On the orbital stability of a family of traveling waves for the cubic Schrödinger equation on the Heisenberg group

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On the orbital stability of a family of traveling waves for the cubic Schrödinger equation on the Heisenberg group

Louise Gassot

To cite this version:

Louise Gassot. On the orbital stability of a family of traveling waves for the cubic Schrödinger equation

on the Heisenberg group. 2019. �hal-02288854�

On the orbital stability of a family of traveling waves for the cubic Schrödinger equation on the Heisenberg group.

Louise Gassot

Abstract

We consider the focusing energy-critical Schrödinger equation on the Heisenberg group in the radial case

i∂

t

u − ∆

_H1

u = |u|

²

u, ∆

_H1

= 1

4 (∂

²_x

+ ∂

_y²

) + (x

²

+ y

²

)∂

²_s

, (t, x, y, s) ∈ R × H

¹

, which is a model for non-dispersive evolution equations. For this equation, existence of smooth global solutions and uniqueness of weak solutions in the energy space are open problems. We are interested in a family of ground state traveling waves parametrized by their speed β ∈ (−1, 1). We show that the traveling waves of speed close to 1 present some orbital stability in the following sense. If the initial data is radial and close enough to one traveling wave, then there exists a global weak solution which stays close to the orbit of this traveling wave for all times. A similar result is proven for the limiting system associated to this equation.

Contents

1 Introduction 2

1.1 Motivation . . . . 2

1.2 Main results . . . . 2

1.3 Comparison with other equations . . . . 5

2 Notation 6 2.1 The Heisenberg group . . . . 6

2.2 Decomposition along the Hermite functions . . . . 7

3 Conditional orbital stability of the ground state Q in the limiting equation 8 3.1 Weighted Bergman spaces . . . . 8

3.2 Construction of approximate solutions . . . . 9

3.3 Weak convergence . . . . 11

3.4 Modulation . . . . 12

4 Conditional orbital stability of the ground states Q

_β

in the Schrödinger equation 15 4.1 Construction of approximate solutions . . . . 16

4.2 Limit γ → 1 for the n-th partial sum . . . . 19

4.3 Weak convergence . . . . 23

5 Appendix : Rate of convergence of Q

β

to Q 24

1 Introduction

1.1 Motivation

We are interested in the Schrödinger equation on the Heisenberg group ( i∂

t

u − ∆

_H1

u = |u|

²

u

u(t = 0) = u

₀

, (t, x, y, s) ∈ R × H

¹

. (1) The operator ∆

_H1

denotes the sub-Laplacian on the Heisenberg group. When the solution is radial, in the sense that it only depends on t, |x + iy| and s, the sub-Laplacian writes

∆

_H1

= 1

4 (∂

_x²

+ ∂

_y²

) + (x

²

+ y

²

)∂

_s²

.

The Heisenberg group is a typical case of sub-Riemannian geometry where dispersive properties of the Schrödinger equation disappear (see Bahouri, Gérard and Xu [3]). To take it further, Del Hierro [9] proved sharp decay estimates for the Schrödinger equation on H-type groups, depending on the dimension of the center of the group. More generally, Bahouri, Fermanian and Gallagher [2] proved optimal dispersive estimates on stratified Lie groups of step 2 under some property of the canonical skew-symmetric form. In contrast, they also give a class of groups without this property displaying total lack of dispersion, which includes the Heisenberg group.

Dispersion impacts the way one can address the Cauchy problem for the Schrödinger equation. Indeed (see Burq, Gérard and Tzvetkov [6], remark 2.12), the existence of a smooth local in time flow map on the Sobolev space H

^s

(M ) for the Schrödinger equation on a Riemaniann manifold M with Laplace-Beltrami operator ∆

( i∂

_t

u − ∆u = |u|

²

u u(t = 0) = u

0

implies the following Strichartz estimate

k e

^it∆

f k

L⁴([0,1]×M)

≤ Ckf k

_H^s_2(M)

.

The argument also applies for the Heisenberg group with the homogeneous Sobolev spaces H ˙

^s

( H

¹

), for which the inequality holds if and only if s ≥ 2 [12]. In particular, uniqueness of weak solutions in the energy space H ˙

¹

( H

¹

) is an open problem, whereas without a conser- vation law controlling the H ˙

²

-norm, there is no existence existence result of smooth global solutions. Constructing weak global solutions to the Schrödinger equation on the Heisenberg group is still possible through a compactness argument, nevertheless, this method does not ensure that these solutions are relevant. Indeed, the energy of the solution is only bounded above by the initial energy. Therefore, the cancellation of the energy of the solution at some time may not imply that the solution is identically zero, and we do not exclude the possibility of non-uniqueness of weak solutions as in the 2D incompressible Euler equation [8].

The aim of this paper is to construct some weak global solutions with a prescribed behaviour. More precisely, given an initial data close to some ground state traveling wave solution for the Schrödinger equation on the Heisenberg group, we want to construct a weak global solution which stays close to the orbit of the traveling wave for all times. Combined with an uniqueness result, this would lead to the orbital stability of this ground state traveling wave.

1.2 Main results

We consider a family of traveling waves with speed β ∈ (−1, 1) under the form u

_β

(t, x, y, s) = p

1 − βQ

_β

(x, y, s + βt).

The profile Q

_β

satisfies the following stationary hypoelliptic equation

− ∆

_H1

+ βD

s

1 − β Q

_β

= |Q

_β

|

²

Q

_β

.

Because of the scaling invariance, it would have been equivalent in the rest of the study to define Q

_β

as

u

β

(t, x, y, s) = Q

β

x

√ 1 − β , y

√ 1 − β , s + βt 1 − β

.

From [11], we know that as β tends to 1, the ground state solutions of speed β converge up to symmetries in H ˙

¹

( H

¹

) to some profile Q. Moreover, Q is solution to a limiting equation

D

s

Q = Π

⁺₀

(|Q|

²

Q), (2)

for which the ground state solution is unique up to symmetries, equal to Q(x, y, s) = i √

2

s + i(x

²

+ y

²

) + i .

The operator Π

⁺₀

is an orthogonal projector onto a relevant space for our analysis denoted by V

₀⁺

. For more details, see the Notation part 2.

We first prove the conditional orbital stability of the ground state Q in the limiting equation, and then focus on the conditional orbital stability of the ground states Q

β

in the Schrödinger equation on the Heisenberg group as β is close to 1.

