Stability of Solitary Waves for a System of Nonlinear Schrödinger Equations With Three Wave Interaction

www.elsevier.com/locate/anihpc

Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction

M. Colin

, Th. Colin

, M. Ohta

aInstitut Mathématiques de Bordeaux, Université de Bordeaux and INRIA Bordeaux-Sud Ouest, EPI MC2, 351 cours de la libération, 33405 Talence Cedex, France

bDepartment of Mathematics, Faculty of Science, Saitama University, Saitama 338-8570, Japan Received 4 November 2008; received in revised form 26 January 2009; accepted 27 January 2009

Available online 24 February 2009

Abstract

In this paper we consider a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability of scalar solutions of the form(e^2iωtϕ,0,0),(0, e^2iωtϕ,0),(0,0, e^2iωtϕ), whereϕis a ground state of the scalar nonlinear Schrödinger equation.

©2009 Elsevier Masson SAS. All rights reserved.

Résumé

Dans cet article, on s’intéresse à un système de trois équations de Schrödinger qui modélise le phénomène d’amplification Raman dans les plasmas. On étudie la stabilité orbitale de solutions scalaires de la forme(e^2iωtϕ,0,0),(0, e^2iωtϕ,0),(0,0, e^2iωtϕ), oùϕ est l’état fondamental d’une équation de Schrödinger scalaire.

©2009 Elsevier Masson SAS. All rights reserved.

MSC:35Q55; 35A15; 35B35

Keywords:Solitary wave; Orbital stability; NLS

1. Introduction

In this paper, we investigate the stability properties of solitary waves for a Schrödinger-type system related to the Raman amplification in a plasma. The study of laser-plasma interactions is an active area of interest. The main goal is to simulate nuclear fusion in a laboratory. In order to simulate numerically these experiments we need some accurate models. The kinetic ones are the most relevant but very difficult to deal with for practical computations. The fluids ones like bifluid Euler–Maxwell system seem more convenient but still inoperative in practice because of the high frequency motion and the small wavelength involved in the problem. This is why we need some intermediate models which are reliable from a numerical point of view. In [5], a new set of equations describing nonlinear interaction between a laser beam and a plasma has been derived. This model describes the Raman process which is a nonlinear instability

* Corresponding author.

E-mail addresses:mcolin@math.u-bordeaux1.fr (M. Colin), colin@math.u-bordeaux1.fr (Th. Colin), mohta@mail.saitama-u.ac.jp (M. Ohta).

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.01.011

phenomenon. The physical situation is the following. When an incident laser field enters a plasma, it is backscattered by a Raman type process. These two waves interact to create an electronic plasma wave. The three waves combine to create a variation of the density of the ions which has itself an influence on the three preceedings waves. The system describing this phenomenon is composed by three Schrödinger equations coupled to a wave equation and reads in a suitable dimensionless form:

i(∂_t+v_C∂_y)+α₁∂_y²+α_2⊥

A_C=b²

2 nA_C−γ (∇ ·E)A_Re^−iθ, (1.1)

i(∂t+vR∂y)+β1∂_y²+β2_⊥

AR=bc

2 nAR−γ (∇ ·E^∗)ACe^iθ, (1.2)

(i∂_t+δ₁)E=b

2nE+γ∇

A^∗_RA_Ce^iθ

, (1.3)

(∂_t²−v_s²)n=a

|E|²+b|A_C|²+c|A_R|²

, (1.4)

whereA₀,A_RandEare complex vectors and are respectively the incident laser field, the backscattered Raman field and the electronic plasma-wave whereas n is the variation of the density of the ions and θ=(k₁y−ω₁t )where ω₁=k₁²δ₁. For a complete description of this model as well as a precise description of the physical coefficients, we refer to [5] and [6]. In [5], it is proved that system (1.1)–(1.4) is locally well-posed in suitable Sobolev spaces. It is then natural to investigate the global well-posedness theory. The existence of global solutions is still an open problem since no suitable conservation laws have been yet derived to handle this question. From this point of view, solitary waves play a crucial role and the study of their dynamics represents an important step toward the global well-posedness. The study of their behaviour is the main motivation of this paper. Unfortunately, we are not able to perform such analysis on (1.1)–(1.4). We will study a subsystem that includes the spacial dynamics and that is obtained as follows.

