
w3
,
∞
−∞
∂ ¯R
∂c− PL
dx = 0,
∞
−∞
∂ ¯R
∂ξ− PL
dx =
8
3
a4c
w
,
∞
−∞
∂ ¯R
∂w− PL
dx = 0,
∞
−∞
∂ ¯R
∂φ− PL
dx = −
8
3
a4
w
.

Combining the conservative and non-conservative contributions, the equations of motion
from the NCVA for the NLS with density dependent loss are the following:



˙a = −2
3 a3 − ab − 2
π2 a3,
˙b = 2
π2 w4 − 2
π2 a2w2 − 2b2,
˙c = 0,
˙ξ = c,
˙w = −2bw − 4
π2 a2w,
˙φ = 5
6a2 − 1
3w2 + 1
2c2,
(3.24)
which correspond to the same dynamics as the conservative case (3.14) with the added
nonlinear loss terms −(2/3+2/π2) a3 for the evolution of the amplitude and −4/π2 a2w
for the evolution of the inverse width.
3.2.2 Numerical Results: NLS with Density Dependent Loss
Figure 3.4 depicts a numerical comparison between full integration of the NLS with den-
sity dependent loss and the NCVA [Eq. (3.24)] for = 0.1. The same numerical approach,
ansatz and initial conditions were taken as in Sect. 3.1.2. For = 0.1, the numerical in-
tegration temporal domain was changed to t ∈ [0, 10], with the same discretization dx =
0.05 and dt = 0.001.
Similar to linear loss, the density dependent loss NCVAs are in good agreement to the
full NLS dynamics even in the presence of a nonlinear loss. In general, the dynamics
of the PDE solution agrees well with the parameters in the coupled ODEs that ﬁt to
the ansatz. The speed c is constant in time and ξ(t) = c t as a linear increase in the
test cases. The chirp parameter b has more complex dynamics in agreement between
the PDE and ODEs. The main discrepancy between the PDE and the ODE is that the

dissipation of the height a, described by ˙a = −4
3 a3 in the NCVA, follows a power law
rather than an exponential as expected.
0
0.5
1
|u|2
−10 −8 −6 −4 −2 0 2 4 6
0
0.5
x
|u|2
0.5
1
a
0
2
4
b
0.5
1
w
−0.1
0
0.1
φ
0
0.2
c
0 2 4 6 8 10
−5
−4.5
−4
t
ξ
Figure 3.4: Evolution of an NLS bright soliton under the presence of nonlinear loss of
strength = 0.1. The NCVA results are obtained from Eq. (3.24) (red lines) while the
and layout of panels as in previous ﬁgures.

3.3 Exciton-Polariton Condensate - NLS with Linear Gain
and Density Dependent Loss
The third dynamical system is based on exciton-polariton condensates. In exciton-
polariton condensates, the condensing “entities” are excitons, namely bound electron-
hole pairs. These excitons strongly couple with light when confined in quantum wells
placed in high-finesse microcavities, forming exciton-photon mixed quasi-particles known
as polaritons [38]. These condensates exist at finite temperatures, even near room tem-
perature, which means the the polaritons can only exist for a few picoseconds in the
cavity before they decay into photons. The finite lifetime of the polaritons precludes the
system from reaching thermal equilibrium, in fact, the system is a genuinely far-from-
equilibrium condensate which requires an external pump from a reservoir of excitons to
counter the loss of polaritons.
Exciton-polariton condensates offer numerous key features of the superfluid character
including: the flow without scattering (analog of the flow without friction) [39], the
existence of vortices [40] and their interactions [41, 42], the collective superfluid dynamics
[43], as well as remarkable applications such as spin switches [44], and light emitting
diodes [45] operating even near room temperatures.
There is a wide variety of different types of models for polaritons to describe the associate
pumping and damping mechanisms. One of these models, proposed in Refs. [46–48],
suggests the use of a single NLS-type equation for the polariton condensate wavefunction
which incorporates a gain-loss mechanism for the decay of polaritons to photons and
pumping of excitons from an external reservoir. Specifically, this model, based on a
repulsive (g = −1) NLS equation with linear gain (iχ(x)u) and density dependent loss