Definition 1.1. Fix u ∈ H ˙

¹

( H

¹

) and X = (s

0

, θ, α) ∈ R × T × R

^∗+

. We denote by T

X

u the element of H ˙

¹

( H

¹

) satisfying

T

X

u(x, y, s) := e

^iθ

αu(αx, αy, α

²

(s − s

0

)), (x, y, s) ∈ H

¹

. We also define

d(u, M) = inf

X∈R×T×R^∗+

kT

X

u − Qk

H˙¹(H¹)

, as the distance of u to the orbit M of Q

M = {T

X

Q | X ∈ R × T × R

^∗+

}, and

d(u, p

1 − βQ

β

) = inf

X∈R×T×R^∗+

kT

X

u − p

1 − βQ

β

k

H˙¹(H¹)

, as the distance of

^√_1−β^u

to the orbit Q

_β

of Q

_β

Q

β

= {T

X

Q

β

| X ∈ R × T × R

^∗+

}.

Our first result is that the profile Q is conditionally orbitally stable for the evolution problem linked to the limiting equation

( i∂

_t

u = Π

⁺₀

(|u|

²

u) u(t = 0) = u

0

. (3)

Theorem 1.2 (Conditional orbital stability of Q). For some c

₀

> 0 and r

₀

> 0 the following holds. Let r ≤ r

₀

and u

₀

∈ H ˙

¹

( H

¹

) ∩ V

₀⁺

such that

ku

0

− Qk

H˙¹(H¹)

< r

²

. (4) Then there exists a weak solution u ∈ C( R , H ˙

¹

( H

¹

)) (with the weak topology) to equation (3) such that for all t ∈ R,

d(u(t), M) ≤ c

0

r.

Using the links between the limiting equation and the Schrödinger equation, we deduce our second result : the conditional orbital stability of the profiles Q

β

for the Schrödinger equation when β is close to 1 in the radial case.

Theorem 1.3 (Conditional orbital stability of Q

_β

). For some c

₀

> 0 and r

₀

> 0 the

following holds. Let r ∈ (0, r

₀

) . Then there exists β

_∗

∈ (0, 1) such that if β ∈ (β

_∗

, 1) , and if

u

₀

∈ H ˙

¹

( H

¹

) is radial and satisfies

• if u

₀

∈ H ˙

¹

( H

¹

) ∩ V

₀⁺

:

ku

0

− p

1 − βQ

_β

k

H˙¹(H¹)

< p 1 − βr

²

• in the general case :

ku

₀

− p

1 − βQ

_β

k

H˙¹(H¹)

< (1 − β)r, (5) then there exists a weak radial solution u ∈ C( R , H ˙

¹

( H

¹

)) (with the weak topology) to the Schrödinger equation on the Heisenberg group (1)

( i∂

t

u − ∆

_H1

u = |u|

²

u u(t = 0) = u

₀

such that for all t ∈ R,

^√u(t)_1−β

is close to the orbit of Q

β

:

d u(t), p

1 − βQ

β

≤ c

0

p 1 − βr.

Note that, unlike the weak solutions discussed in the first part 1.1, the energy of the weak solutions from Theorem 1.2 (resp. Theorem 1.3) is controlled, indeed, this energy is very close to the one of the ground state Q (resp. √

1 − βQ

_β

). Furthermore, these two theorems would imply the orbital stability of Q and Q

β

in the radial case in both situations if we had a uniqueness result for the solutions.

The assumption required on a general initial condition for the Schrödinger equation (5) is owing to the estimates on the component of the initial condition on the orthogonal of V

₀⁺

. In the assumptions of Theorem 1.2, this component is naturally zero, leading to a weaker assumption (4) on the initial condition.

The key point in both proofs is the following local stability estimate for Q, which comes from the invertibility of the linearized operator around Q for the limiting equation (2) on a subspace of V

₀⁺

of finite co-dimension.

Definition 1.4. For u ∈ H ˙

¹

( H

¹

) ∩ V

₀⁺

, we define δ(u) :=

kuk

²_H˙1(

H¹)

−kQk

²_H˙1(

H¹)

+

kuk

⁴_L4(H¹)

−kQk

⁴_L4(H¹)

= |P(u) − P(Q)|+|E(u) − E (Q)|.

Proposition 1.5. [11] There exist δ

0

> 0 and C > 0 such that for all u ∈ H ˙

¹

( H

¹

) ∩ V

₀⁺

, if δ(u) ≤ δ

0

, then

d(u, M)

²

≤ Cδ(u).

In order to prove Theorem 1.2, we construct the weak solution for the limiting initial value problem (3) as a limit of smooth functions. The approximating functions solve slightly modified equations where we have cut frequencies, so that the Cauchy problem is globally well-posed. We show that we can control their distance to the orbit of the ground state Q using Proposition 1.5. Finally, we build modulation parameters which stay bounded on finite time intervals for the approximate solutions, and through a compactness argument, we control the distance of the weak solution to the orbit of Q when passing to the limit.

For Theorem 1.3, the idea for the construction is the same, however we only have at

our disposal the information on the limiting equation from Proposition 1.5. Therefore, we

need to take advantage of the fact that Q

β

is close to Q when β is close to 1. In this spirit,

in order to tackle Theorem 1.3 for the speed β, we first introduce Cauchy problems for

the Schrödinger equation (1) with a parameter γ increasing from β to 1. We display some

continuity between the Cauchy problems, therefore it is possible to show their convergence

to a Cauchy problem for the limiting equation as γ tends to 1. In the proof, we combine this

strategy with the above method : we approximate by smooth functions the weak solutions

to the Cauchy problems with parameter γ by cutting frequencies. Finally, we are able to

get back to the problem with speed β by continuity and conclude in the same way as the

proof of Theorem 1.2, by constructing bounded modulation parameters for the approximate

solutions.

1.3 Comparison with other equations

Concerning the focusing energy-critical Schrödinger equation on the Euclidean plane R

^N

i∂

t

u − ∆u = |u|

^pc−1

u,

where N ≥ 3 and p

c

=

^N+2_N−2

, there exists an explicit stationary solution

W (x) = 1

(1 +

_N(N−2)^|x|2

)

^N−22

.

The orbit {x 7→ CW (

^x+x_λ⁰

) | (C, x

0

, λ) ∈ R × R × R

^∗+

} of W is the set of minimizers for the Sobolev embedding H ˙

¹

( R

^N

) , → L

^2∗

( R

^N

) (see the work of Talenti [17] and Aubin [1]).

The energy E(W ) =

¹₂

kW k

H˙¹(R^N)

−

_p¹

c+1

kW k

_Lpc+1(R^N)

and the H ˙

¹

norm kW k

H˙¹(R^N)

play an important role in the dynamical behaviour of the solutions. Kenig and Merle [10] proved in the radial case that if N ∈ {3, 4, 5} and the initial condition u

0

∈ H ˙

¹

( R

^N

) satisfies E(u

0

) < E(W ) and ku

0

k

H˙¹(R^N)

< kW k

H˙¹(R^N)

, then the solution is global and scatters in H ˙

¹

( R

^N

), whereas if E(u

0

) < E(W ) and ku

0

k

H˙¹(R^N)

> kW k

H˙¹(R^N)

, then the solution must blow up in finite time.

