Orbital stability of standing waves for the nonlinear Schr¨ odinger equation with attractive delta potential and double power repulsive nonlinearity

Orbital stability of standing waves for the nonlinear Schr¨ odinger equation with attractive delta potential and double power repulsive nonlinearity

Ram´on G. Plaza

Institute of Applied Mathematics (IIMAS)

Universidad Nacional Aut´onoma de M´exico (UNAM) July 3, 2019

ZumbrunFest - Stability of Nonlinear Waves: Analysis and Computation. Institut Henri Poincar´e.

Joint work with: Jaime Angulo Pava(Univ. Sao Paulo) and C´esar Hern´andez Melo(Univ. Maring´a)

Sponsors:

• DGAPA-UNAM, program PAPIIT, grant no. IN-100318.

• FAPESP, S˜ao Paulo, processo 12543-4.

Table of contents

1. Introduction

2. Local and global well posedness for the NLS 3. Existence of standing waves

4. Orbital stability

2

Introduction

Nonlinear Schr¨ odinger equation (NLS)

NLS equation withattractive delta potentialandrepulsive double power nonlinearity:

iu_t+u_xx+Zδ(x)u+λ₁u|u|^p−1+λ2u|u|^2p−2= 0

• Unknown: u=u(x,t)∈C, forx,t∈R.

• Parameters: λ1≤0,λ2<0,Z>0,p>1.

• δ :H¹(R)→C,hδ,gi=g(0) (Dirac deltacentered atx= 0.)

• Linear interaction: ∂_x²+Zδ(x).

• Nonlinear term: λ1u|u|^p−1+λ2u|u|^2p−2.

• i²=−1.

3

Physical applications

• We recall that the general NLS model

iu_t+u_xx+V(x)u+f(|u|²)u= 0,

represents a trapping (wave-guiding) structure for light beams, induced by an inhomogeneity of the local refractive index.

• The delta-function termV(x) =Zδ(x) represents anarrow trap which is able to capture broad solitonic beams.

• It models a spatially localized point defectof the medium in which the soliton travels (localized attractive “impurity”).

• The non linear termf(x) =λ1x^(p−1)/2+λ2x^p−1is well known in optical media.

Standing waves

Standing wavesare solutions to the NLS model of the form u(x,t) =e^−iωtφ(x),

whereω∈Rand theprofile of the waveφ:R→Rsatisfies (

φ⁰⁰+Zδ(x)φ+ω φ+f(|φ|²)φ= 0,

φ∈H¹(R), (ODE)

wheref =f(·) is an arbitrary function satisfying f ∈C¹((0,+∞);R) with f(0) = 0,

f⁰(x)<0 for allx>0. (H_f)

Example: if 1<p<∞,λ1≤0 andλ2<0 then f(x) =λ1x^(p−1)/2+λ2x^p−1 satisfies (H_f).

5

Standing waves

Standing wavesare solutions to the NLS model of the form u(x,t) =e^−iωtφ(x),

whereω∈Rand theprofile of the waveφ:R→Rsatisfies (

φ⁰⁰+Zδ(x)φ+ω φ+f(|φ|²)φ= 0,

φ∈H¹(R), (ODE)

wheref =f(·) is an arbitrary function satisfying f ∈C¹((0,+∞);R) with f(0) = 0,

f⁰(x)<0 for allx>0. (H_f) Example: if 1<p<∞,λ1≤0 andλ2<0 then

f(x) =λ1x^(p−1)/2+λ2x^p−1 satisfies (H_f).

δ -interaction quantum operator

Theδ-interaction quantum operatorA_Z is defined as A_Z:=−∂_x²−Zδ(x) (A_Zf(x) =−f⁰⁰(x), x6= 0,

D(A_Z) ={f ∈H¹(R)∩H²(R\{0}) :f⁰(0+)−f⁰(0−) =−Zf(0)},

6

Orbital stability

Definition (orbital stability)

The standing wavee^−iωtφ_ω is orbitally stableby the flow of the NLS equation nH¹(R), if for anyε>0 there existsδ>0 such that if ku0−φωk_H1<δ then

inf

θ∈R

ku(t)−e^iθφωk_H1<ε, for all t∈R,

whereu(t) denotes the solution to the NLS equation with initial data u(0) =u0∈H¹(R). Otherwise,e^−iωtφω is said to beorbitally unstablein H¹(R).