WritingE=F e^iθ and takingn=0, (1.1)–(1.4) reads i(∂_t+v_C∂_y)+α₁∂_y²+α_2⊥

A_C= −γ∇ ·F A_R−ik₁γ F A_R, (1.5)

i(∂t+vR∂y)+β1∂_y²+β2_⊥

AR=γ ik1F^∗AC−γ∇ ·F^∗AC, (1.6)

i∂_t+ω₁+δ₁+2ik1∂_y−δ₁k²₁

F=ik₁γ A^∗_RA_C+γ∇ A^∗_RA_C

. (1.7)

Now, in the right-hand side, we neglect the∇terms in front ofik₁(it is an envelope approximation). In the left-hand side, we neglect the longitudinal dispersion terms∂_y²in front of the transverse ones_⊥. We also use the dispersion relationω₁=k₁²δ₁in (1.7) and we study some stationary version∂_t=0. The system reads

(iv_C∂_y+α_2⊥)A_C= −γ ik₁F A_R, (iv_R∂_y+β_2⊥)A_R=γ ik₁F^∗A_C, (2ik1∂_y+δ_1⊥)F=ik₁γ A^∗_RA_C.

Let us introducew1=AC,w2=A^∗_Randw3=F, letting the coefficients to 1 (note thatvR<0 andvC>0) leads to (i∂y+_⊥)w1= −iγ w3w^∗₂,

(i∂_y+_⊥)w₂= −iγ w₃w^∗₁, (i∂y+_⊥)w3=iγ w1w2. Takingv_j=iw_j forj=1,2,3 gives

(i∂_y+_⊥)v₁= −γ v₃v₂^∗, (i∂_y+_⊥)v2= −γ v3v₁^∗, (i∂_y+_⊥)v₃= −γ v₁v₂.

In order to model nonlinear effects, we add some nonlinear terms and we switch to the usual notation using t as evolution variable instead of y. The system that we consider in this paper is the following simplified system of nonlinear Schrödinger equations:

i∂_tu₁= −u₁− |u₁|^p−1u₁−γ u₃u₂, (1.8)

i∂_tu₂= −u₂− |u₂|^p−1u₂−γ u₃u₁, (1.9)

i∂_tu₃= −u₃− |u₃|^p−1u₃−γ u₁u₂, (1.10)

for(t, x)∈R×R^N, whereN=1,2,3, 1< p <1+4/N,γ >0, andu₁,u₂andu₃are complex valued functions of (t, x)∈R×R^N. We assume furthermore thatγ >0. Indeed, the caseγ <0 is obtained by replacingu3by−u3in system (1.8)–(1.10). Note that the values ofNcorresponding to physical cases areN=1 or 2.

Here and hereafter, we putu=(u₁, u₂, u₃). We introduce the following quantities E(u) =

3 j=1

1

2∇u_j²_L2− 1

p+1u_j^p_L⁺p+¹1

−γ

R^N

u₁u₂u₃dx,

Q₁(u) = u₁²_L2+ u₃²_L2 and Q₂(u) = u₂²_L2+ u₃²_L2,

where(z)denotes the real part of a complex numberz. Note that (1.8)–(1.10) can be written as

∂_tu(t ) = −iE u(t )

, and that

E

e^iθ1u1, e^iθ2u2, e^i(θ1+θ2)u3

=E(u), E u(· +y)

=E(u),

for any(θ₁, θ₂)∈R²,y∈R^N andu∈H¹(R^N,C³). The Cauchy problem for (1.8)–(1.10) is globally well-posed in H¹(R^N,C³)(see [2]).

Proposition 1.LetN=1,2,3,1< p <1+4/N,γ >0, andu₀∈H¹(R^N,C³). Then there exists a unique global solutionu∈C(R;H¹(R^N,C³))to system(1.8)–(1.10)satisfyingu(0) = u₀. Furthermore, the solutionu(t ) satisfies the conservation laws

E u(t )

=E(u0), (1.11)

Q1

u(t )

=Q1(u0), Q2

u(t )

=Q2(u0) (1.12)

for allt∈R.