The situation is different for the Schrödinger equation on the Heisenberg group. Indeed, from the equation satisfied Q

β

, one can see that the traveling waves

u

_β

(t, x, y, s) = p

1 − βQ

_β

(x, y, s + βt) have a vanishing energy as β tends to 1 :

E(u

β

(t)) = 1

2 ku

β

(t)k

²_H˙1(

H¹)

− 1

4 ku

β

(t)k

⁴_L4(H¹)

∼ (1 − β) π

²

2 → 0, therefore there exists solutions that do not scatter with arbitrary small energy.

A better parallel would be the mass-critical focusing half-wave equation on the real line i∂

t

u + |D|u = |u|

²

u, (t, x) ∈ R × R , (6) where D = −i∂

_x

, |D|f [ (ξ) = |ξ| f b (ξ). The half-wave equation in one dimension also presents some lack of dispersion, and admits traveling waves with speed β ∈ (−1, 1) (see Krieger, Lenzmann and Raphaël [14])

u(t, x) = Q

β

x + βt 1 − β

e

^−it

, where the profile Q

β

is a solution to

|D|−βD

1 − β Q

β

+ Q

β

= |Q

β

|

²

Q

β

.

The profiles Q

β

in the half-wave equation converge [13] as β tends to 1 in H

¹²

( R ) to a ground state solution Q to some limiting equation

DQ + Q = Π(|Q|

²

Q), D = −i∂

x

.

From Q, we recover a traveling wave solution to the cubic Szegő equation

i∂

_t

u = Π(|u|

²

u) (7)

by setting u(t, x) = Q(x−t) e

^−it

. Moreover, the linearized operator around Q is coercive [16], and in particular, the Szegő profile is orbitally stable in the relevant space for Q

H

1 2

+

( R ) = {u ∈ H

¹²

( R ) | Supp( u) b ⊂ R

+

}.

Theorem 1.6 (Orbital stability of Q for the Szegő equation). There exist ε

₀

> 0 and C > 0 such that for all solution u of the Szegő equation (7) with initial condition u

₀

∈ H

₊¹²

( R ) , if

ku

0

− Qk

H¹2(R)

≤ ε

0

, then

sup

t∈R

inf

(γ,y)∈T×R

ke

^−iγ

u(t, · − y) − Qk

²

H¹2(R)

≤ Cku

0

− Qk

H¹2(R)

.

Gérard, Lenzmann, Pocovnicu and Raphaël [13] deduced the invertibility of the linearized operator for the half-wave equation around the profiles Q

β

when β is close enough to 1, and their estimates imply the orbital stability of theses profiles.

Theorem 1.7 (Orbital stability of Q

β

for the half-wave equation). There exists β

_∗

∈ (0, 1) such that the following holds. Let β ∈ (β

_∗

, 1) . Then there exist ε

0

(β ) > 0 and C(β) > 0 such that for all solution u of the half-wave equation (6) with initial condition u

₀

∈ H

¹²

( R ) , if

ku

0

− Q

β

( · 1 − β )k

H¹2(R)

≤ ε

0

(β), then

sup

t∈R

inf

(γ,y)∈T×R

ke

^−iγ

u(t, · − y) − Q

β

( · 1 − β )k

²

H¹2(R)

≤ Cku

0

− Q

β

( · 1 − β )k

H¹2(R)

. In higher dimensions d ≥ 2, traveling waves for the half-wave equation on R

^d

i∂

t

u + √

−∆u = |u|

^p−1

u, (t, x) ∈ R × R

^d

,

are also orbitally stable in the radial case for mass-subcritical non-linearities 1 < p < 1 +

_n²

, but orbitally unstable in the mass-supercritical regime 1 +

²_n

< p < 1 +

_n−1²

[5]. Moreover, in the energy-critical and subcritical case, Bellazzini, Georgiev, Lenzmann and Visciglia [4]

proved that there can be no small data scattering in the energy space because of the existence of traveling waves with arbitrary small energy.

As we will see in this paper, one cannot directly adapt the proofs for the half-wave equation because we lack information on the Cauchy problem. A second complication arising in comparison to the half-wave equation is the fact that only two conservation laws are available (energy and momentum), because the masses of the ground states may be infinite (this fact is easy to check for Q for instance). The method both for the Schrödinger equation on the Heisenberg group and for its limiting system is the construction of some weak solutions as a limit of smooth functions, and show that we can pass to the limit on their stability properties.

The paper is organized as follows. We first prove the orbital stability of Q for the limiting equation in Section 3. Then, we assess how close the solutions are to the limiting equation as β tends to 1, in order to study the orbital stability of Q

β

for the Schrödinger equation in Section 4.

Acknowledgements The author is grateful to her PhD advisor P. Gérard for his gen- erous advice and encouragement.

2 Notation

2.1 The Heisenberg group

Let us now recall some facts about the Heisenberg group. We use coordinates and identify the Heisenberg group H

¹

with R

³

. The group multiplication is given by

(x, y, s) · (x

⁰

, y

⁰

, s

⁰

) = (x + x

⁰

, y + y

⁰

, s + s

⁰

+ 2(x

⁰

y − xy

⁰

)).

The Lie algebra of left-invariant vector fields on H

¹

is spanned by the vector fields X =

∂

x

+ 2y∂

s

, Y = ∂

y

− 2x∂

s

and T = ∂

s

=

¹₄

[Y, X]. The sub-Laplacian is defined as L

0

:= 1

4 (X

²

+ Y

²

) = 1

4 (∂

_x²

+ ∂

_y²

) + (x

²

+ y

²

)∂

_s²

+ (y∂

x

− x∂

y

)∂

s

.

When u is a radial function, the sub-Laplacian coincides with the operator

∆

_H1

:= 1

4 (∂

_x²

+ ∂

_y²

) + (x

²

+ y

²

)∂

_s²

.

The space H

¹

is endowed with a smooth left invariant measure, the Haar measure, which in the coordinate system (x, y, s) is the Lebesgue measure dλ

3

(x, y, s). Sobolev spaces of positive order can then be constructed on H

¹

from powers of the operator −∆

_H1

, for example, H ˙

¹

( H

¹

) is the completion of the Schwarz space S ( H

¹

) for the norm

kuk

H˙¹(H¹)

:= k(−∆

_H1

)

¹²

uk

_L2(H¹)

.