Main theorem

Theorem (Angulo Pava, Hernandez Melo, P (2019))

Let1<p<∞,λ1≤0,λ2<0and Z>0 in the NLS equation. Then for all values ofω<0 satisfying

− pλ₁² (p+ 1)²λ2

<−ω<Z² 4

the family of standing wave solutions, u(x,t) =e^−iωtφ_ω, withφ_ω given by

φ_ω= α

−ω +

√v

−ωsinh

(p−1)√

−ω

|x|+R₁⁻¹ Z

2√

−ω

−_p−1¹

where v=ω β−α², areorbitally stablesolutions in H¹(R)under the flow of the NLS equation.

8

Previous results

• CaseZ = 0: Ohta (1995), double power nonlinearity;Maeda (2008), multiple power nonlinearity.

• Repulsive δ potential: Fukuizumi, Jeanjean (2008)

• Attractiveδ potential: Fukuizumi et al. (2008)

• Kaminaga, Ohta (2009): attractiveδ with repulsive single power nonlinearity.

Multibody interactions ofsame sign(repulsive, double power

nonlinearity) appear in the study ofBose-Einstein condensates: Brazhnyi, Konotop (2004); Belobo Belobo et al. (2014); Kamchatnov, Salerno (2009); Kamchatnov, Korneev (2010) (dark solitons).

Concentration-compactness method: Cazenave, Lions (1982)

• The Cauchy problem: The initial value problem associated to the NLS equation isglobally well-posedinH¹(R) for 1<p<+∞, λ1≤0,λ2<0 andZ>0.

• Existence of profile solution φω: for parameter valuesp,λ1,λ2,Z andω satisfying the assumptions there exists a profile solutionφω of the elliptic equation (ODE) (explicit construction).

• The stationary problem: The setAω ofnon-trivial solutionsof the equation for the profiles inH¹(R) will be characterized, via

uniqueness, by

Aω={v:G_ω⁰(v) = 0,v6= 0}={e^iθφω:θ∈R}, where

G_ω(v) =E(v)−ω 2kvk²_L2, E(v) :=1

2kv_xk²_L2−Z

2|v(0)|²− λ1

p+ 1kvk^p+1

L^p+1−λ2

2pkvk^2p

L^2p, forv∈H¹(R).

10

Concentration-compactness method (ii)

• The minimization problem: Forp,λ1,λ2,Z andω satisfying the assumptions of our main theorem, the quantity

m(ω) = inf{G_ω(v) :v∈H¹(R)}, satisfies the following properties:

(a) (boundedness below) −∞<m(ω)<0; and, (b) (compactness) any sequenceh_n∈H¹(R) such that

lim_n→∞G_ω(h_n) =m(ω) admits a subsequence converging to some h∈H¹(R) withG_ω(h) =m(ω).

Local and global well posedness

for the NLS

Preliminaries (i)

The formal expressionA_Z :=−∂_x²−Zδ(x) represents all the self-adjoint extensions (von Neumann theory) associated to the following closed, symmetric, densely defined linear operator:

( A₀=−∂_x²

D(A0) ={g∈H²(R) :g(0) = 0}.

More precisely, the quantum operatorA_Z =−∂_x²−Zδ(x) is given by (A_Zf(x) =−f⁰⁰(x) x6= 0,

D(A_Z) ={f ∈H¹(R)∩H²(R\{0}) :f⁰(0+)−f⁰(0−) =−Zf(0)}.

Preliminaries (ii)

Upon application of theFirst Representation Form Theorem(cf. Kato), it is possible to show that the associated form toA_Z is given by

F_Z[u,v] = Re Z +∞

−∞

u⁰(x)v⁰(x)dx−ZRe(u(0)v(0)),

where (u,v)∈D(F_Z) =H¹(R)×H¹(R). The bilinear form defined above is closed and bounded below. In addition, operatorA_Z =−∂_x²−Zδ(x) can be extended as a linear bounded operatoru→A_ZufromH¹(R) to H⁻¹(R). This action is defined by

hA_Zu,vi=F_Z[u,v], for u,v∈H¹(R).

13

Spectral properties of A

• Essential spectrum: Σ_ess(A_Z) = [0,+∞), for allZ∈R.