Forω >0, letϕ∈H¹(R^N)be a positive radial solution of

−v+2ωv− |v|^p−1v=0, x∈R^N. (1.13)

It is known thatϕis unique (see [15]). Then, we see that e^2iωtϕ,0,0

,

0, e^2iωtϕ,0

,

0,0, e^2iωtϕ

(1.14) solve (1.8)–(1.10). Note thatuω(t )=e^2iωtϕis a standing wave solution of the single nonlinear Schrödinger equation

i∂_tu= −u− |u|^p−1u, (t, x)∈R×R^N, (1.15)

and it is well known that forω >0,u_ω(t )is orbitally stable if 1< p <4/N. When 1+4/Np <1+4/(N−2), u_ω(t )is strongly unstable in the sense that for anyλ >1 the solutionw_λ(t )of (1.15) with initial dataw_λ(0)=λϕ blows up in finite time (see [1,3,18,21] and also [2,20,12,13]).

The purpose in this paper is to study the stability properties of the solitary wave solutions (1.14) for the coupled system (1.8)–(1.10). We first introduce the following definition.

Definition.We say that the solitary wave solution(e^2iωtϕ,0,0)of (1.8)–(1.10) isorbitally stableif for anyε >0 there existsδ >0 such that ifu₀∈H¹(R^N,C³)andu₀−(ϕ,0,0)_H¹< δ, then the solutionu(t ) of (1.8)–(1.10) withu(0) = u₀satisfies

sup

t0

inf

θ∈R,y∈R^Nu(t ) −

e^iθϕ(· +y),0,0

H¹< ε.

Otherwise, (e^2iωtϕ,0,0) is called orbitally unstable. The orbital stability and instability of (0, e^2iωtϕ,0) and (0,0, e^2iωtϕ)are defined analogously.

Here, we remark that when 1+4/Np <1+4/(N−2), the solitary wave solution(e^2iωtϕ,0,0)of (1.8)–(1.10) is strongly unstable, since for anyλ >1,(w_λ(t ),0,0)is a solution of (1.8)–(1.10) and blows up in finite time, where w_λ(t )is the blowup solution of (1.15) withw_λ(0)=λϕ. By the same reason, the solitary wave solutions(0, e^2iωtϕ,0) and (0,0, e^2iωtϕ)of (1.8)–(1.10) are also strongly unstable when 1+4/Np <1+4/(N−2). Thus, in what follows, we consider the case 1< p <1+4/N only. The main results of this paper are the following. The first one is concerned with the stability of the first two solitary waves(e^2iωtϕ,0,0)and(0, e^2iωtϕ,0), and reads as follows.

Theorem 2.Let1N3,1< p <1+4/N,γ >0,ω >0, and letϕbe the positive radial solution of (1.13). Then, the solitary wave solutions(e^2iωtϕ,0,0)and(0, e^2iωtϕ,0)of(1.8)–(1.10)are orbitally stable.

The proof of Theorem 2 is very classical and follows the argument introduced in [3], the key point being the variational characterization of the ground states(e^2iωtφ,0,0)and(0, e^2iωtφ,0)given in Lemma 4. The second result deals with the third wave (0,0, e^2iωtϕ). For this case, the analysis is much more delicate. Indeed, it is proved that there exists a critical valueγ^∗ofγ such that the solitary wave is stable for 0< γ < γ^∗, whereas the wave is unstable forγ > γ^∗. We notice that we do not prove any result forγ=γ^∗, and we leave this question as an open problem.

Theorem 3.Let1N3,1< p <1+4/N,ω >0, and letϕbe the positive radial solution of(1.13). Then, there exists a positive constantγ^∗satisfying the following.

(i) If0< γ < γ^∗, then the solitary wave solution(0,0, e^2iωtϕ)of (1.8)–(1.10)is orbitally stable.

(ii) Assume further thatN2andp >2. Ifγ > γ^∗, then the solitary wave solution(0,0, e^2iωtϕ)of (1.8)–(1.10)is orbitally unstable.