2.2 Decomposition along the Hermite functions

In order to study radial functions valued on the Heisenberg group H

¹

, it is convenient to use their decomposition along Hermite-type functions (see for example [15], Chapters 12 and 13). The Hermite functions

h

_m

(x) = 1

π

¹⁴

2

^m2

(m! )

¹²

(−1)

^m

e

^x

2

2

∂

_x^m

( e

^−x2

), x ∈ R , m ∈ N ,

form an orthonormal basis of L

²

( R ). In L

²

( R

²

), the family of products of two Hermite functions (h

m

(x)h

p

(y))

_m,p∈N

diagonalizes the two-dimensional harmonic oscillator : for all m, p ∈ N ,

(−∆

x,y

+ x

²

+ y

²

)h

m

(x)h

p

(y) = 2(m + p + 1)h

m

(x)h

p

(y).

Given u ∈ S ( H

¹

), we will denote by u b its usual Fourier transform under the s variable, with corresponding variable σ

u(x, y, σ) = b 1

√ 2π Z

R

e

^−isσ

u(x, y, s) ds.

For m, p ∈ N , set h d

m,p

(x, y, σ) := h

m

( p

2|σ|x)h

p

( p

2|σ|y). Then the family (h

m,p

)

m,p∈N

diagonalizes the sub-Laplacian in the following sense :

∆ \

_H¹

h

m,p

= −(m + p + 1)|σ| h d

m,p

.

Let k ∈ {−1, 0, 1}, and denote by H ˙

^k

( H

¹

) ∩ V

_n^±

the subspace of H ˙

^k

( H

¹

) spanned by {h

m,p

| m, p ∈ N , m + p = n}. Some u

^±_n

∈ H ˙

^k

( H

¹

) belongs to H ˙

^k

( H

¹

) ∩ V

_n^±

if there exists a family (f

_m,p^±

)

_m+p=n

such that

u c

^±n

(x, y, σ) = X

m,p∈N; m+p=n

f

_m,p^±

(σ) h d

_m,p

(x, y, σ) 1

σ≷0

.

For u

^±_n

∈ H ˙

^k

( H

¹

) ∩ V

_n^±

, the H ˙

^k

-norm of u

^±_n

writes ku

^±_n

k

²_H˙k(

H¹)

= Z

R±

((n + 1)|σ|)

^k

Z

R²

| u c

^±n

(x, y, σ)|

²

dx dy dσ

= X

m,p∈N; m+p=n

Z

R±

((n + 1)|σ|)

^k

|f

_m,p^±

(σ)|

²

dσ 2|σ| .

Any function u ∈ H ˙

^k

( H

¹

) admits a decomposition along the orthogonal sum of the subspaces H ˙

^k

( H

¹

) ∩ V

_n^±

. Let us write u = P

±

P

n∈N

u

^±_n

where u

^±_n

∈ H ˙

^k

( H

¹

)∩ V

_n^±

for all (n, ±). Then kuk

²_H˙k(

H¹)

= X

±

X

n∈N

ku

^±_n

k

²_H˙k(

H¹)

.

For k = 0, we get an orthogonal decomposition of the space L

²

( H

¹

), and denote by Π

^±_n

the associated orthogonal projectors.

The particular space H ˙

^k

( H

¹

) ∩ V

₀⁺

is spanned by a unique radial function h

⁺₀

, satisfying h c

⁺₀

(x, y, σ) = 1

√ π e

^−(x2+y2)σ

1

^σ≥0

.

Set u ∈ H ˙

^k

( H

¹

) ∩ V

₀⁺

, then there exists f such that

b u(x, y, s) = f(σ) h c

⁺₀

(x, y, σ), and

kuk

²_H˙k(

H¹)

= Z

R+

|f(σ)|

²

dσ 2σ

^1−k

.

3 Conditional orbital stability of the ground state Q in the limiting equation

In this section, we prove Theorem 1.2 on the conditional orbital stability of the ground state Q in the limiting equation (3)

( i∂

_t

u = Π

⁺₀

(|u|

²

u)

u(t = 0) = u

0

∈ H ˙

¹

( H

¹

) ∩ V

₀⁺

, (t, x, y, s) ∈ R × H

¹

.

For convenience, we replace in this part the elements u ∈ H ˙

^k

( H

¹

) ∩ V

₀⁺

, k ∈ {−1, 0, 1} with the corresponding holomorphic function on the complex upper half-plane F

u

∈ H ˙

^k2

( C

+

) ∩ Hol( C

+

), defined as

F

_u

(s + i(x

²

+ y

²

)) := u(x, y, s),

which transforms the Cauchy problem for u in a Cauchy problem for F

_u

written as ( i∂

t

u = P

0

(|u|

²

u)

u(t = 0) = u

₀

∈ H ˙

¹²

( C

+

) ∩ Hol( C

+

) , (t, z) ∈ R × C

+

, (8) P

₀

being a Bergman projector. We recall such a correspondence in part 3.1. Then we construct some smooth functions approximating a weak solution of equation (8) in part 3.2, prove their weak convergence in part 3.3, and deduce from their distance to the orbit of Q an upper bound on the distance of the weak limit to this orbit in part 3.4.

3.1 Weighted Bergman spaces

Recall that if u ∈ H ˙

^k

( H

¹

) ∩ V

₀⁺

, then F

_u

∈ H ˙

^k2

( C

+

) ∩ Hol( C

+

), and the Fourier transform of u along the s variable corresponds to a function in L

²

( R

⁺

, σ

^k−1

dσ) : for some f ∈ L

²

( R

+

, σ

^k−1

dσ),

F

_u

(z) = 1 π √

2 Z

+∞

0

e

^izσ

f (σ) dσ where

kuk

²_H˙k(

H¹)

= πkF

u

k

²_˙

H^k2(C+)

= πk(−i∂

z

)

^k2

F

u

k

²_L2(C+)

= 1 2

Z

+∞

0

|f(σ)|

²

σ

^k−1

dσ.

For k < 1, F

u

belongs to the weighted Bergman space A

²_1−k

.

Definition 3.1 (Weighted Bergman spaces). Given k < 1 , the weighted Bergman space A

²_1−k

is the subspace of L

²_1−k

:= L

²

( C

+

, Im(z)

^−k

dλ(z)) composed of holomorphic functions of the complex upper half-plane C

+

:

A

²_1−k

:=

F ∈ Hol( C

+

) | kFk

²_L2 1−k

:=

Z

+∞

0

Z

R

|F (s + it)|

²

ds dt t

^k

< +∞

.

Indeed, recall the Paley-Wiener theorem for Bergman spaces [7].

Theorem 3.2 (Paley-Wiener). Let k < 1 . Then for every f ∈ L

²

( R

+

, σ

^k−1

dσ) , the follow- ing integral is absolutely convergent on C

+

F(z) = 1

√ 2π Z

+∞

0

e

^izσ

f (σ) dσ, (9)

and defines a function F ∈ A

²_1−k

which satisfies

kFk

²_L2

1−k

= Γ(1 − k) 2

^1−k

Z

+∞

0

|f (σ)|

²

σ

^k−1

dσ. (10) Conversely, for every F ∈ A

²_1−k

, there exists f ∈ L

²

( R

+

, σ

^k−1

dσ) such that (9) and (10) hold.