(−iσ|u|2u) terms, can be written in the following non-dimensional form [46, 49]:
iut +
1
2
uxx − |u|2
u − V (x)u = i χ(x) − σ|u|2
u, (3.25)
where σ is the strength of the density dependent loss and χ is considered the localized,
spatially dependent gain given by
χ(x) = α exp −
x2
2β2
, (3.26)
describing a laser pump of amplitude α and width β. The potential V is a general
harmonic potential of strength Ω:
V (x) =
1
2
Ω2
x2
. (3.27)
For the application of variational approximations, we define the Gaussian ansatz
uA(x, t; p) = ae− x2
2w2 ei(bx2+φ), (3.28)
where the ansatz parameter pi = (ai, wi, bi, φi) for i = 1 and 2 represent, respectively,
the amplitude, width, chirp, and phase of the ansatz solution. The departure from a
sech-type ansatz is based on two reasons: (i) the Hamiltonian NLS has a Gaussian-type
solution for a low density condensate, and (ii) given a Gaussian-type gain, this ansatz
allows us to obtain explicit ODEs through the NCVA.
In the NCVA, we use two ansätze ¯u1 = uA(x, t; p1) and ¯u2 = uA(x, t; p2) as defined by
the Gaussian profile of Eq. (3.28). The selection of a Gaussian profile is to characterize

the breathing motion of a ground state inside the trap, rather than to characterize the
translational dynamics of the wavefunction. In order to find translational modes, we
would require a different ansatz with an added degree of freedom corresponding to a
center position parameter of the wavefunction.
According to the NCVA method, the Lagrangian is ¯L = ¯L1 − ¯L2 + ¯R, where the conser-
vative terms have the Lagrangian densities for i = 1, 2 given by
¯Li = i
2 ¯ui ¯u∗
i,t − ¯u∗
i ¯ui,t + 1
2|¯ui,x|2 + 1
2|¯ui|4 + V (x)|¯ui|2, (3.29)
and ¯R has the same type of density dependent loss [see Section 3.2.1] and a linear
gain (equivalent to the negative of linear loss) [see Section 3.1.1] shown in the previous
examples. The non-conservative terms are defined as follows:
¯R = ¯Pu∗
− + ¯P∗
u−, (3.30)
= −iχ(x) (¯u2 ¯u∗
1 − ¯u1 ¯u∗
2) (3.31)
+iσ[|¯u1|2
(¯u2 ¯u∗
1 − ¯u∗
2 ¯u1) + |¯u2|2
(¯u2 ¯u∗
1 − ¯u∗
2 ¯u1) + ¯u2 ¯u2¯u∗
1¯u∗
1 − ¯u1¯u1 ¯u∗
2 ¯u2
2].
For all the parameters we made the substitutions of ± coordinates into the expression
for the total Lagrangian and from the ¯L1 and ¯L2 parts we recover the conservative
Euler-Lagrange equations for a Gaussian ansätz with four-parameters. From the non-
conservative term ¯R, we expand in the ± coordinate systems and find the integrals,
which are combinations of the integrals for linear gain and density dependent loss [see

Sections 3.1.1 and 3.2.1]:
∞
−∞
∂ ¯R
∂a− PL
dx = 0,
∞
−∞
∂ ¯R
∂b− PL
dx = −
√
2π
4
σa4
w3
+
2
√
2παβ3a2w3
(w2 + 2β2)3/2
,
∞
−∞
∂ ¯R
∂w− PL
dx = 0,
∞
−∞
∂ ¯R
∂φ− PL
dx = −
√
2πσa4
w +
2
√
2παβa2w
w2 + 2β2
.
Finally, combining non-conservative and conservative terms, the NCVA yields the ap-
proximate equations of (breathing) motion for the exciton-polariton ground-state con-
densate of the form:



˙a =
√
2
8 σa3 − 3
√
2
4
σa3w2
w2+2β2 + 3
√
2
2
αβaw2
(w2+2β2)3/2 − 3
√
2
2
σβ2a3
w2+2β2 + 2
√
2αβ3a
(w2+2β2)3/2 − ab,
˙b =
√
2
4
a2
w2 + 1
2w4 − 1
2Ω2 − 2b2,
˙w = −5
√
2
4 σa2w + 3
√
2
2
σa2w3
w2+2β2 −
√
2αβw3
(w2+2β2)3/2 + 3
√
2σβ2a2w
w2+2β2 + 2wb,
˙φ = −5
√
2
8 a2 − 1
2w2 .
(3.32)
3.3.2 Numerical Results: Exciton-Polariton Condensate
Figures 3.5 and 3.6 depict the numerical comparison between direct integration of the
NLS with linear gain and density dependent loss and the NCVA for the exciton-polariton
condensate example using initial conditions below and above the equilibrium for the NLS,
respectively. In order to simulate solutions below and above equilibrium, the initial
solution amplitudes are perturbed below and above the theoretical equilibrium values.
In the exciton-polariton example we use coeﬃcients σ = 0.37, α = 2, β = 2, and Ω =
√
2
based on Ref. [49] to guarantee that the solution state with no excitations (bright soliton)