We give the outline of the proof. First of all, the variational method used to prove Theorem 2 does not apply to the case of Theorem 3(i), because the conservation laws (1.11)–(1.12) are not the suitable ones. To prove Theorem 3(i), we first introduce the actionS(see (3.1)) associated with system (1.8)–(1.10), so that the third solitary wave(0,0, e^2iωtϕ) is a critical point ofS. Following the general theory developed by Grillakis, Shatah and Strauss in [12], the proof is based on a carefull study of the linearized operator S . The key point is to show thatS is an elliptic operator on H¹(R^N,C³)which is a sufficient criteria to obtain the stability of(0,0, e^2iωtϕ). For that purpose, we decomposeS into two partsB1andB2(see (3.2)) whereB1depends onv₁andv₂whereasB2depends only onv₃. The operatorB2

is elliptic under some orthogonality conditions (see Lemma 5). In Lemma 7, we prove thatB1is elliptic ifγ is less than a critical valueγ^∗,γ^∗being closely related to a minimization problem (see Lemma 6). The end of the proof is classical and follows the arguments proposed in [12].

The proof of Theorem 3(ii) is more delicate. Since system (1.8)–(1.10) is symmetric with respect tou₁andu₂, we first perform the change of variable(4.1)to obtain system(4.2). The main idea is to construct an unstable direction zγ =(ζ_γ,0)that is ifvδis the solution of system(4.2)with the initial conditionvδ(0)=δzγ whereδcan be choosen arbitrarily small, thenvδ does not stay in the orbit of the third solitary wave(0,0, e^2iωtϕ). This contruction is based on a careful study of the spectrum of the two linearized operator L1 andL2 (see (4.3)–(4.4) for the definition of L1 andL2). More precisely, we show in Proposition 11 that if γ > γ^∗, then the upper bounds of the real part of spectras of L=(L2,L2)andL1are equal and that it is an eigenvalue ofLandL1. Then the unstable direction is constructed from the corresponding eigenvector. We have to notice that the critical valueγ^∗is that of Theorem 3(i).

This comes from the fact thatγ^∗is closely related to a minimization problem (see Lemma 7). For all 0< γ < γ^∗, we haveΛ_γ+ω >0 whereas ifγ > γ^∗thenΛ_γ+ω <0 (see Lemma 10). This explains why the behaviour of the third solitary wave(0,0, e^2iωtφ)is different between part (i) and part (ii) of Theorem 3.

Remarks.

(1) The additional assumption N 2 and p >2 in Theorem 3(ii) is related to the regularity of the nonlinearity

|u|^p−1u, and it is needed to estimate the nonlinear terms of the linearized equation (4.2) in Lemmas 12 and 13 essentially. Note that when N 2, the function z→ |z|^p−1zis not so smooth under our assumptionp <

1+4/N3. The nonlinear estimate for the caseN=2 and 2< p <3 in Lemma 13 is due to Kenji Nakanishi and Tetsu Mizumachi.

(2) The study of the form and the orbital stability of the general standing waves of system (1.8)–(1.9) is a difficult question to deal with. Our aim is to proceed step by step and so to begin with the particular cases described here.

The paper is organized as follows. In Section 2, we prove Theorem 2, the orbital stability of the first two solitary waves, using the variational method by Cazenave and Lions [3]. In Section 3, we prove the first part of Theorem 3 concerning the stability of the third solitary wave for γ small, whereas in Section 4, the orbital instability of this solitary wave is established for largeγ.

2. Proof of Theorem 2

In this section, we prove Theorem 2 using the variational method by Cazenave and Lions [3]. We put E₀(u)=1

2∇u²_L2− 1

p+1u^p_L⁺p+¹1.

We recall the variational characterization of the positive radial solutionϕto (1.13) (see [3]).

Lemma 4.LetN1,1< p <1+4/N,ω >0, and letϕbe the positive radial solution of (1.13). Then, E0(ϕ)=inf E0(v): v∈H¹

R^N

, vL²= ϕL²

.

Moreover, if{v_n} ⊂H¹(R^N)satisfies

v_nL²→ ϕL², E0(v_n)→E0(ϕ),

then there exist a subsequence{v_nk}and a sequence{(θ_k, y_k)}inR×R^Nsuch thate^iθkv_nk(· +y_k)→ϕinH¹(R^N).