For k = 1, F

_u

belongs to the Hardy space H

²

( C

+

).

Definition 3.3. The Hardy space H

²

( C

+

) space of holomorphic functions of the upper half- plane C

+

such that the following norm is finite :

kFk

²_H2(C+)

:= sup

t>0

Z

R

|F (s + it)|

²

ds < +∞.

Theorem 3.4 (Paley-Wiener). For every f ∈ L

²

( R

+

) , the following integral is absolutely convergent on C

+

F(z) = 1

√ 2π

Z

+∞

0

e

^izσ

f (σ) dσ, (11)

and defines a function F in the Hardy space H

²

( C

⁺

) which satisfies

kF k

²_H2(C+)

= Z

+∞

0

|f (σ)|

²

dσ. (12)

Conversely, for every F ∈ H

²

( C

+

) , there exists f ∈ L

²

( R

+

) such that (11) and (12) hold.

In the following, we will work with the holomorphic representations, the solutions being valued in the Hardy space H

²

( C

+

) = ˙ H

¹²

( C

+

) ∩ Hol( C

+

).

3.2 Construction of approximate solutions

Given an initial data u

0

∈ H

²

( C

+

) = ˙ H

¹²

( C

+

) ∩ Hol( C

+

) close enough to the ground state Q(z) =

√ 2 z + i ,

we want to construct a global solution to the Cauchy problem (8) ( i∂

t

u = P

0

(|u|

²

u), (t, z) ∈ R × C

+

u(t = 0) = u

0

which stays close to Q (up to symmetries) for all times. The Bergman projection P

0

from L

²

( C

+

) to A

²₁

writes (see eg [7])

P

0

(u)(z) = − 1 π

Z

R+

Z

R

1

(z − s + it)

²

u(s + it) ds dt, z ∈ C

+

.

We approximate u by functions with higher regularity, satisfying equations for which we

can use a classical global well-posedness result.

Construction of smoothing projectors P e

_ε,M

: For ε, M > 0, we define the projec- tor P e

_ε,M

as follows. Write u ∈ H ˙

^k

( C

+

) ∩ Hol(C

₊

), k ≤

¹₂

(or u ∈ H

^k

( C

+

) ∩ Hol( C

+

), k ≥ 0) as

u(z) = 1

√ 2π Z

+∞

0

e

^izσ

f (σ) dσ, then

P e

ε,M

(u)(z) := 1

√ 2π

Z

M

ε

e

^izσ

f (σ) dσ.

This projector cuts the high and low frequencies of u, in order to add some regularity on the solutions. It defines a bounded projector from H ˙

^k

( C

+

) ∩ Hol( C

+

) to itself for k ≤

¹₂

, and from H

^k

( C

+

) ∩ Hol( C

+

) to itself for k ≥ 0.

Construction of a sequence of approximate solutions (u

_n

)

_n

: We consider f ∈ L

²

( R

⁺

) such that for all z ∈ C

⁺

,

u

₀

(z) = 1

√ 2π Z

+∞

0

e

^izσ

f (σ) dσ, which satisfies

ku

₀

k

²

H˙¹2(C+)

= 1

2 kf k

²_L2(R+)

.

Let us fix a sequence of positive numbers (ε

n

)

n

going to zero, and consider the following initial data belonging to H

²

( C

+

) ∩ Hol( C

+

)

u

ⁿ₀

(z) := P e

_ε

n,¹

εn

u

₀

(z) = 1

√ 2π Z

1/εn

ε_n

e

^izσ

f (σ) dσ.

We denote by H

_ε²

( C

+

)∩ Hol( C

+

) the space of functions u ∈ H

²

( C

+

) ∩ Hol( C

+

) satisfying P e

_ε,1

ε

(u) = u. On this space, the H ˙

^k

-norms, k ≥ 0, are equivalent : ε

^2k

kuk

²_L2(C+)

≤ kuk

²_H˙k(

C+)

= 1 2

Z

1/ε

ε

σ

^2k−1

|f (σ)|

²

dσ ≤ 1

ε

^2k

kuk

²_L2(C+)

. Define the projection P

₀ⁿ

as

P

₀ⁿ

= P e

_εn,1 εn

◦ P

0

. We consider the following Cauchy problem

( i∂

t

u

n

= P

₀ⁿ

(|u

n

|

²

u

n

)

u

_n

(t = 0) = u

ⁿ₀

, (13)

which is globally well-posed in H

_ε²

n

( C

+

) ∩ Hol( C

+

).

Proposition 3.5. Let u

ⁿ₀

∈ H

_ε²

n

( C

+

) ∩ Hol( C

+

) . Then there exists a unique solution u

_n

∈ C

^∞

( R , H

_ε²

n

( C

+

) ∩ Hol( C

+

)) of (13) in the distribution sense.

Proof. The local existence comes from the Cauchy-Lipschitz theory for PDEs. Indeed, H

^k

( C

+

) ∩ Hol( C

+

) is an algebra as soon as k > 1, and in this case, P

0

extends to a bounded projector from H

^k

( C

+

) ∩ Hol( C

+

) to itself preserving H

_ε^k_n

( C

+

) ∩ Hol( C

+

), therefore P

₀ⁿ

ex- tends to a bounded projector from H

^k

( C

+

) ∩ Hol( C

+

) onto H

_ε^k

n

( C

+

) ∩ Hol( C

+

). Moreover, the time of existence of the solution only depends on the norm of the initial data in H

²

( C

+

).

In order to prove that local solutions extend globally in time, it is now enough to show that the H

²

-norm of the solution stays bounded. Thanks to the equivalence of the H ˙

^k

-norms in H

_ε²

n

( C

+

) ∩ Hol( C

+

), this lies in the fact that equation (13) has conserved momentum P (u) := (u, −iu

z

)

_L2(C+)

= kuk

²_˙

H¹2(C+)

.

The energy E(u) = kuk

⁴_L4(C+)

is also conserved.

We now show that u

n

(t) is close to the orbit M of the ground state Q. Thanks to Proposition 1.5, it is enough to focus on δ(u

n

(t)). But using the conservation laws, we know that for all t ∈ R ,

δ(u

_n

(t)) = δ(u

ⁿ₀

).

Moreover, by construction of u

ⁿ₀

, we know that ku

ⁿ₀

− u

0

k

_˙

H¹2(C+)

tends to 0 as n tends to +∞, therefore δ(u

ⁿ₀

) tends to δ(u

0

). Assume that δ(u

0

) < δ

0

, then δ(u

ⁿ₀

) < δ

0

after some rank N. Thanks to Proposition 1.5, we deduce that for all n ≥ N and t ∈ R ,

d(u

_n

(t), M)

²

≤ Cδ(u

ⁿ₀

). (14)

3.3 Weak convergence

In this part, we show that u

n

has a weak limit u, which is a weak solution to equation (8).

