We study the dynamics of a soliton in the generalized NLS with a small external potential. We prove that there exists an effective mechanical system describing the dynamics of the soliton and that, for any positive integer r, the energy of such a mechanical system is almost conserved up to times of order ϵ^{−r}. In the rotational invariant case we deduce that the true orbit of the soliton remains close to the mechanical one up to times of order ϵ^{−r}.

1. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Freezing of energy of a soliton in an external
potential
A. Maspero∗
joint work with D. Bambusi
∗
Laboratoire de Math´ematiques Jean Leray - Universit´e de Nantes, Nantes
November 27, 2015, Nantes

2. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Outline
1 The problem: Dynamics of soliton in NLS with external potential
2 Main Theorem
3 Scheme of the proof

3. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
The equation
NLS with small external potential V
i∂tψ = −∆ψ − β (|ψ|2
)ψ + V (x)ψ , x ∈ R3
,
V Schwartz class
β focusing nonlinearity, β ∈ C∞
(R, R)
β(k)
(u) ≤ Ck u
1+p−k
, β (0) = 0 p < 2/3 ,
Under this assumptions: global unique solution for initial data in H1
Problem: effective dynamics of solitary waves solutions

4. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Case = 0
Look for moving solitary waves:
ψ(x, t) = eiγ(t)
eip(t)·(x−q(t))/m(t)
ηm(t)(x − q(t))
where ηm is the ground state of NLS with mass m, and p, q ∈ R3
, γ ∈ R
time-dependent parameter which fulfill



˙p = 0
˙q = p
m
˙m = 0
˙γ = E(m) + |p|2
4m
(1)
Invariant 8-dimensional symplectic manifold of solitary waves:
T =
p,q∈R3
γ∈R,m∈I
eiγ
eip·(x−q)/m
ηm(x − q)
Solitary wave travels through space with constant velocity
hamiltonian equations of free mechanical particle

5. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Case = 0
T is not longer invariant. ”Symplectic Decomposition”:
ψ = eiγ
eip·(x−q)/m
ηm(· − q) + φ(x, t)
with time-dependent parameter p(t), q(t), γ(t), m(t) fulfilling



˙p = − V eff
(q) + O( 2
)
˙q = p
m + O( 2
)
˙m = O( 2
)
˙γ = E(m) + |p|2
4m − V eff
(q) + O( 2
)
(2)
with V eff
(q) = R3 V (x + q) η2
m dx
first two equations are hamiltonian equations of mechanical particle interacting
with external field
Hmech(p, q) =
|p|2
2m
+ V eff
(q)
Solitary waves moves ”like a mechanical particle in an external potential”
for which interval of time is the argument rigorous?

6. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Case = 0
T is not longer invariant. ”Symplectic Decomposition”:
ψ = eiγ
eip·(x−q)/m
ηm(· − q) + φ(x, t)
with time-dependent parameter p(t), q(t), γ(t), m(t) fulfilling



˙p = − V eff
(q) + O( 2
)
˙q = p
m + O( 2
)
˙m = O( 2
)
˙γ = E(m) + |p|2
4m − V eff
(q) + O( 2
)
(2)
with V eff
(q) = R3 V (x + q) η2
m dx
first two equations are hamiltonian equations of mechanical particle interacting
with external field
Hmech(p, q) =
|p|2
2m
+ V eff
(q)
Solitary waves moves ”like a mechanical particle in an external potential”
for which interval of time is the argument rigorous?

7. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Previous results
Fr¨ohlich-Gustafson-Jonsson-Sigal ’04, ’06, Holmer-Zworski ’08:
dist (p(t), q(t)), (pmech(t), qmech(t)) 1/2
for times |t| ≤ T −3/2
Question: can we do better? Answer: NO!

8. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Previous results
Fr¨ohlich-Gustafson-Jonsson-Sigal ’04, ’06, Holmer-Zworski ’08:
dist (p(t), q(t)), (pmech(t), qmech(t)) 1/2
for times |t| ≤ T −3/2
Question: can we do better? Answer: NO!

9. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Previous results
Fr¨ohlich-Gustafson-Jonsson-Sigal ’04, ’06, Holmer-Zworski ’08:
dist (p(t), q(t)), (pmech(t), qmech(t)) 1/2
for times |t| ≤ T −3/2
Question: can we do better? Answer: NO!
true motions of the soliton are actually different from the
mechanical ones and the difference becomes macroscopic after a
quite short time scale

10. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
New point of view
Not surprising! in classical mechanics motions starting nearby get far
away after quite short time scales.
to control the dynamics for longer times, control only on some
relevant quantities: actions or energy of subsystem
in our case: system is composed by two subsystems evolving on
different time-scales:
HNLS = Hmech(p, q) + Hfield (φ) + high order coupling
New Question: can we say that Hmech does not change for longer times?

11. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
New point of view
Not surprising! in classical mechanics motions starting nearby get far
away after quite short time scales.
to control the dynamics for longer times, control only on some
relevant quantities: actions or energy of subsystem
in our case: system is composed by two subsystems evolving on
different time-scales:
HNLS = Hmech(p, q) + Hfield (φ) + high order coupling
New Question: can we say that Hmech does not change for longer times?

12. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
New point of view
Not surprising! in classical mechanics motions starting nearby get far
away after quite short time scales.
to control the dynamics for longer times, control only on some
relevant quantities: actions or energy of subsystem
in our case: system is composed by two subsystems evolving on
different time-scales:
HNLS = Hmech(p, q) + Hfield (φ) + high order coupling
New Question: can we say that Hmech does not change for longer times?

13. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
New point of view
Not surprising! in classical mechanics motions starting nearby get far
away after quite short time scales.
to control the dynamics for longer times, control only on some
relevant quantities: actions or energy of subsystem
in our case: system is composed by two subsystems evolving on
different time-scales:
HNLS = Hmech(p, q) + Hfield (φ) + high order coupling
New Question: can we say that Hmech does not change for longer times?

14. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Outline
1 The problem: Dynamics of soliton in NLS with external potential
2 Main Theorem
3 Scheme of the proof

15. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Assumptions
Linearization at the soliton: ˙χ = L0χ
We assume
1 σd (L0) = {0}, σc (L0) = ± ±i[E, ±∞)
2 Ker(L+) = span(ηm) , Ker(L−) = span(∂xj
ηm)j=1,...,3, 0 has
multiplicity 8
3 ±iE are not resonances, i.e. L0χ = iEχ has no solution with
x
−δ
χ ∈ L2
, ∀δ > 1/2

16. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Theorem (Bambusi, M.)
Fix arbitrary r ∈ N. Then ∃ r s.t. for 0 ≤ < r , it holds the following:
∀ψ0 ∈ H1
with
ψ0 − ei¯α
η ¯m(¯p, ¯q) H1 ≤ K1
1/2
Hmech(¯p, ¯q) < K2 ,
(3)
then the solution ψ(t) admits the decomposition in H1
ψ(t) := eiα(t)
ηm(p(t), q(t)) + φ(t) , (4)
with a constant m and smooth functions p(t), q(t), α(t) s.t.
|Hmech(p(t), q(t)) − Hmech(p(0), q(0))| ≤ C1
3/2
, |t| ≤
T0
r
. (5)
Furthermore, for the same times one has
φ(t) H1 ≤ C1
1/2
.

17. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Some remarks
the energy of the soliton freezes for times ∼ −r
Key point: frequency of the soliton ∼ 1/2
, frequency of the field ∼ 1
assume that V and ψ0 are symmetric around an axis, then the
mechanical hamiltonian is 1-dimensional. Then one can control also
the orbit of the soliton

18. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Outline
1 The problem: Dynamics of soliton in NLS with external potential
2 Main Theorem
3 Scheme of the proof

19. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Main steps
1) Canonical variables (Darboux theorem ):
- ψ = eqj
JAj (ηp + φ), but p, q, φ not canonical
- Darboux theorem: canonical p , q , φ in a neighborhood of T
2) Birkhoff normal form (continuous spectrum)
- in Darboux coordinates
H = 1/2
Hmech(p , q ) +
1
2
EL0φ , φ + 3/4
Ψ(p , q ), φ + h.o.t.
- cancel terms linear in φ up to order r
3) dispersive estimates (Strichartz)
- ˙φ = L0φ + B(t)φ + r Ψ + h.o.t.
with B(t) time-dependent unbounded operator
- Strichartz stable: φ L2
t [0,T]W
1,6
x
+ φ L∞
t [0,T]H1
x
≤ K 1/2 , ∀|t| ≤ T −r
- almost conservation of energy
|HL(φ(t)) − HL(φ(0))| ≤ ∀|t| ≤ T −r

20. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Main steps
1) Canonical variables (Darboux theorem ):
- ψ = eqj
JAj (ηp + φ), but p, q, φ not canonical
- Darboux theorem: canonical p , q , φ in a neighborhood of T
2) Birkhoff normal form (continuous spectrum)
- in Darboux coordinates
H = 1/2
Hmech(p , q ) +
1
2
EL0φ , φ + 3/4
Ψ(p , q ), φ + h.o.t.
- cancel terms linear in φ up to order r
3) dispersive estimates (Strichartz)
- ˙φ = L0φ + B(t)φ + r Ψ + h.o.t.
with B(t) time-dependent unbounded operator
- Strichartz stable: φ L2
t [0,T]W
1,6
x
+ φ L∞
t [0,T]H1
x
≤ K 1/2 , ∀|t| ≤ T −r
- almost conservation of energy
|HL(φ(t)) − HL(φ(0))| ≤ ∀|t| ≤ T −r

21. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Main steps
1) Canonical variables (Darboux theorem ):
- ψ = eqj
JAj (ηp + φ), but p, q, φ not canonical
- Darboux theorem: canonical p , q , φ in a neighborhood of T
2) Birkhoff normal form (continuous spectrum)
- in Darboux coordinates
H = 1/2
Hmech(p , q ) +
1
2
EL0φ , φ + 3/4
Ψ(p , q ), φ + h.o.t.
- cancel terms linear in φ up to order r
3) dispersive estimates (Strichartz)
- ˙φ = L0φ + B(t)φ + r Ψ + h.o.t.
with B(t) time-dependent unbounded operator
- Strichartz stable: φ L2
t [0,T]W
1,6
x
+ φ L∞
t [0,T]H1
x
≤ K 1/2 , ∀|t| ≤ T −r
- almost conservation of energy
|HL(φ(t)) − HL(φ(0))| ≤ ∀|t| ≤ T −r

22. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Framework
Phase space:
Hs,k , scale of Hilbert spaces, ψ Hs,k = x s
(∆ − 1)k/2ψ L2
ψ1; ψ2 = 2Re ψ1
¯ψ2
ω(ψ1, ψ2) := Eψ1; ψ2 , with E = i, J = E−1 Poisson tensor
Symmetries:
Aj := i∂xj , j = 1, 2, 3, A4 = 1,
eqj JAj ψ ≡ ψ(· − qj ej )
Soliton manifold: T :=
q,p
eqj
JAj ηp, where ηp := e
i
pk xk
p4 ηp4
restriction ω|T = dp ∧ dq
Natural decomposition:
L2
≡ Tηp L2
Tηp T ⊕ T∠
ηp
T
with T∠
ηp
T := U : ω(U; X) = 0 , ∀X ∈ Tηp T
Let Πp : L2 → T∠
ηp
T the projector on the symplectic orthogonal

23. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Framework
Phase space:
Hs,k , scale of Hilbert spaces, ψ Hs,k = x s
(∆ − 1)k/2ψ L2
ψ1; ψ2 = 2Re ψ1
¯ψ2
ω(ψ1, ψ2) := Eψ1; ψ2 , with E = i, J = E−1 Poisson tensor
Symmetries:
Aj := i∂xj , j = 1, 2, 3, A4 = 1,
eqj JAj ψ ≡ ψ(· − qj ej )
Soliton manifold: T :=
q,p
eqj
JAj ηp, where ηp := e
i
pk xk
p4 ηp4
restriction ω|T = dp ∧ dq
Natural decomposition:
L2
≡ Tηp L2
Tηp T ⊕ T∠
ηp
T
with T∠
ηp
T := U : ω(U; X) = 0 , ∀X ∈ Tηp T
Let Πp : L2 → T∠
ηp
T the projector on the symplectic orthogonal

24. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Framework
Phase space:
Hs,k , scale of Hilbert spaces, ψ Hs,k = x s
(∆ − 1)k/2ψ L2
ψ1; ψ2 = 2Re ψ1
¯ψ2
ω(ψ1, ψ2) := Eψ1; ψ2 , with E = i, J = E−1 Poisson tensor
Symmetries:
Aj := i∂xj , j = 1, 2, 3, A4 = 1,
eqj JAj ψ ≡ ψ(· − qj ej )
Soliton manifold: T :=
q,p
eqj
JAj ηp, where ηp := e
i
pk xk
p4 ηp4
restriction ω|T = dp ∧ dq
Natural decomposition:
L2
≡ Tηp L2
Tηp T ⊕ T∠
ηp
T
with T∠
ηp
T := U : ω(U; X) = 0 , ∀X ∈ Tηp T
Let Πp : L2 → T∠
ηp
T the projector on the symplectic orthogonal

25. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Framework
Phase space:
Hs,k , scale of Hilbert spaces, ψ Hs,k = x s
(∆ − 1)k/2ψ L2
ψ1; ψ2 = 2Re ψ1
¯ψ2
ω(ψ1, ψ2) := Eψ1; ψ2 , with E = i, J = E−1 Poisson tensor
Symmetries:
Aj := i∂xj , j = 1, 2, 3, A4 = 1,
eqj JAj ψ ≡ ψ(· − qj ej )
Soliton manifold: T :=
q,p
eqj
JAj ηp, where ηp := e
i
pk xk
p4 ηp4
restriction ω|T = dp ∧ dq
Natural decomposition:
L2
≡ Tηp L2
Tηp T ⊕ T∠
ηp
T
with T∠
ηp
T := U : ω(U; X) = 0 , ∀X ∈ Tηp T
Let Πp : L2 → T∠
ηp
T the projector on the symplectic orthogonal

26. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Step 1: Canonical variables
Coordinate system: Fix p0 := (0, 0, 0, m) and Vs,k := Πp0 Hs,k . Define
F : J × R4
× V1,0
→ H1,0
, (p, q, φ) → eqj
JAj (ηp + Πpφ)
Problems:
p, q, φ are not canonical variables
F not smooth, only continuous. Indeed
R4
× H1
(q, φ) → eqj
JAj φ ∈ H1
only continuous
Darboux theorem
There exists a map of the form
D(p , q , φ ) = p − N + P, q + Q, Πp0 eαj JAj (φ + S) ,
with the following properties
1. S : J × R4 × V−∞ → V∞ is smooth.
2. P, Q, αj : J × R4 × V−∞ → R4 are smooth.
3. Nj := 1
2
Aj φ , φ
4. p , q , φ are canonical.

27. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Step 1: Canonical variables
Coordinate system: Fix p0 := (0, 0, 0, m) and Vs,k := Πp0 Hs,k . Define
F : J × R4
× V1,0
→ H1,0
, (p, q, φ) → eqj
JAj (ηp + Πpφ)
Problems:
p, q, φ are not canonical variables
F not smooth, only continuous. Indeed
R4
× H1
(q, φ) → eqj
JAj φ ∈ H1
only continuous
Darboux theorem
There exists a map of the form
D(p , q , φ ) = p − N + P, q + Q, Πp0 eαj JAj (φ + S) ,
with the following properties
1. S : J × R4 × V−∞ → V∞ is smooth.
2. P, Q, αj : J × R4 × V−∞ → R4 are smooth.
3. Nj := 1
2
Aj φ , φ
4. p , q , φ are canonical.

28. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Proof of the Darboux theorem
Moser deformation argument:
Interpolation of symplectic forms
Ω1 := F∗
ω , Ω0 := Ω1|T , Ωt = Ω0 + t(Ω1 − Ω0)
look for (Dt )t with D0 = 1 and d
dt
D∗
t Ωt = 0.
Dt = flow map of time dependent vector field Yt : one has
Ωt (Yt , ·) = θ0 − θ1
Yt = (Y p
t , Y q
t , Y φ
t ) has the structure
Yt = P, Q, S + wj
∂xj φ
with smoothing wj , P, Q and S.
Smoothing maps: S : J × R4 × V−∞ → S(p, q, N, φ) ∈ V∞ or R, smooth
Prove that it generates flow and study its analytic properties!
Almost smooth maps: A map of the form
A(p, q, φ) = p + P, q + Q, Πp0 eαj
JAj (φ + S)
with smoothing αj , P, Q and S is said to be almost smooth.

29. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Step 2: The Hamiltonian in Darboux coordinates and normal form
Hamiltonian in Darboux coordinates: Let D be the Darboux map. Then
H ◦ F ◦ D = 1/2
h + HL + 3/4
HR
with
h =
p2
2m
+ V eff
(q), HL =
1
2
EL0φ, φ
and
dφHR (0)[φ] = S, φ , S(p, q, N) smoothing (Nj =
1
2
i∂xj φ, φ )
Birkhoff normal form: eliminate terms linear in φ up to order r in .
Lie transform: flow map of χr = r χ(r)(N, p, q), φ
Homological equation: L0χ(r) = S, L0 with continuous spectrum
Flow: hamiltonian vector field of χr is not smooth
Xχr = P, Q, S + wj
∂xj φ
It generates an almost smooth map, furthermore H ◦ F ◦ D ◦ A has the same
structure, but with terms linear in φ at order r+1

30. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Step 2: The Hamiltonian in Darboux coordinates and normal form
Hamiltonian in Darboux coordinates: Let D be the Darboux map. Then
H ◦ F ◦ D = 1/2
h + HL + 3/4
HR
with
h =
p2
2m
+ V eff
(q), HL =
1
2
EL0φ, φ
and
dφHR (0)[φ] = S, φ , S(p, q, N) smoothing (Nj =
1
2
i∂xj φ, φ )
Birkhoff normal form: eliminate terms linear in φ up to order r in .
Lie transform: flow map of χr = r χ(r)(N, p, q), φ
Homological equation: L0χ(r) = S, L0 with continuous spectrum
Flow: hamiltonian vector field of χr is not smooth
Xχr = P, Q, S + wj
∂xj φ
It generates an almost smooth map, furthermore H ◦ F ◦ D ◦ A has the same
structure, but with terms linear in φ at order r+1

31. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Step 3: Analysis of the normal form
We have H ◦ T r+2 in normal form at order r + 2.
˙φ = L0φ + 3/4
B(t)φ + r+2
S + h.o.t.
and B(t) = wj (t)∂xj + V (x, t) time-dependent unbounded linear operator
Aim: Nonlinear stability for φ!
Remark: linearized equation has time-dependent unbounded operator: a disaster on a
compact manifold!
Miracle: on Rn, Strichartz
Strichartz stable under some unbounded small perturbations (Perelman,
Beceanu):
φ L2
t [0,T]W
1,6
x
+ φ L∞
t [0,T]H1
x
≤ K 1/4
, ∀|t| ≤ T −r
Using this, one proves that
|HL(φ(t)) − HL(φ(0))| ≤ ∀|t| ≤ T −r
conservation of energy implies conservation of Hmech

32. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Step 3: Analysis of the normal form
We have H ◦ T r+2 in normal form at order r + 2.
˙φ = L0φ + 3/4
B(t)φ + r+2
S + h.o.t.
and B(t) = wj (t)∂xj + V (x, t) time-dependent unbounded linear operator
Aim: Nonlinear stability for φ!
Remark: linearized equation has time-dependent unbounded operator: a disaster on a
compact manifold!
Miracle: on Rn, Strichartz
Strichartz stable under some unbounded small perturbations (Perelman,
Beceanu):
φ L2
t [0,T]W
1,6
x
+ φ L∞
t [0,T]H1
x
≤ K 1/4
, ∀|t| ≤ T −r
Using this, one proves that
|HL(φ(t)) − HL(φ(0))| ≤ ∀|t| ≤ T −r
conservation of energy implies conservation of Hmech

33. The problem: Dynamics of soliton in NLS with external potential Main Theorem Scheme of the proof
Thanks for your attention!