Proof of Theorem 2. Suppose that(e^2iωtϕ,0,0)is not orbitally stable. Then, there exist a constantδ >0, a sequence {u_n(t )}of solutions of (1.8)–(1.10) and a sequence{t_n}in(0,∞)such that

u_n(0)−(ϕ,0,0)

H¹→0, (2.1)

inf

θ∈R,y∈R^Nun(tn)−

e^iθϕ(· +y),0,0

H¹=δ. (2.2)

We denoteu_n(t )=(u_n1(t ), u_n2(t ), u_n3(t )). By (2.1) and the conservation laws (1.11) and (1.12), we have u_n2(t_n)²

L²+u_n3(t_n)²

L²=u_n2(0)²

L²+u_n3(0)²

L²→0, (2.3)

u_n1(t_n)²

L²+u_n3(t_n)²

L²=u_n1(0)²

L²+u_n3(0)²

L²→ ϕ²_L2, (2.4)

E u_n(t_n)

=E u_n(0)

→E(ϕ,0,0)=E₀(ϕ). (2.5)

Since{u_n(t_n)}is bounded inH¹(R^N,C³), it follows from (2.3)–(2.5) that u_n2(t_n)

L²→0, u_n3(t_n)

L²→0, (2.6)

un1(tn)

L²→ ϕL², (2.7)

E0

un1(t_n) +1

2 3 j=2

∇u_nj(t_n)²

L²→E0(ϕ). (2.8)

Then, by (2.7), (2.8) and Lemma 4, we have E₀(ϕ)lim inf

n→∞ E₀ u_n1(t_n)

lim sup

n→∞ E₀ u_n1(t_n)

lim

n→∞

E0

un1(tn) +1

2 3 j=2

∇unj(tn)²

L²

=E0(ϕ),

which implies

E₀ u_n1(t_n)

→E₀(ϕ), (2.9)

∇u_n2(t_n)

L²→0, ∇u_n3(t_n)

L²→0. (2.10)

By (2.7), (2.9) and Lemma 4, there exist a subsequence{u_nk1(t_nk)}and a sequence{(θ_k, y_k)}inR×R^Nsuch that e^iθkun_k1(tn_k,· +yk)−ϕ

H¹→0. (2.11)

Moreover, by (2.6) and (2.10), we have u_n2(t_n)

H¹+u_n3(t_n)

H¹→0. (2.12)

By (2.11) and (2.12), we have inf

θ∈R,y∈R^Nu_nk(t_nk)−

e^iθϕ(· +y),0,0

H¹→0,

which contradicts (2.2). Hence,(e^2iωtϕ,0,0)is orbitally stable. The stability of(0, e^2iωtϕ,0)can be proved in the same way. 2

3. Proof of Theorem 3(i)

In this section, we prove the first part of Theorem 3. We putΦ=(0,0, ϕ). We regardΦ as a critical point of the functionalSdefined by

S(v) =E(v) +ωQ(v), (3.1)

Q(v) =Q1(v) +Q2(v) = v1²_L2+ v2²_L2+2v3²_L2

forv∈H¹(R^N,C³). Then we haveS(Φ)=0 and a direct computation gives S (Φ)v, v

=B1(v₁, v₂)+B2(v₃) (3.2)

forv=(v1, v2, v3)∈H¹(R^N,C³), where B1(v1, v2)=

2 j=1

∇vj²_L2+ωvj²_L2

−2γ

R^N

ϕv1v2dx

for(v1, v2)∈H¹(R^N,C²), and B2(v)= ∇v²_L2+2ωv²_L2−p

R^N

ϕ^p−1(v)²dx−

R^N

ϕ^p−1(v)²dx

for v= v+iv ∈H¹(R^N,C), where (v) denotes the imaginary part of v. The following positivity of B2 is well-known.

Lemma 5.LetN1,1< p <1+4/N,ω >0, and letϕbe the positive radial solution of (1.13). Then, there exists a constantδ2>0such thatB2(v)δ2v²_H1 for anyv∈H¹(R^N,C)satisfying(v, ϕ)_L2 =0,(v, iϕ)_L2 =0and (v,∇ϕ)_L2=0.

Proof. We put

L₊= −+2ω−pϕ^p−1, L₋= −+2ω−ϕ^p−1. Then one has

B2(v)= L₊v,v + L₋v,v,

and the result follows from Proposition 1 in [14] and Lemma 4.2 in [10] (see also [22, Section 2], [12,13] and [9, Section 3]). 2

In order to prove the positivity ofB1, we first need the following.