In order to do so, we first prove that t 7→ ∂

t

u

n

(t) is uniformly bounded in H ˙

⁻¹²

( C

⁺

), then use Ascoli’s theorem.

Because of the conservation of the momentum and the fact that P (u

ⁿ₀

) ≤ P (u

0

) for all n ∈ N , we know that for all n ∈ N and t ∈ R ,

ku

n

(t)k

_˙

H¹2(C+)

≤ ku

0

k

_˙

H¹2(C+)

. Using the equation satisfied by u

n

, we also know that

k∂

_t

u

_n

(t)k

_˙

H⁻¹2(C+)

≤ kP

₀ⁿ

(|u

_n

|

²

u

_n

)(t)k

_˙

H⁻¹2(C+)

.

By the dual Sobolev embedding L

⁴³

( C

+

) , → H ˙

⁻¹²

( C

+

) and the fact that P

₀

extends to a bounded projector from L

^p

( C

+

) to itself as soon as 1 < p < +∞ (see for example [7]), we can estimate

kP

₀ⁿ

(|u

n

|

²

u

n

)(t)k

_˙

H⁻¹2(C+)

≤ kP

0

(|u

n

|

²

u

n

)(t)k

_˙

H⁻¹2(C+)

≤ CkP

0

(|u

n

|

²

u

n

)(t)k

L⁴3(C+)

≤ C

⁰

k|u

n

|

²

u

_n

(t)k

L⁴3(C+)

≤ C

⁰

ku

n

(t)k

³_L4(C+)

.

Since u

n

(t) is uniformly bounded in H ˙

¹²

( C

+

) and therefore in L

⁴

( C

+

), we conclude that the term k∂

t

u

n

(t)k

_˙

H⁻¹2(C+)

is also uniformly bounded.

We now prove that that up to a subsequence, (u

n

)

n

converges in C([−T, T ], H ˙

¹²

( C

⁺

) ∩ Hol( C

+

)) (with the weak topology) to a function u for all T > 0.

We know that H ˙

⁻¹²

( C

+

)∩Hol( C

+

) is separable, since it is isometric to L

²

( R

+

). Moreover, by cutting the Fourier function f at infinity, one can see that H ˙

⁻¹²

( C

+

)∩ H ˙

¹²

( C

+

) ∩ Hol( C

+

) is dense in H ˙

⁻¹²

( C

+

) ∩ Hol( C

+

). We can therefore consider a countable sequence (ϕ

_k

)

_k

in H ˙

⁻¹²

( C

+

) ∩ H ˙

¹²

( C

+

) ∩ Hol( C

+

) such that every function in H ˙

⁻¹²

( C

+

) ∩ Hol( C

+

) can be approximated by a subsequence of (ϕ

_k

)

_k

for the H ˙

⁻¹²

-norm.

Fix k ∈ N . Since (t 7→ ∂

t

u

n

(t))

n

and (t 7→ u

n

(t))

n

are uniformly bounded in H ˙

⁻¹²

( C

+

) and in H ˙

¹²

( C

+

) respectively, the sequence `

_n

(·, ϕ

k

) : t ∈ [−T, T ] 7→ (u

_n

(t), ϕ

_k

) is equicontin- uous and equibounded : for all n and t,

|∂

t

`

n

(t, ϕ

k

)|= |(∂

t

u

n

(t), ϕ

k

)|≤ k∂

t

u

n

(t)k

_˙

H⁻¹2(C+)

kϕ

k

k

_˙

H¹2(C+)

and

|`

n

(t, ϕ

k

)|= |(u

n

(t), ϕ

k

)|≤ ku

n

(t)k

_˙

H¹2(C+)

kϕ

k

k

_˙

H⁻¹2(C+)

.

Applying Ascoli’s theorem, for every k ∈ N , there is a subsequence (n

_p

)

_p

such that

(`

_np

(·, ϕ

_k

))

_p

converges in C([−T, T ], C ) to some continuous function `(·, ϕ

_k

) as p tends to

+∞. By a diagonal argument, we can use the same subsequence for all k ∈ N . Using a

second diagonal argument on a sequence of times (T

_n

)

_n

going to +∞, we can assume that

for all k, there exists `(·, ϕ

k

) ∈ C ( R , C ) such that for all T > 0, the sequence (`

_np

(·, ϕ

k

))

_p

converges in C([−T, T ], C ) to `(·, ϕ

_k

)|

_{[−T ,T]}

.

By density, ` extends to a bounded linear map ` ∈ C( R , ( ˙ H

⁻¹²

( C

+

) ∩ Hol( C

+

))

^∗

) (with the weak topology). Now, by duality, ` can be represented by u ∈ C( R , H ˙

¹²

( C

+

)∩ Hol( C

+

)) : for all ϕ ∈ H ˙

⁻¹²

( C

+

) ∩ Hol( C

+

),

`(t, ϕ) = (u(t), ϕ).

To conclude, by construction, for all T > 0, the sequence (`

_np

|

_{[−T ,T]}

)

_p

converges weakly to

`|

_{[−T ,T}]

in the space C([−T, T ], ( ˙ H

⁻¹²

( C

+

) ∩ Hol( C

+

))

^∗

), therefore (u

n

)

n

converges weakly to u in C([−T, T ], H ˙

¹²

( C

+

) ∩ Hol( C

+

)). Passing to the limit, we conclude that u is a global solution to the original equation (8) in the distribution sense.

We deduce that

d(u(t), M)

²

= inf

X∈R×T×R^∗₊

ku(t) − T

X

Qk

²_˙

H¹2(C+)

≤ inf

X∈R×T×R^∗₊

lim inf

n→+∞

ku

_n

(t) − T

_X

Qk

²

H˙¹2(C+)

.

Since X is not compact, this inequality is not sufficient if we want to apply inequality (14) to estimate d(u(t), M). In the following part, we construct a map t 7→ X

n

(t), such that for all t ∈ R , u

n

(t) is close to T

_Xn(t)

Q and (X

n

(t))

_n∈N

stays bounded, then use a compactness argument.

3.4 Modulation

Recall the notation. Fix u ∈ H

²

( C

+

) = ˙ H

¹²

( C

+

) ∩ Hol( C

+

), X = (s, θ, α) ∈ R × T × R

^∗+

, we denote by T

X

u the element of H

²

( C

+

) satisfying

T

X

u(z) := e

^iθ

αu(α

²

(z − s)), z ∈ C

+

. We write X

⁻¹

= (−s, −θ, α

⁻¹

) and

|X |:= |s|+|θ|+|log(α)|.