• Discrete spectrum:

Σ_dis(A_Z) = (

∅, Z≤0, −Z²/4 , Z>0, ForZ>0, Ψ_Z(x) =

qZ

2e^−Z2|x| is thenormalized eigenfunction

associated to theunique negative simple eigenvalue−Z²/4. In addition, the operatorsA_Z are bounded from below:

(A_Z≥ −Z²/4, Z>0, A_Z≥0, Z<0 Ref.: Albeverio, Gesztesy, Høegh-Kron (2005).

The Cauchy problem

Consider theCauchy problem,

( iu_t−A_Zu+ (λ₁|u|^p−1+λ2|u|^2p−2)u= 0, u(0) =u0∈H¹(R).

Ref. Cazenave, Courant LN, vol. 10 (2003).

15

Local well-posedness

Theorem (local well-posedness)

For any u0∈H¹(R)and Z∈R, there exists T>0and a unique solution u∈C([−T,T];H¹(R))∩C¹([−T,T];H⁻¹(R))to the NLS equation with u(0) =u₀such that

lim

t→T⁻ku(t)k_H1= +∞, if T <∞.

Moreover, the solution u(t)satisfies conservation of charge and energy:

ku(t)k_L2=ku0k_L2, E(u(t)) =E(u₀), for all t∈[−T,T], where the energy functional E is defined as

E(v) :=1

2kvxk²_L2−Z

2|v(0)|²− λ1

p+ 1kvk^p_L⁺¹_p+1−λ2

2pkvk²_L^p_2p, for v∈H¹(R).

Proof sketch

• The nonnegative self-adjoint operatorA ≡A_Z+β on the space X =L²(R), withβ =Z²/4 for Z>0 andβ= 0 forZ ≤0, and domainD(A) =D(A_Z), induces a norm

kuk²_X

A =ku_xk²_L2+ (β+ 1)kuk²_L2−Z|u(0)|², which is equivalent to the usual norm inH¹(R).

• The self-adjoint operator A_Z generates a strongly continuous group of unitary operatorsT(t)g=e^−itAZg.

• Duhamel integral u(t) =T(t)u₀+

Z t 0

T(t−s)(λ₁|u(s)|^p−1u(s) +λ₂|u(s)|^2p−2u(s))ds Direct application of Thm. 3.7.1 in Cazenave.

17

Remark: Gagliardo-Nirenberg interpolation inequality

For anyp>1,H¹(R)⊂L²(R)∩L^p+1(R)∩L^2p(R), inasmuch as the Gagliardo-Nirenberg interpolation inequality(cf. Leoni, 2017) yields

kuk_Lp+1≤C₁kuk^θ_L¹₂kuxk¹_L^−θ₂ ¹, kuk_L2p≤C2kuk^θ2

L²ku_xk^1−θ2

L² ,

with uniform constantsC_j>0 andθ1= (p+ 2)/(2p+ 2)∈(0,1), θ2= (p+ 1)/2p∈(0,1).

Conservation of charge/energy

• Conservation of charge:

d

dt||u(t)||²_L2= 2Re Z

R

u_tudx

= 2Re Z

R

−i(A_Zu)u+if(|u|²)|u|²dx= 0.

• Conservation of Energy: NLS equation can be written in the Hamiltonian form u_t=−iE⁰(u(t)), then

d

dtE(u(t)) = Re Z

R

E⁰(u(t))utdx= Re Z

R

i|E⁰(u(t))|²dx= 0.

19

Auxiliary bound

Let us define the followingC¹functional inH¹(R), R(v) :=1

2kv_xk²_L2−Z

2|v(0)|²−λ2

2pkvk^2p

L^2p =E(v) + λ1

p+ 1kvk^p+1

L^p+1.

Lemma (Auxiliary bound)

Let1<p<∞,λ1≤0,λ2<0and Z>0. Then there exists a uniform constant C=C(p,Z)>0such that

Z

2|v(0)|²≤R(v) +C, for all v∈H¹(R).