is stable. The initial condition is designed below and above equilibrium amplitude by first
computing the steady state of the NLS (3.25) and projecting (with least-squares fitting)
into the Gaussian ansatz (3.28) gives the equilibrium amplitude parameter ae ≡ 2.6431.
The other initial parameters are width w(0) = 1.5583, chirp b(0) = −0.1563, and phase
φ(0) = 0.2415. Figure 3.5 is simulated with an initial amplitude below the equilibrium
a(0) = 0.6608 = ae/4, i.e., four times smaller than the equilibrium solution. Figure 3.6
is simulated with an initial amplitude above equilibrium a(0) = 7.9292 = 3ae, i.e., three
times larger than the equilibrium solution.
The equations of motion in the NCVA Eq. (3.32) are numerically solved and the NLS is
fully integrated by the same methods described in Section 3.1.2. To compare the NCVA
evolution of the parameters to the NLS numerics, the integrated solutions are projected
onto the variational ansatz uA at discrete time intervals (blue dots). For the numerics
the spatial domain is x ∈ [−40, 40] with spatial step size dx = 0.05 and dt = 0.001 over
t ∈ [0, 50]. Similar to the previous figures, the top two panels are the spatial profiles of
the densities |u|2 for the NLS and NCVA solutions at the initial time (t = 0) and the final
time t = 50, and the bottom four panels depict the dynamics of the ansatz parameters.
The NCVA system for the exciton-polariton condensate and the NLS dynamics are in
very good qualitative agreement and good quantitative agreement as observed in Figs. 3.5
and 3.6. The discrepancies in the quantitative agreement are caused by the choice
of ansatz. The original NLS solution is well approximated with a Gaussian only for
small atom number. As the atom number increases, the atomic density approaches the
Thomas-Fermi limit (inverted parabola) profile which is apparent in the density |u|2 dis-
crepancy between the converged full NLS and NCVA state in the second subpanel at
t = 50 in Figs. 3.5 and 3.6. For the breathing motion of a ground state inside the trap,
the Gaussian ansatz leads to dynamics of the NCVA which converge (in an oscillatory

manner) to the stable solution in agreement with the dynamics of the NLS convergence
to the stable equilibrium solution. As stated previously, to more accurately capture the
dynamics of the NLS with the NCVA (i.e. translational dynamics of the wavefunction)
one needs to use a better suited ansatz such as the q-Gaussian proposed in Ref. [50].
However, increasing the number of variational parameters is at the expense of more
complicated equations of motion.
0
5
|u|2
−5 0 5
0
5
x
|u|2
2.5
a
−0.5
0
0.5
b
0.5
1
1.5
2
w
0 5 10 15 20
−100
0
t
φ
Figure 3.5: Evolution of the ground state of Eq. (3.25) starting below equilibrium
in the presence of a linear spatially dependent gain (3.26) with α = 2 and β = 2, and
density dependent loss of strength σ = 0.37, as well as a harmonic potential (3.27) of
strength Ω =
√
2. To craft initial conditions with amplitudes below the equilibrium
amplitudes we first computed the steady state of the NLS (3.25) which, after projection,
using least-squares fitting, into the Gaussian ansatz (3.28) yields the following initial
parameters: amplitude: a(0) = 0.6608 = ae/4 (four times smaller than the equilibrium
solution), width: w(0) = 1.5583, chirp: b(0) = −0.1563, and phase: φ(0) = 0.2415.
Depicted are the comparison of the NCVA approximation of Eq. (3.32) (red lines) with
the full, numerical, NLS evolution of Eq. (3.25) (blue dots). The top two panels depict
the density |u|2
at the initial time (top subpanel) and at time t = 50 (second subpanel).
The bottom four subpanels depict the evolution of the NCVA ansatz parameters a, b,
w, and φ. For the full NLS evolution the parameters are extracted by projecting the
current solution into the NCVA ansatz using least-squares fitting.