We have also defined

d(u, M) = inf

X=(s,θ,α)∈R×T×R^∗+

kT

X

u − Qk

_˙

H¹2(C+)

, as the distance of u to the orbit of Q

M = {T

X

Q | X ∈ R × T × R

^∗+

}.

We choose 0 < r < 1, and assume that ku

0

− Qk

_˙

H¹2(C+)

< r

²

. For n ≥ N large enough and K > 0, the regularized initial data u

ⁿ₀

satisfies δ(u

ⁿ₀

) < Kr

²

. Applying Proposition 1.5, there exist c

0

> 0 and r

0

> 0 such that if 0 < r < r

0

, then d(u

n

(t), M) < c

0

r for all t ∈ R .

We start from the observation that around time t = 0, one can choose X

n

(t) = (0, 0, 1) for all n ≥ N since ku

ⁿ₀

−Qk

_˙

H¹2(C+)

< c

0

r. By continuity, we know that ku

n

(t) − Qk

_˙

H¹2(C+)

≤ (1 + ε)c

0

r on some small time interval, which can be taken independently of n. Indeed,

ku

n

(t) − Qk

²_˙

H¹2(C+)

= ku

n

(t)k

²_˙

H¹2(C+)

+kQk

²_˙

H¹2(C+)

−2(u

n

(t), Q)

_˙

H¹2(C+)

= ku

ⁿ₀

k

²_˙

H¹2(C+)

+kQk

²_˙

H¹2(C+)

−2(u

n

(t), −iQ

z

)

L²(C+)

, therefore the derivative of ku

_n

(t) − Qk

²

H˙¹2(C+)

is bounded by

d

dt ku

n

(t) − Qk

²

H˙¹2(C+)

=

2(∂

t

u

n

(t), −iQ

z

)

_L2(C+)

≤ 2k∂

t

u

n

(t)k

_˙

H⁻¹²(C+)

k−iQ

z

k

_˙

H¹²(C+)

.

But we have already seen that k∂

t

u

_n

(t)k

_˙

H⁻¹2(C+)

is bounded independently of t and n, therefore there exists K > 0 such that for n ≥ N and t ∈ R

ku

n

(t) − Qk

²_˙

H¹2(C+)

≤ ku

ⁿ₀

− Qk

²_˙

H¹2(C+)

+K|t|

≤ (c

₀

r)

²

+ K|t|.

For fixed ε > 0, we conclude that ku

_n

(t)−Qk

_˙

H¹2(C+)

≤ (1+ε)c

₀

r as long as |t|≤

^(1+ε)_K²⁻¹

(c

₀

r)

²

. Set ε > 0 and t

1

:=

^(1+ε)_K²⁻¹

(c

0

r)

²

. Assume that at time t

0

, there exists a bounded sequence (X

_n⁰

)

n

in R × T × R

^∗+

such that for all n, ku

n

(t

0

) − T

X_n⁰

Qk

_˙

H¹2(C+)

< c

0

r. By the above method, one can show that ku

n

(t) − T

X_n⁰

Qk

_˙

H¹2(C+)

≤ (1 + ε)c

0

r on [t

0

− t

1

, t

0

+ t

1

].

Indeed, let v

n

:= T

_(X0

n)⁻¹

u

n

. The equation satisfied by u

n

is not invariant by scaling, but one can explicit which equation is satisfied by v

n

. Recall that if

u(z) = 1

√ 2π

Z

+∞

0

e

^izσ

f (σ) dσ, then

P e

ε,M

u(z) = 1

√ 2π Z

M

ε

e

^izσ

f (σ) dσ.

Write (X

_n⁰

) =: (s

⁰_n

, θ

⁰_n

, α

⁰_n

) and P f

₀ⁿ

:= P e

εn (α0

n)2, ¹

εn(α0 n)2

◦ P

0

, then v

n

= T

_(X0

n)⁻¹

u

n

satisfies i(v

_n

)

_t

= P f

₀ⁿ

(|v

_n

|

²

v

_n

),

moreover kv

n

(t

0

) − Qk

_˙

H¹2(C+)

< c

0

r. But we have the same inequalities as above k∂

t

v

n

(t)k

_˙

H⁻¹²(C+)

≤ kf P

0

(|v

n

|

²

v

n

)(t)k

_˙

H⁻¹²(C+)

≤ Ckf P

0

(|v

n

|

²

v

n

)(t)k

L⁴3(C+)

≤ C

⁰

k|v

n

|

²

v

_n

(t)k

L⁴3(C+)

≤ C

⁰

kv

n

(t)k

³_L4(C+)

.

Since kv

n

(t)k

L⁴(C+)

= ku

n

(t)k

L⁴(C+)

is uniformly bounded by conservation of the L

⁴

-norm, we conclude that kv

n

(t)− Qk

_˙

H¹2(C+)

= ku

n

(t) −T

X_n⁰

Qk

_˙

H¹2(C+)

≤ (1 +ε)c

₀

r as long as |t −t

0

|≤ t

₁

. We construct X

_n

as a piecewise C

¹

functional on R as follows. For k ∈ Z , X

_n

is constant on [kt

₁

, (k + 1)t

₁

[, equal to some X

_n^k

∈ R × T × R

^∗+

to be chosen. We first set X

_n⁻¹

= X

_n⁰

= (0, 0, 1). Then, at time t

_k

= kt

₁

, k ≥ 1, we use the fact that d(u

_n

(t

_k

), M) < r and choose X

_n^k

such that ku

_n

(t

_k

) − T

_Xk

n

Qk

_˙

H¹2(C+)

< c

₀

r. Then from the above paragraph, ku

n

(t) −T

_Xk

n

Qk

_˙

H¹2(C+)

≤ (1 + ε)c

₀

r on [t

_k

, t

_k

+ t

₁

]. We do a similar construction for negative times. The map X

n

satisfies

ku

n

(t) − T

_Xn(t)

Qk

_˙

H¹2(C+)

≤ (1 + ε)c

0

r, t ∈ R .

It remains to show that X

_n

is bounded independently of n on bounded intervals. In order to do so, it is enough to control the gap between X

_n^k−1

and X

_n^k

. By construction, at time t

_k

,

ku

n

(t

_k

) − T

_Xk−1 n

Qk

_˙

H¹2(C+)

≤ (1 + ε)c

₀

r and

ku

n

(t

k

) − T

_Xk n

Qk

_˙

H¹2(C+)

< c

0

r, therefore

kT

_X^k−1

n

Q − T

_Xk n

Qk

_˙

H¹2(C+)

≤ (2 + ε)c

₀

r.