Lemma 6.LetN1,1< p <1+4/(N−2),ω >0, and letϕbe the positive radial solution of (1.13). Forγ >0, let

Λγ=inf Bγ(v): v∈H¹ R^N,C

,vL²=1 , B_γ(v)= ∇v²_L2−γ

R^N

ϕ|v|²dx.

Then we have

(i) −γϕL^∞Λγ(∇ϕ²_L2−γϕ³_L3)/ϕ²_L2for anyγ >0.

(ii) IfΛ_γ<0, then there existsχ_γ∈H¹(R^N)such thatχ_γL²=1and−χ_γ−γ ϕχ_γ =Λ_γχ_γ. (iii) If0< γ1< γ2andΛγ₁<0, thenΛγ₁> Λγ₂.

Proof. (i) Sinceϕ∈L^∞(R^N), we have Bγ(v)−γ

R^N

ϕ|v|²dx−γϕL∞v²_L2

for anyv∈H¹(R^N), which showsΛ_γ−γϕL^∞. Moreover, sinceϕis positive, we have Λ_γB_γ

ϕ ϕL²

∇ϕ²_L2−γϕ³_L3

ϕ²_L2

. (ii) See Lieb and Loss [16, Section 11.5].

(iii) SinceΛ_γ1 <0, by (ii) there existsχ_γ1 ∈H¹(R^N)such thatB_γ1(χ_γ1)=Λ_γ1 andχ_γ1L² =1. Then, since γ1< γ2, we haveΛγ₂Bγ₂(χγ₁) < Bγ₁(χγ₁)=Λγ₁. 2

We are now able to prove the positivity ofB1for smallγ. Lemma 7.Under the same assumptions as in Lemma6, let

γ^∗=inf{γ >0: Λ_γ<−ω}. (3.3)

Then,0< γ^∗<∞. Moreover, if0< γ < γ^∗, then there exists a constantδ1>0such thatB1(v1, v2)δ1(v1, v2)²_H1

for any(v₁, v₂)∈H¹(R^N,C²).

Proof. By Lemma 6, we have 0< ω

ϕL^∞ γ^∗∇ϕ²_L2+ωϕ²_L2

ϕ³_L3

,

which shows 0< γ^∗<∞. Moreover, if 0< γ < γ^∗, then by Lemma 6 and (3.3), we see thatΛ_γ+ω >0, and B_γ(v)+ωv²_L2(Λ_γ+ω)v²_L2

for anyv∈H¹(R^N,C), from which it follows that there existsδ₁>0 such thatB_γ(v)+ωv²_L2δ₁v²_H1 for any v∈H¹(R^N,C). Thus, we have

B1(v₁, v₂)= 2 j=1

B_γ(v_j)+ωv_j²_L2

+γ

R^N

ϕ|v₁−v₂|²dx

δ₁(v₁, v₂)²

H¹

for any(v₁, v₂)∈H¹(R^N,C²). 2

Using Lemmas 5 and 7, one can prove that under suitable restrictions, the linearized energyS (Φ)controls the H¹-norm.

Proposition 8.Let1N3,1< p <1+4/N,ω >0, and letΦ=(0,0, ϕ), whereϕis the positive radial solution of (1.13). Assume0< γ < γ^∗, whereγ^∗is the positive constant defined by(3.3). Then, there exists a constantδ >0 such that

S (Φ)v, v

δv²_H1

for anyv=(v₁, v₂, v₃)∈H¹(R^N,C³)satisfying(v₃, ϕ)_L2=0,(v₃, iϕ)_L2=0and(v₃,∇ϕ)_L2=0.

As a consequence, we state the following lemma which is at the heart of Theorem 3(i).

Lemma 9.Under the same assumption as in Proposition8, there exist positive constantsCandεsuch that E(v) −E(Φ)Cd(v, Φ) ²

for anyv∈H¹(R^N,C³)satisfyingd(v, Φ) < ε andQ(v) =Q(Φ), where we put d(v, Φ) = inf

θ∈R, y∈R^Nv−

0,0, e^iθϕ(· +y)

H¹.