0
50
|u|2
−5 0 5
0
50
x
|u|2
2
4
6
8
a
−1
−0.5
0
0.5
b
0.5
1.5
2.5
w
0 5 10 15 20
−100
0
t
φ
Figure 3.6: Evolution of the ground state of Eq. (3.25) with the same coeﬃcients
as Fig. 3.5 starting above equilibrium a(0) = 7.9292 = 3ae (three times larger than
the equilibrium solution), width: w(0) = 1.5583, chirp: b(0) = −0.1563, and phase:
φ(0) = 0.2415. The layout of the panels is the same as in the previous Fig. 3.5.

Chapter 4
Spontaneous Symmetry Breaking of
the Lugiato-Lefever Equation
The following chapter is based on Ref. [51] coauthored with Ricardo Carretero-González,
Panayotis G. Kevrekidis, and Mariana Haragus. The aim of the chapter is to further
extend the NCVA approach to a variant of the NLS equation: the mean-ﬁeld Lugiato-
Lefever (LL) model [52, 53]. Experimentally [16], temporal spontaneous symmetry break-
ing (SSB) is found in passive Kerr resonators described by the LL equation. We examine
this SSB-induced instability interval in the the passive Kerr resonator modeled by the
Lugiato-Lefever equation by means of the NCVA [54] described in Chapter 2, and further
through a center manifold reduction [55] enabling the analysis of the dominant associated
eigenmodes (responsible for determining the spectral stability of the system). It is rele-
vant to mention at this point that a thorough bifurcation analysis for a LL equation in
the case of constant external pumping was recently carried out in Ref. [56], showing quite
complex bifurcation scenarios in both the anomalous and normal dispersion regimes. In
the NCVA context, our aim is to apply a variational method based on well-informed
ansätze in the corresponding Lagrangian of the system. The ansätze reduce the com-
plexity of the original inﬁnite-dimensional problem to a few degrees of freedom capturing
51

Chapter 4. SSB of the LL Equation 52
the principal, static and dynamic characteristics of the system. This method attempts
to project the infinite-dimensional dynamics of the Lugiato-Lefever equation into a low-
dimensional dynamical system that qualitatively and, to some extent, quantitatively
captures SSB bifurcations an the solutions emanating from it. Based on Galley’s [14]
approach to extend variational approximation method to open, non-conservative dissi-
pative systems we developed the NCVA, which in turn was generalized to dissipative
(containing gain and loss) NLS-type systems in Ref. [54]. This was inspired by the work
of Ref. [57] on the extension of Galley’s formalism to PT-symmetric variants of field
theories. It is this variant of the NCVA that we will explore in the present setting.
The chapter is organized as follows. In Sec. 4.1 we introduce SSB and setup the LL
model. In Sec. 4.1.1 we identify the equilibria and study their stability by means of a
spectral analysis of the linearization problem; this is a perspective that was absent in the
original work of Ref. [16] and which, we argue, provides a more systematic insight into
the stability (and the potential instabilities) of the system. In doing so, we recover the
forward and reverse pitchfork bifurcations (i.e., a pitchfork loop) observed in Ref. [16]
as well as identify a Hopf bifurcation for larger pump power giving rise to asymmetric,
stable, periodic solutions; the latter is an important feature of dynamical interest in its
own right. Section 4.1.3 is devoted to the application of the NCVA to capture the SSB
bifurcation for physically relevant parameters values of the system as in Ref. [16]. In
Sec. 4.1.4 we complement our understanding of the pitchfork loop bifurcation by giving
the local bifurcation analysis which is effective towards qualitatively and quantitatively
describing the emerging asymmetric solutions close to the pitchfork bifurcation points.
Finally, in Sec. 4.1.5 we summarize our findings.