Proof sketch

By Sobolev and Young’s inequalities, for anyZ>0 there exists C₁=C₁(Z)>0 such that forv∈H¹(R)

Z|v(0)|²≤1

2kvxk²_L2+C₁kvk²_L2(−1,1). Apply H¨older’s and Young’s inequalities to estimate

kvk²_L2(−1,1)≤2^(p−1)/p _{Z 1}

−1

|v|^2pdx 1/p

≤δkvk^2p

L^2p(−1,1)+ 2C_δ, for anyδ >0. Since λ2<0, chooseδ =−λ2/(2pC₁)>0 to obtain

Z|v(0)|²≤1

2kv_xk²_L2−λ2

2pkvk^2p

L^2p+ 2C1C_δ.

21

Global well-posedness

Theorem (global well-posedness)

For every p>1, Z>0,λ1≤0andλ2<0the Cauchy problem is globally well-posed in H¹(R).

Proof. Letu∈C([−T,T];H¹(R))∩C¹([−T,T];H⁻¹(R)) be the local solution to the Cauchy problem fort∈(−T,T).

1

2ku_xk²_L2=E(u) +Z

2|u(t)|²+ λ1

p+ 1kuk^p+1

L^p+1+λ2

2pkuk^2p

L^2p

≤E(u(t)) +Z 2|u(t)|²

≤E(u(t)) +R(u(t)) +C

Thus, we arrive at 1

2ku_x(t)k²_L2≤E(u(t)) +R(u(t)) +C≤2E(u(t)) +C.

In view thatu conserves charge and energy we finally conclude that ku(t)k²_H1≤4E(u(0)) +ku(0)k²_L2+ 2C,

which implies, together with lim

t→T⁻ku(t)k_H1= +∞, if T<∞, that the time of existence of the solutionuisT = +∞.

23

Existence of standing waves

ODE problem

Recall theprofile equation (

φ⁰⁰+Zδ(x)φ+ω φ+f(|φ|²)φ= 0,

φ∈H¹(R), (ODE)

Hypothesis onf:

f ∈C¹((0,+∞);R) with f(0) = 0,

f⁰(x)<0 for allx>0. (H_f) φ∈H¹(R) is a solution in thedistributional senseif for everyχ∈H¹(R)

0 = Re _{Z +∞}

−∞ φ⁰(x)χ⁰(x)dx−Zφ(0)χ(0)−ω Z +∞

−∞

φ(x)χ(x)dx

− Z +∞

−∞

f(|φ|²(x))φ(x)χ(x)dx

.

24

Analysis of the (ODE) (i)

Lemma

Letφ∈H¹(R), withφ⁰⁰+Zδ(x)φ+ω φ+f(|φ(x)|²)φ(x) = 0in the distributional sense, then

φ∈C^j(R\{0})∩C(R), j= 1,2. (1a) φ⁰⁰(x) +ω φ(x) +f(|φ(x)|²)φ(x) = 0, for x6= 0. (1b)

φ⁰(0+)−φ⁰(0−) =−Zφ(0). (1c)

φ⁰(x),φ(x)→0, if |x| →∞. (1d)

|φ⁰(x)|²+ω|φ(x)|²+g(|φ(x)|²) = 0, for x6= 0. (1e) where g(s) =

Z s 0

f(s)ds.

Analysis of the (ODE) (ii)

Lemma

Let p>1,ω,λ1,λ2∈Rand Z∈R\ {0}. Letφ be a non-trivial solution to(1a) -(1e). Thenφ(x)6= 0for all x∈Rand|φ|>0. −φ is also a solution.

Lemma (Useful)

Let p>1,ω,λ1,λ₂∈Rand Z∈R\ {0}. Letφ be a non-trivial solution to(1a) -(1e). Then we have either one of the following:

(i) Im(φ(x)) = 0 for all x∈R; or,

(ii) there exists c∈Rsuch that Re(φ(x)) =c Im(φ(x))for all x∈R.

26

Explicit profile construction for ω 6= 0, Z = 0, λ

≤ 0, λ

< 0

By usingφ,φ⁰→0 asx→∞we obtain

[φ⁰]²+ω φ²+ 2α φ^p+1+β φ^2p= 0, withα=λ1/(p+ 1),β=λ2/p. Then,

φ(x) =

"

−α ω +

pω β−α²

ω sinh (p−1)√

−ωx

#−_p−1¹

,

is the profile of the standing wave solution provided that

− pλ₁² (p+ 1)²λ2

<−ω.