Proof. This result follows from Proposition 8. Since it is classical, we refer to Theorem 3.4 in [12] for a complete proof (see also [14, Proposition 3] and [10, Lemma 2.1]). 2

Proof of Theorem 3(i). We repeat briefly the argument of the proof of Theorem 3.5 in [12]. We use the notation in Lemma 9. Assume that the conclusion of Theorem 3(i) is not true, then there exist a sequence of initial data{un(0)} inH¹,μ∈(0, ε)and{tn}in(0,+∞)such that

un(0)−Φ

H¹→0, (3.4)

d

un(tn), Φ

=μ. (3.5)

By the conservation laws (1.11)–(1.12), and by (3.4) and the continuity ofEandQonH¹, we have E

u_n(t_n)

=E u_n(0)

→E(Φ), (3.6)

Q u_n(t_n)

=Q u_n(0)

→Q(Φ). (3.7)

Here we putv_n=(Q(Φ)/Q(u_n(t_n)))^1/2u_n(t_n). Then, by (3.7), (3.5) and (3.6), we haveQ(v_n)=Q(Φ),d(v_n, Φ) < ε for largen, andE(v_n)→E(Φ). We then apply Lemma 9 to obtain

Cd(v_n, Φ)²E(v_n)−E(Φ)→0.

Thus, we haved(u_n(t_n), Φ)→0, and this contradicts (3.5). 2 4. Proof of Theorem 3(ii)

In this section, we prove the second part of Theorem 3. For that purpose, we make a special change of variables

u(t )=

e^iωtv₁(t ), e^iωtv₁(t ), e^2iωt

ϕ+v₂(t )

(4.1) in (1.8)–(1.10). Note that (1.8)–(1.10) is symmetric with respect tou1andu2. Then, the equations for(v1, v2)read

∂_tv₁=L1v₁+F₁(v₁, v₂), ∂_tv₂=L2v₂+F₂(v₁, v₂), (4.2) where the linear termsL1v₁andL2v₂are given by

L1v= −i(−v+ωv−γ ϕv), (4.3)

L2v= −i

−v+2ωv−p+1

2 ϕ^p−1v−p−1 2 ϕ^p−1v

, (4.4)

and the nonlinear termsF₁(v₁, v₂)andF₂(v₁, v₂)are given by

F₁(v₁, v₂)=i

γ v₁v₂+ |v₁|^p−1v₁ , F₂(v₁, v₂)=i

γ v₁²+ |ϕ+v₂|^p−1(ϕ+v₂)−ϕ^p−p+1

2 ϕ^p−1v₂−p−1 2 ϕ^p−1v₂

. We also write (4.2) as

∂_tv=Lv+F (v), (4.5)

wherev=(v₁, v₂),Lv=(L1v₁,L2v₂), andF (v)=(F₁(v₁, v₂), F₂(v₁, v₂)). It is convenient to let L1v=(−+ω+γ ϕ)v−i(−+ω−γ ϕ)v,

L2v=(−+2ω−ϕ^p−1)v−i(−+2ω−pϕ^p−1)v.

We considerL1andL2as linear operators inL²(R^N,C)with domainsD(L1)=D(L2)=H²(R^N,C). It is known that the spectrumσ (L2)ofL2satisfiesσ (L2)⊂iRifp <1+4/N (see, e.g., [4, Summary 2.5]). Furthermore, we define

Q= −+ω−γ ϕ, P = −+ω+γ ϕ.

Note that

Qv, v =B_γ(v)+ωv²_L2, v∈H¹ R^N

,

and the operatorP :H¹(R^N)→H⁻¹(R^N)is bounded, and there exist positive constantsc1andc2such that c₁v²_H1P v, vc₂v²_H1, v∈H¹

R^N .

Thus, the inverseP⁻¹:H⁻¹(R^N)→H¹(R^N)exists and is bounded, and there exist positive constantsc3andc4such that

c₃f²_H−1

f, P⁻¹f

c₄f²_H−1, f∈H⁻¹ R^N

. (4.6)

Lemma 10.Let μ_γ =inf

Qv, v

P⁻¹v, v: v∈H¹ R^N

\ {0}

, (4.7)

andγ^∗defined as in Lemma7. Ifγ > γ^∗, then−∞< μγ <0, and there existsξγ∈H²(R^N)\ {0}such that Qξ_γ=μ_γP⁻¹ξ_γ.