4.1 Spontaneous Time-Reversal Symmetry Breaking in
Synchronously-Pumped Passive Kerr Resonators
Spontaneous symmetry breaking (SSB) is the basis for many phase transitions and ac-
count for effects including ferromagnetism, superconductivity, and convection cells [58,
59]. SSB has been widely observed in nonlinear optics and is at the heart of numerous
fundamental phenomena including, but not limited to, asymmetric dynamics in coupled
mode models [60], optical wave guide arrays [61], coupled nonlinear micro-cavities [62],
photonic lattices [63]. For a detailed exposition of numerous recent directions within the
subject from the perspective of nonlinear phenomena, see Ref. [64]. SSB is not restricted
to Hamiltonian (conservative) systems. For instance, over the past few years, it has
also played a prominent role in the context of parity-time, so-called PT, symmetric sys-
tems [65, 66] bearing a balanced interplay between gain and loss. There, it is responsible
for the emergence of novel “ghost” states both in the case of dimers [67], but also in that
of continuous media [68], where they can be responsibility for the destabilization and
bifurcations associated with solitary waves and vortices.
A remarkable example of SSB in a dissipative system was observed by Xu and Coen in
Ref. [16] where a system using an optical fiber ring cavity composed of a synchronously-
pumped passive optical resonator filled with a Kerr nonlinear material was experimentally
explored. This system exhibits a temporal SSB instability in which the discrete time-
reversal symmetry is broken and symmetric states become unstable in favor of stable
asymmetric states. It is the purpose of the present chapter to complement the exper-
imental and numerical analysis of Ref. [16] by putting forward a thorough analytical
(and partially numerically assisted) understanding of the origin and manifestation of
SSB bifurcations in this system.

We consider, as in Ref. [16], a model for a passive Kerr resonator in an optical fiber ring
cavity described by a single PDE, resulting from an averaging procedure, of the NLS
equation-type, known as the mean-field Lugiato-Lefever (LL) model [52, 53]. The LL
equation, taking into account gain and loss in the system, can be cast, in non-dimensional
form, as [16, 69, 70]:
∂E(z, τ)
∂z
= −1 + i(|E|2
− ∆) − iη
∂2
∂τ2
E + S(τ), (4.1)
where z is the slow evolution variable of the intracavity field E over successive normal-
ized cavity round-trips and τ describes the temporal variable in the dependence of the
intracavity pulse envelope. The terms in the right-hand-side of Eq. (4.1) correspond,
respectively, to cavity losses (−E), Kerr nonlinearity (i |E|2
E), cavity phase detuning
(−i∆E), chromatic dispersion (−iη ∂2
∂τ2 E), and external pumping (S(τ)). Within this
non-dimensional form [69, 70], the cavity phase detuning corresponds to ∆ = δ0α, where
α is half the fraction of power lost per round-trip and the cavity finesse is F = π/α, and
δ0 = 2mπ −φ0 where φ0 is the overall cavity round-trip phase shift and m is the order of
the closest cavity resonance. The sign of the group-velocity dispersion coefficient of the
fiber is η which is taken as η = −1 for our analysis with self-focusing nonlinearity. The
field envelope of the external pump pulses, S(τ), is modeled by a symmetric chirp-free
Gaussian pulse given by S(τ) =
√
X exp −(τ/T0)2 , with T0 = 2.3 as in the experiments
of Ref. [16].
For the SSB instability of the passive Kerr cavity, the pump pulse field profile is tem-
porally symmetric, S(τ) = S(−τ), and the model is symmetric under a time reversal
transformation, τ → −τ, yet it admits asymmetric solutions, as described in Ref. [16].
The associated pitchfork bifurcation illustrates that at low pump peak power X, the

solutions are symmetric in time; however, above a certain pump peak power thresh-
old the symmetric states become unstable while stable asymmetric states emerge. The
particular experimental parameters of Ref. [16] generate, as X is increased further, a
reverse pitchfork as well, in which the asymmetric states collide and disappear while the
symmetric state recovers its stability.
4.1.1 The Full Lugiato-Lefever Model: Equilibria, Stability and Bifur-
cations
In this section, we follow the various equilibria of Eq. (4.2) as the peak pump power, X,
is varied and determine their stability. Let us recast Eq. (4.1) into the simpler form
iuz + uττ + (|u|2
− ∆)u = −iu + iS(τ), (4.2)
which corresponds to the NLS with additional non-conservative terms (namely the terms
in the right-hand side). In what follows, we identify stationary solutions, u(z, τ) = u0(τ)
of Eq. (4.2) by numerically solving the steady-state equation
u0,ττ + (|u0|2
− ∆)u0 = −iu0 + iS(τ). (4.3)
It is relevant to mention that since the forcing (pump) term in Eq. (4.1) is independent
of the ﬁeld’s wavefunction, it is necessary for the steady state to be independent of z
(i.e., here the detuning parameter ∆ plays the role of the frequency). It is also worth
mentioning that the steady state is, in general, complex which, as we will see below, is
crucial for the steady state to sustain itself through a stationary ﬂow from the gain to
the loss portions of the solution.