Explicit construction

The function φ1(x) :=φ(−|x| −d), −l<d,

satisfies all the properties of our first lemma except possibly the jump condition: φ⁰(0+)−φ⁰(0+) =−Zφ(0). If we consider

R1: (−l,∞)→(1,∞) the diffeomorphism defined by R1(d) =

pω β−α²cosh((p−1)√

−ωd) pω β−α²sinh((p−1)√

−ωd) +α .

then, we get d=R₁⁻¹

Z 2√

−ω

, with Z>0 and −ω<Z² 4 .

28

Profile existence theorem; case ω 6= 0

Theorem

Let p>1,λ1≤0,λ2<0and Z>0 in the NLS equation. Then for all values ofω<0satisfying

− pλ₁² (p+ 1)²λ2

<−ω<Z² 4

the familiy of standing wave solutions, u(x,t) =e^−iωtφω, withφω given by

φω= α

−ω +

√v

−ωsinh

(p−1)√

−ω

|x|+R₁⁻¹ Z

2√

−ω

⁻_p−1¹

are solutions to the NLS equation. Here v=ω β−α².

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� ϕω

Figure 1: Profile functionφ_ω=φ_ω(x) for parameter valuesω=−0.25,Z= 2, p= 3,λ1=λ2=−1.

30

Figure 2: Time evolution of the standing wave solutionu(x,t) =e^−iωtφω(x) withω=−0.25,Z= 2 and in the case of a quintic/cubic (p= 3), doubly repulsive (λ1=λ2=−1) nonlinearity.

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Figure 3: Dynamics in the (φ,φ⁰)-plane forf(x) =−x(1 +x²), that is, for λ1=λ2=−1 andω=−0.25, in the case of a quintic/cubic nonlinearity with p= 3.

32

Orbital stability

Critical points

Let us consider the functionalG_ω:H¹(R)→Rfor valuesω≤0, defined as

G_ω(v) =1

2kv_xk²_L2−Z

2|v(0)|²−ω

2kvk²_L2−1 2

Z ∞

−∞

g(|v(x)|²)dx, and theset of critical pointsassociated toG_ω as

Aω={v∈H¹(R) :G_ω⁰(v) = 0,v6= 0}.

Hereg =g(·) is the antiderivative off =f(·). Forφ∈Aω we have the relation

G_ω⁰(φ) =A_Zφ−ω φ−f(|φ|²)φ

33

Properties of the set of critical points (i)

Lemma

Let1<p<∞, Z>0and let ω∈Rbe such that ω+^Z₄² ≤0. Then the setAω is empty.

Proof. If there existsh∈H¹(R)\ {0} satisfyingG_ω⁰(h) = 0, then 0 = d

dsG(sh)

s=1, and since hA_Zh,hi ≥ −Z² 4 khk²_L2

for allh∈H¹(R), we then obtain

0 =kh_xk²_L2−Z|h(0)|²−ωkhk²_L2− Z ∞

−∞

f(|h(x)|²)|h(x)|²dx

≥ −(Z²/4 +ω)khk²_L2− Z _∞

−∞

f(|h(x)|²)|h(x)|²dx

≥ − Z ∞

−∞

f(|h(x)|²)|h(x)|²dx>0,

Properties of the set of critical points (ii)

Lemma

Let1<p<∞and Z∈R. Ifω>0then Aω=∅. Lemma

Letω∈Rand Z<0. Then, we have thatAω=∅.

Proofs by contradiction.

35

Properties of the set of critical points (ii)

Lemma

Let1<p<∞and Z∈R. Ifω>0then Aω=∅. Lemma

Letω∈Rand Z<0. Then, we have thatAω=∅. Proofs by contradiction.

Properties of the set of critical points (iii)

Lemma

Let p>1,λ1<0,λ2<0, Z >0andω such that −_(p+1)^pλ12₂

λ₂ <−ω<^Z₄². Considering f(x) =λ1x^(p−1)/2+λ₂x^p−1, then

Aω={e^iθφω:θ∈R}.

Proof. It is clear that for allθ∈R, e^iθφω∈Aω. Conversely, ifg∈Aω, theng satisfies all the necessary conditions to be a solution of the Euler-Lagrange equation and|g|>0. Goal: to show that there exist θ∈Rsuch that g(x) =e^iθφω(x) for allx∈R.