Using the following Lemma, we conclude that if r is chosen small enough, then there exists a constant c

1

> 0 such that for all n ≥ N and k ∈ Z ,

|X

_n^k−1

(X

_n^k

)

⁻¹

|≤ c

₁

.

Lemma 3.6. For some constants c

₁

, r

₁

> 0 , the following holds. Let X ∈ R × T × R

^∗+

such that

kT

_X

Q − Qk

_˙

H¹2(C+)

≤ r

₁

. Then

|X |≤ c

1

.

Proof. Thanks to the invariance of the H ˙

¹²

-norm by symmetries, one can assume that X = (s, θ, α) with α ≥ 1 up to exchanging X and X

⁻¹

. We develop

kT

X

Q − Qk

²_˙

H¹2(C+)

= kT

X

Qk

²_˙

H¹2(C+)

+kQk

²_˙

H¹2(C+)

−2(T

X

Q, Q)

_˙

H¹2(C+)

= 2π − 2(T

X

Q, Q)

_˙

H¹2(C+)

. Now, recall that

Q(z) =

√ 2 z + i = 1

√ 2π Z

+∞

0

e

^izσ

f (σ) dσ with

f(σ) = −2i √ πe

^−σ

, and

kQk

²_˙

H¹2(C+)

= 1 2

Z

+∞

0

|f (σ)|

²

dσ = π.

With this notation, the function corresponding to T

_X

Q is g(σ) = −2i √

πe

^iθ

e

^−isσ

e

^−ασ2_α¹₂

, therefore

kT

_X

Q − Qk

²

H˙¹2(C+)

= 2π − 4πRe Z

+∞

0

e

^iθ

e

^−isσ

e

^−ασ2

1

α

²

e

^−σ

dσ

= 2π − 4πRe e

^iθ

α

²

is +

_α¹2

+ 1

! . Set α = 1 + β with β ≥ 0. We want to bound s and β. By assumption,

Re e

^iθ

is

^(1+β)₂ ²

+ 1 + β +

^β₂²

!

− 1

≤ r

²₁

2π =: δ

1

. Denote z :=

^eiθ

is^(1+β)2₂ +1+β+^β₂²

. The fact |Re(z) − 1|≤ δ

1

implies that |z|≥ Re(z) ≥ 1 − δ

1

, and if δ

1

< 1, that

1

|z| =

is (1 + β)

²

2 + 1 + β + β

²

2

≤ 1 1 − δ

1

. On the one hand, taking the real part,

1 + β + β

²

2 ≤ 1

1 − δ

1

.

Since β ∈ R

+

7→ 1 + β +

^β₂²

is strictly increasing and going to +∞ as β goes to +∞, there exists some constant c > 0 such that β ≤ c, or in other terms 0 ≤ log α ≤ log(1 + c). On the other hand, since β ≥ 0, the bound on the imaginary part implies that

|s|≤ 2 1 − δ

1

.

Using the Lemma, assume that 3c

0

r < r

1

and fix t ∈ R . We now know that (X

n

(t))

n

takes values in a compact set : up to extraction, one can assume that (X

n

(t))

n

converges to some X (t). Moreover, for all t ∈ R and n ∈ N , ku

n

(t)−T

_Xn(t)

Qk

_˙

H¹2(C+)

≤ (1+ε)c

0

r, therefore passing to the weak limit n → +∞ we conclude that ku(t) − T

_X(t)

Qk

_˙

H¹2(C+)

≤ (1 + ε)c

0

r.

Since ε > 0 can be taken arbitrarily small, we have proven the following reformulation of

Theorem 1.2.

Theorem 3.7. For some c

₀

> 0 and r

₀

> 0 , the following holds. Let r ≤ r

₀

and u

₀

∈ H ˙

¹²

( C

+

) ∩ Hol( C

+

) such that ku

₀

− Qk

_˙

H¹2(C+)

< r

²

. Then there exists a weak solution u ∈ C( R , H ˙

¹²

( C

+

) ∩ Hol( C

+

)) (with the weak topology) to equation (8)

( i∂

t

u = P

0

(|u|

²

u) u(t = 0) = u

0

, (t, z) ∈ R × C

+

such that for all t ∈ R,

d(u(t), M) ≤ c

₀

r.

4 Conditional orbital stability of the ground states Q β in the Schrödinger equation

We now consider the Schrödinger equation on the Heisenberg group (1) ( i∂

t

u − ∆

_H1

u = |u|

²

u

u(t = 0) = u

₀

, (t, x, y, s) ∈ R × H

¹

.

For β ∈ (β

_∗

, 1), we are interested in solutions with initial data u

0

∈ H ˙

¹

( H

¹

) satisfying ku

0

− p

1 − βQ

β

k

H˙¹(H¹)

< (1 − β)r.

Let u be an eventual solution, and set u(t, x, y, s) = p

1 − βU ((1 − β)t, x, y, s + βt), so that U is a solution to

i∂

t

U − ∆

_H1

+ βD

_s

1 − β U = |U |

²

U, (t, x, y, s) ∈ R × H

¹

. (15) The initial data U

0

satisfies

kU

0

− Q

_β

k

H˙¹(H¹)

< p 1 − βr.

There are two relevant conserved quantities for this equation : the energy E

_β

(V ) := 1

2 (− ∆

_H1

+ βD

s

1 − β V, V )

_L2(H¹)

− 1

4 kV k

⁴_L4(H¹)

, and the momentum

P (V ) := (D

s

V, V )

_L2(H¹)

, V ∈ H ˙

¹

( H

¹

).

Theorem 1.3 is equivalent to prove that if β is large, then one can construct a weak global solution U to equation (15) which stays close to the orbit of Q

β

for all times, which leads to the following reformulation.

Theorem 4.1. For some constants c

0

∈ R

^∗+

and r

0

∈ R

^∗+

, for all r ∈ (0, r

0

) , there exists β

^∗

(r) ∈ (0, 1) such that the following holds. Let β ∈ (β

^∗

(r), 1) and U

0

∈ H ˙

¹

( H

¹

) satisfying

• if U

0

∈ H ˙

¹

( H

¹

) ∩ V

₀⁺

:

kU

0

− Q

β

k

H˙¹(H¹)

< r

²

• in the general case :

kU

0

− Q

_β

k

H˙¹(H¹)

< p 1 − βr.

Then there exists a global weak solution U

_β

∈ C( R , H ˙

¹

( H

¹

)) (with the weak topology) to equation (15)

( i∂

_t

U

_β

−

^∆H1_1−β^+βDs

U

_β

= |U

_β

|

²

U

_β

U

β

(t = 0) = U

0

such that for all t ∈ R, U

_β

(t) is close to the orbit Q

_β

= {T

_X

Q

_β

| X ∈ R × T × R

^∗+

} of Q

_β

:

d(U

_β

(t), Q

_β

) ≤ c

₀

r.