Let us now consider the stability of the steady state u0 by means of a spectral stability
analysis. Speciﬁcally, small perturbations of order O( ), with 0 < 1, to the stationary
solutions are introduced in the form:
u(z, τ) = u0(τ) + [a(τ)eλz
+ b∗
(τ)eλ∗z
],
and substituted into Eq. (4.2). Then, the ensuing linearized equations are solved to O( ),
leading to the eigenvalue problem:
iλ




a(z)
b(z)



 =




M1 M2
−M∗
2 −M∗
1








a(z)
b(z)



 , (4.4)
for the eigenvalues λ and associated eigenvector ξ = (a(z), b(z))T, where (·)∗ denotes
complex conjugation and M1 and M2 are the following operators:
M1 = −∂2
τ − 2|u0|2
+ (∆ − i),
M2 = −u2
0. (4.5)
The stationary solutions are linearly unstable provided Re(λ) > 0. When unstable,
the dynamics of the respective instabilities can be monitored through direct numerical
simulations of Eq. (4.2). It is relevant to mention at this point that a thorough (Turing)
stability analysis for frequency combs in both the anomalous and normal dispersion
regimes was recently carried out in Ref. [56].
Figure 4.1 depicts the linearization spectrum for the symmetric stationary solution [see
(red) dashed line in panels (c) and (d) of Fig. 4.2] as a function of the pump peak power.
The spectrum in Fig. 4.1 evidences the existence of two unstable branches: (i) a pitchfork
bifurcation loop containing a forward pitchfork bifurcation, see point P1 at X ≈ 4.6, and

−2
−1
0
Re(λ)
P1 P2
H
0 2 4 6 8 10 12 14 16
−2
−1
0
1
2
X
Im(λ)
P1 P2
H
H
Figure 4.1: Linearization spectrum for the symmetric and asymmetric steady state
solutions of the Lugiato-Lefever equation (4.2) as the pump power X is varied for
∆ = 0.92 and T0 = 2.3. The top and bottom panels depict, respectively, the real
and imaginary parts of the eigenvalues. Stable symmetric solutions bearing Re(λ) < 0
are depicted by small (red) dots in the top panel while unstable symmetric solutions
are depicted with thick solid lines. The thick (green) solid line between the points
P1 and P2 represents the unstable solutions through a forward (P1) and reverse (P2)
pitchfork bifurcations. The thin (black) curve between the points P1 and P2 corresponds
to the stable asymmetric solution branches created through the pitchfork bifurcation.
(The small black dot next to the point P1 is the stable eigenvalue used for the slope
computation in Fig. 4.10.) The thick (magenta) solid line to the right of the Hopf
bifurcation point H indicates the onset of instability for the symmetric state and the
existence of an asymmetric periodic solution.

a reverse pitchfork bifurcation, see point P2 at X ≈ 10.6, and (ii) a Hopf bifurcation,
see point H at X ≈ 15.1. The pitchfork bifurcation, see thick (green) line between the
points P1 and P2 in Fig. 4.1, is responsible, as the pump power is increased, for the loss
of stability of the symmetric state towards a pair of asymmetric states (one to the left
and one to the right) at P1. As the pump power is increased, a reverse pitchfork at P2
is responsible for the collision (and annihilation) of the two asymmetric states towards
the symmetric state that recovers its stability. A sample of the dynamic destabilization
of the (unstable) symmetric state for a pump strength X = 8, namely between the
two pitchfork points, is depicted in Fig. 4.2(a). As the ﬁgure shows, the symmetric
state [see dashed (red) line in Fig. 4.2(c)] destabilizes towards the stable, asymmetric
state [see solid (blue) line in Fig. 4.2(c)]. On the other hand, the instability due to the
Hopf bifurcation branch, see the thick (magenta) line emanating from the point H in
Fig. 4.1, is responsible for the instability of the symmetric state towards a periodic (in z)
solution. A sample of the evolution for the symmetric state towards the stable periodic
solution is depicted in Fig. 4.2(b). The periodic solution contains three “humps” in its
τ dependence: a central one performing left-to-right oscillations while the side “humps”
oscillate alternatively up-and-down. Snapshots for the asymmetric states when the side
“humps” have the largest magnitude are depicted in panel (d) corresponding to the times
depicted by horizontal white lines in panel (b).
It is important to mention that, due to the cavity loss term (−iu), the real part of the
spectrum is symmetric with respect to Re(λ) = −1 (see Sec. 4.1.4 for details). Therefore,
tuning the cavity loss parameter is crucial to the existence of the SSB bifurcation as
higher values of this parameter shift the real part of the spectrum down precluding the
possibility of eigenvalues crossing the origin and leading to such bifurcations. By the
same token, reducing the value of the cavity loss parameter will induce more eigenvalues