Proof. By definition (3.3) ofγ^∗and Lemma 6(iii), we see that ifγ > γ^∗, thenΛ_γ<−ω <0. Then, by Lemma 6(ii), there existsχ_γ ∈H¹(R^N)such thatχ_γL²=1 andQχ_γ, χ_γ =B_γ(χ_γ)+ωχ_γ²_L2=Λ_γ+ω <0. Thus, we have

μγ Qχ_γ, χ_γ P⁻¹χ_γ, χ_γ<0.

Moreover, forv∈H¹(R^N), by (4.6) we have γ

R^N

ϕ|v|²dxγϕL∞v²_L2γϕL∞vH⁻¹vH¹

1

2

∇v²_L2+ωv²_L2

+C

P⁻¹v, v

, (4.8)

which implies

0Qv, v +C

P⁻¹v, v ,

and thenμ_γ >−∞. Next, by the definition ofμ_γ, there exists a sequence{v_n}inH¹(R^N)such that P⁻¹v_n, v_n

=1, Qv_n, v_n →μ_γ.

Then, by (4.8), we see that 1

2

∇v_n²_L2+ωv_n²_L2

Qv_n, v_n +C

P⁻¹v_n, v_n ,

and so {vn}is bounded inH¹(R^N). Thus, there exist a subsequence of{vn}(still denoted byvn) andw∈H¹(R^N) such thatvn wweakly inH¹(R^N). Then, we have

R^N

ϕ|v_n|²dx→

R^N

ϕ|w|²dx

(see [16, Section 11.4]), and Qw, wlim inf

n→∞Qv_n, v_n =μ_γ<0, (4.9)

P⁻¹w, w

lim inf

n→∞

P⁻¹v_n, v_n

=1. (4.10)

By (4.9) and (4.10), we see thatw=0 and Qw, w

P⁻¹w, w=μγ.

Since w∈H¹(R^N)\ {0}attains the infimum in (4.7), it is easy to see that w satisfies Qw=μ_γP⁻¹w, andw∈ H²(R^N). 2

In the next proposition, we study the upper bound of the real part of spectra ofL1andL.

Proposition 11.Letμ_γ be in Lemma10. Ifγ > γ^∗, thenλ_γ:= √−μ_γ is a positive eigenvalue ofL1andL, and sup λ: λ∈σ (L)

=sup λ: λ∈σ (L1)

=λ_γ.

Proof. We use the same notation as in Lemma 10. Sinceμγ<0, we haveλγ= √−μγ>0. Let ηγ=λγP⁻¹ξγ, ζγ=ξγ+iηγ.

Then, we see that P η_γ =λ_γξ_γ,Qξ_γ = −λ_γη_γ, and ζ_γ ∈H²(R^N,C), so that L1ζ_γ =P η_γ −iQξ_γ =λ_γζ_γ and ζ_γ=0. Thus,λ_γ is a positive eigenvalue ofL1with eigenvectorζ_γ, and it is also an eigenvalue ofLwith eigenvector (ζ_γ,0). It is known that the essential spectrumσ_ess(L1) of L1 satisfies σ_ess(L1)⊂iR, and that σ (L1)\σ_ess(L1) consists of finitely many eigenvalues. Letλ∈C\iRbe an eigenvalue ofL1with eigenvectorw∈H²(R^N,C). Then, we have

−λ²w= −L²₁w=P Qw+iQPw.

SinceP andQare self-adjoint operators inL²(R^N), we have

−λ²w²_L2=(P Qw,w)_L2+(QPw,w)_L2+i(QPw,w)_L2−i(P Qw.w)_L2

=(Qw, Pw)_L2+(Pw, Qw)_L2+i(Pw, Qw)_L2−i(Qw, Pw)_L2

=(Qw, Pw)_L2+(Pw, Qw)_L2.

Since w=0, we see that λ²∈R. Moreover, since λ /∈iR, we have λ∈R\ {0}. Thus, byL1w=λw, we have Pw=λw andQw= −λw. Then, we havew=0 andP Qw= −λ²w. SinceP is invertible, we have Qw= −λ²P⁻¹w, and

−λ²= (Qw,w)_L2

(P⁻¹w,w)_L2 μ_γ.

Therefore, we have λ√−μ_γ =λ_γ, which shows sup{λ: λ∈σ (L1)} =λ_γ. Finally, sinceσ (L2)⊂iRif p <

1+4/N, we see that sup{λ: λ∈σ (L)} =sup{λ: λ∈σ (L1)} =λ_γ. 2