36

• φ_ω∈D(A_Z) is the unique positive solution of the Euler-Lagrange equation. Indeed, ifv∈H¹(R) is a positive solution thenv satisfies the IVP

(−ψ⁰⁰(x) =ω ψ(x) +f(ψ²(x))ψ(x) :=H(ψ(x)), x>0, ψ(0) =c0, ψ⁰(0) =−Zc0/2,

where c0 is the unique positive root of Φ_ω(c,Zc/2) =Z²

4 c²+ωc²+g(c²).

SinceHis locally Lipschitz around zero the IVP has a unique positive solution given byφω. Thus,v≡φω on (0,∞). Similar arguments show that v≡φω on (−∞,0). Hence,v(x) =φω(x) for allx∈R.

• Ifg(x) =e^iθ(x)ρ(x) then θ,ρ>0 satisfy

( θ⁰⁰ρ+ 2θ⁰ρ⁰= 0, x>0,

−(θ⁰)²ρ+ρ⁰⁰+ω ρ+f(|ρ|²)ρ= 0, x>0.

The first equation together with the boundedness of|g⁰|imply that g(x) =e^iθ0ρ(x) for allx∈(0,+∞). Then, from second equation and by the analysis above we necessarily have thatg(x) =e^iθ0φω(x) for allx∈(0,∞). A similar analysis shows thatg(x) =e^iθ1φω(x) for all x∈(−∞,0). Hence,

g(x) =e^iθ0φ_ω(x), for allx∈R.

38

The minimization problem

Let us suppose that 1<p<∞,Z>0,λ1≤0,λ2<0,ω is such that

− pλ₁² (p+ 1)²λ2

<−ω<Z²

4 , and f(x) =λ1x^(p−1)/2+λ₂x^p−1. Minimization problemassociated toG_ω:

m(ω) = inf{G_ω(v) :v∈H¹(R)}, and theminimal set

M(ω) ={u∈H¹(R) :G_ω(u) =m(ω)}.

The set of minima

Lemma

−∞<m(ω)<0and M(ω)⊂Aω.

Proof. First verify that−∞<m(ω). Write G_ω(v) =R(v)−ω

2kvk²_L2− λ1

p+ 1kvk^p+1

L^p+1, v∈H¹(R).

Then, by the auxiliary bound lemma we get G_ω(v)≥R(v)≥Z

2|v(0)|²−C≥ −C,

for allv∈H¹(R) and some uniformC>0, yielding−∞<m(ω).

40

To show thatm(ω)<0, letv(x) :=sh(x)∈H¹(R) withs>0 and where h(x) =e^−Z|x|2 is the eigenfunction of the operatorA_Z associated to the eigenvalue ^−Z₄². Therefore

G_ω(v) =−s² 2

Z² 4 +ω

khk²_L2−1 2

Z ∞

−∞

g(s²h²(x))dx.

Since −g(s²h²(x))<−f(s²)s²h²(x), G_ω(v)≤ −s²

2khk²_L2

Z²

4 +ω+f(s²) .

SinceZ²/4 +ω>0 and lim_s→0+f(s²) = 0 we conclude that there exists s₀>0 such thatZ²/4 +ω>−f(s²)>0 for 0<s≤s₀and so

G_ω(s0h)<0. Lastly, supposeM(ω)6=∅. Then since forh∈M(ω) we haveh6= 0 andG_ω⁰(h) = 0, then by previous Lemmata we obtain M(ω)⊂Aω.

Auxiliary result: Br´ ezis-Lieb lemma

A refinement of Fatou’s lemma:

Lemma (Br´ezis-Lieb, 1983)

Let2≤q<∞and{u_j} be a bounded sequence in L^q(R)such that u_j(x)→u(x)a.e. in x∈Ras j→∞. Then,

ku_jk^q_Lq− ku_j−uk^q_Lq− kuk^q_Lq→0, as j→∞.

42

Compactness

Lemma

Let h_n∈H¹(R)be such thatlimn→∞G_ω(h_n) =m(ω). Then there exists a subsequence h_nj and h∈H¹(R)such that lim_nj→∞h_nj=h in H¹(R)and G_ω(h) =m(ω).