−4 −2 0 2 4
0
2
4
6
τ
|u|2
(c)
−4 −2 0 2 4
τ
(d)
Figure 4.2: (a), (b) Examples for the density evolution of unstable symmetric states
and (c), (d) snapshots for the corresponding states. (a) Evolution of unstable symmetric
state for X = 8 between the two pitchfork bifurcations P1 and P2 depicted in Fig. 4.1.
The initial symmetric state, see dashed (red) line in panel (c) evolves towards the
asymmetric steady state depicted in solid (blue) in panel (c). (b) Evolution of unstable
symmetric state towards a periodic breathing solution for X = 16 (i.e., to the right of
the Hopf bifurcation point H in Fig. 4.1). The initial symmetric state [dashed (red)
line] and two snapshots of the density for the periodic solution [solid (blue and light
blue) lines] separated by half a period, at the times corresponding to the white vertical
lines in panel (b), are depicted in panel (d).
to cross the origin and thus lead to richer and more complicated bifurcation scenarios.
A detailed analysis of the bifurcations as the cavity loss parameter is varied is outside of
the scope of the present dissertation work and will be studied in a future work.

4.1.2 Numerical Convergence of the Stability Spectrum
The purpose of this subsection is to briefly discuss the numerics used for analyzing the
frequency spectrum (see Fig. 4.1) which are dependent on the discretization of fast-time
h = dτ, the domain length L, and the boundary condition. The eigenvalue problem
Eq. (4.4) can be recast as iλξ = Mξ. This is numerically solved using second-order
central differencing in one dimension for the Laplacian given by
2
uj =
∂2u
∂τ2
j
≈
uj+1 − 2uj + uj−1
h2
, (4.6)
and when implemented into the M1 matrix, yields a matrix A which is tridiagonal except
from the matrix elements corresponding to the boundary conditions. Since boundary
conditions of the fast-time differencing in a PDE like the LL model have the potential
to alter the stability, it is necessary to compare the stability for each of the specific
boundary conditions we would like to use. For this discussion we limit ourselves to three
boundary conditions [71]: Dirichlet, Neumann, and periodic.
In our analysis, we consider a uniform grid with spacing h on the interval [−L/2, L/2].
Dirichlet boundary conditions specify a fixed constant value along the boundary of the
domain. For our Dirichlet boundary conditions we define u(−L/2) = u(L/2) = 0 given
that the solution has the form of a bright soliton. Using such a formulation the Laplacian

matrix with these Dirichlet boundary conditions becomes
A =
1
h2
















−2 1 0 · · · 0
1 −2 1 · · · 0
0
...
...
... 0
0 0 1 −2 1
0 · · · 0 1 −2
















. (4.7)
Neumann boundary conditions specify the value of the derivative of a solution at the
boundary of the domain. We use a no flux boundary in which ∂τ u(−L/2) = ∂τ u(L/2) = 0
such that the Laplacian matrix becomes
A =
1
h2
















−2 2 0 · · · 0
1 −2 1 · · · 0
0
...
...
... 0
0 0 1 −2 1
0 · · · 0 2 −2
















. (4.8)
The periodic boundary condition is defined as u(−L/2) = u(L/2) and is justified in
the scenario of a ring cavity. The discretized Laplacian matrix for periodic boundary
conditions is
A =
1
h2
















−2 1 0 · · · 1
1 −2 1 · · · 0
0
...
...
... 0
0 0 1 −2 1
1 · · · 0 1 −2
















. (4.9)

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