Proof. First, notice that for allv∈H¹(R) I_ω(v) :=1

2kv_xk²_L2−ω 2kvk²_L2

=G_ω(v) +Z

2|v(0)|²+ λ1

p+ 1kvk^p+1

L^p+1+λ2

2pkvk^2p

L^2p. Sinceω<0, it follows that I_ω(v) is equivalent tokvk²_H1. From the fact thatλ1,λ2<0, we obtain

1

2kv_xk²_L2−ω

2kvk²_L2≤G_ω(v) +R(v) +C≤2G_ω(v) +C, for some uniformC>0.

Hence, it is clear that if the sequenceG_ω(h_n) converges then the sequenceh_n is bounded inH¹(R). Thus, there exists a subsequenceh_nj andh∈H¹(R) such that {h_nj}converges wealky tohinH¹(R). Since H¹(−1,1) is compactly embedded in C[−1,1], we deduce that

h_nj(0)→h(0). Thus,

m(ω)≤G_ω(h)≤lim inf

n_j→∞G_ω(h_nj) =m(ω), which implies thath∈M(ω).

Now, sinceh_nj*hweakly inH¹(R) we have that h_nj(x)→h(x) a.e. in x∈Rand also that

khn_j−hk²_L2+khk²_L2=khn_jk²_L2+o(1), k∂_xh_nj−h_xk²_L2+kh_xk²_L2=k∂_xh_njk²_L2+o(1), asn_j→∞.

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kh_njk_H1 uniformly bounded⇒ kh_njk_Lp+1 andkh_njk_L2p are uniformly bounded (by Gagliardo-Nirenberg interpolation inequalities). As h_nj(x)→h(x) a.e. inx∈R, by Br´ezis-Lieb lemma we get

kh_nj−hk^p+1

L^p+1+khk^p+1

L^p+1=kh_njk^p+1

L^p+1+o(1), kh_nj−hk^2p

L^2p+khk^2p

L^2p=kh_njk^2p

L^2p+o(1), asn_j→∞.

Combining yields

G_ω(h_nj−h) +G_ω(h) =G_ω(h_nj) +o(1), as n_j→∞.

From the def. ofI_ω,

0≤I_ω(h_nj−h)≤I_ω(h_nj−h)− λ1

p+ 1kh_nj−hk^p+1

L^p+1−λ2

2pkh_nj−hk^2p

L^2p

=G_ω(h_nj−h) +Z

2|hn_j(0)−h(0)|²

=G_ω(h_nj)−G_ω(h) +o(1),

inasmuch ash_nj(0)→h(0). This yieldsh_nj→hinH¹(R).

Characterization of the minimal set

Lemma

M(ω) =Aω={e^iθφ_ω:θ∈R}, whereφ_ω denotes the standing wave profile.

Proof. From the previous lemmas, we infer thatM(ω)6=∅. Then there existsh∈H¹(R) such thatG_ω(h) =m(ω), that is,h∈M(ω). Since M(ω)⊂Aω, h∈Aω. Thus, there existsθ0∈Rsuch thath=e^iθ0φω. Now, sinceφ_ω∈H¹(R) and

G_ω(φ_ω) =G_ω(h) =m(ω),

thenφω∈M(ω). This implies thatAω⊂M(ω). The other inclusion was already proved above.

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Proof of the main theorem

Suppose that the standing wavee^−iωtφω is orbitally unstable. Then there existsε0>0, a sequence{h_n(t)}of solutions of the NLS equation and a sequencet_n>0, such that

n→∞limkh_n(0)−φωk_H1= 0, (2a)

θ∈infR

kh_n(t_n)−e^iθφωk_H1≥ε0. (2b) SinceG_ω is conserved by the flow of the NLS equation, we get that G_ω(h_n(t_n)) =G_ω(h_n(0)) for alln∈N. Then (2a) and continuity ofG_ω yield

n→∞limG_ω(h_n(t_n)) =G_ω(φ_ω) =m(ω).

Henceforth, from the former results there exists a subsequenceh_nj such thath_nj(t_nj)→hwithG_ω(h) =m(ω). Thenh∈Aω andh=e^iθ0φω for someθ0∈R. Therefore,

nlimj→∞h_nj(t_nj) =e^iθ0φω, inH¹(R), which contradicts

θ∈infR

kh_n(t_n)−e^iθφωk_H1≥ε0. Hence, we conclude thate^−iωtφω is orbitally stable.

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Reference

• J. Angulo Pava, C. A. Hern´andez Melo, R. G. P.,J. Math. Phys.

(2019), in press.

Happy birthday Kevin!

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