# Standing Wave Solutions for the Discrete Coupled Nonlinear Schrödinger Equations with Unbounded Potentials

Authors: Meihua Huang; Zhan Zhou
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We demonstrate the existence of standing wave solutions of the discrete coupled nonlinear Schrödinger equations with unbounded potentials by using the Nehari manifold approach and the compact embedding theorem. Sufficient conditions are established to show that the standing wave solutions have both of the components not identically zero.


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## Body

## 1. Introduction

Consider the coupled discrete Schrödinger system(1)idundt=-(𝒜u)n+b1nun-a1|un|2un-a3|vn|2un,idvndt=-(𝒜v)n+b2nvn-a2|vn|2vn-a3|un|2vn,
where ai>0, {bjn} are real valued sequences, i=1,2,3,andj=1,2. 𝒜 is the discrete Laplacian operator defined as (𝒜u)n=un+1+un-1-2un.The system (1) could be viewed as the discretization of the two-component system of time-dependent nonlinear Gross-Pitaevskii system (see [1] for detail)
(2)iℏ∂tu=-ℏ22mΔu+b1(x)u-a1|u|2u-a3|v|2u,iℏ∂tv=-ℏ22mΔv+b2(x)v-a2|v|2v-a3|u|2v.In this paper, we will study the standing wave solutions of (1), that is, solutions of the form
(3)un=exp(-iω1t)ϕn,vn=exp(-iω2t)ψn,n∈ℤ,
where the amplitude ϕnandψn are supposed to be real. Inserting the ansatz of the standing wave solutions (3) into (1), we obtain the following equivalent algebraic equations:
(4)-(𝒜ϕ)n-ω1ϕn+b1nϕn-a1|ϕn|2ϕn-a3|ψn|2ϕn=0,-(𝒜ψ)n-ω2ψn+b2nψn-a2|ψn|2ψn-a3|ϕn|2ψn=0.Since Bose-Einstein condensation for a mixture of different interaction atomic species with the same mass was realized in 1997 (see [2]), this stimulated various analytical and numerical results on the standing wave solutions of the system (2). The discrete nonlinear Schrödinger equations (DNLS) have a crucial role in the modeling of a great variety of phenomena, ranging from solid-state and condensed-matter physics to biology. During the last years, there has been a growing interest in approaches to the existence problem for standing waves. We refer to the continuation methods in [3, 4], which have been proved powerful for both theoretical considerations and numerical computations (see [5]), to [6], which exploits spatial dynamics and centre manifold reduction, and to the variational methods in [7–11], which rely on critical point techniques (linking theorems and Nehari manifold).We noticed that most works on the existence of standing waves solutions are for single discrete nonlinear Schrödinger equation, and less is known for discrete nonlinear Schrödinger system. In the recent paper [12], the authors considered the standing wave solutions of the following system:
(5)idundt=-(𝒜u)n+b1nun-c1nvn-a1|un|2un-a3|vn|2un,idvndt=-(𝒜v)n+b2nvn-c2nun-a2|vn|2vn-a3|un|2vn,
which is more general than the system (1). However, they make a mistake to obtain the equivalent algebraic equations because ω1 may be different from ω2. Hence, there are two ways to correct this mistake. The first method is to study the special standing wave solutions (3) of the system (5) with ω1=ω2. The second method is to study the standing wave solutions (3) of the system (5) with c1n≡0andc2n≡0,n∈ℤ. In this paper, we consider the second method. By the way, the proof of the main results in [12] is also not fully corrected.The paper is organized as follows. In Section2, we introduce some preliminaries and a discrete version of compact embedding theorem. Some key lemmas on the Nehari manifold are proved in Section 3. In Section 4, the main results are stated and proved.


## 2. Preliminaries

In this section we describe the functional setting needed for the treatment of the infinite nonlinear system (4). We first introduce a compact embedding theorem.Consider the real sequence spaces(6)lp={ϕ={ϕn}n∈ℤ:∥ϕ∥lp=(∑n∈ℤ‍|ϕn|p)1/p<∞,ϕn∈ℝ,∀n∈ℤ(∑n∈ℤ‍|ϕn|p)1/p}.
Between lp spaces the following elementary embedding relation holds:
(7)lq⊂lp,∥ϕ∥lp≤∥ϕ∥lq,1≤q≤p≤∞.
For the case p=2, we need the usual Hilbert space of l2, endowed with the real scalar product
(8)(ϕ,ψ)=∑n∈ℤ‍ϕnψn,ϕ,ψ∈l2.
Let us point out that the spectrum of -𝒜 in l2 coincides with the interval [0,4]. Obviously, we have
(9)0≤(-𝒜ϕ,ϕ)≤4∥ϕ∥l22,∀ϕ∈l2.
Assume that the potential Vi={bin}n∈ℤ,i=1,2, satisfies(10)lim|n|→∞bin=∞,i=1,2.
Without loss of generality we assume that Vi≥1,i=1,2; that is bin≥1 for n∈ℤ,i=1,2. Let
(11)Li=-𝒜+Vi,i=1,2,
which are self-adjoint operators defined on l2, and
(12)Ei={ϕ∈l2:Li1/2ϕ∈l2},∥ϕ∥Ei=∥Li1/2ϕ∥l2,i=1,2.The following lemma can be found in [9].Lemma 1.
IfVi,i=1,2, satisfy the condition (10), then for any 2≤p≤∞, E1 and E2 are compactly embedded into lp and denote the best embedding constant αp=max∥ϕ∥lp=11/∥ϕ∥E1 and βp=max∥ϕ∥lp=11/∥ϕ∥E2, respectively. Furthermore, the spectra σ(L1) and σ(L2) are discrete, respectively.By (11), (4) becomes
(13)L1ϕn-ω1ϕn-a1|ϕn|2ϕn-a3|ψn|2ϕn=0,L2ψn-ω2ψn-a2|ψn|2ψn-a3|ϕn|2ψn=0.Now we can define the action functional(14)J(ϕ,ψ)=12((L1-ω1)ϕ,ϕ)+12((L2-ω2)ψ,ψ)-14∑n∈ℤ‍(a1ϕn4+a2ψn4+2a3ϕn2ψn2).
By Lemma 1, it follows that the action functional J(ϕ,ψ)∈C1(E1×E2,ℝ) and (13) corresponds to J′(ϕ,ψ)=0. So we define
(15)I(ϕ,ψ)=(J′(ϕ,ψ),(ϕ,ψ))=((L1-ω1)ϕ,ϕ)+((L2-ω2)ψ,ψ)-∑n∈ℤ‍(a1ϕn4+a2ψn4+2a3ϕn2ψn2),
and the Nehari manifold
(16)N={(ϕ,ψ)∈E1×E2:I(ϕ,ψ)=0,(ϕ,ψ)≠0}.


## 3. Some Lemmas on the Nehari Manifold

Let(17)λi=inf{σ(Li)},i=1,2.To prove the main results, we need some lemmas on the Nehari manifold.Lemma 2.
Assume thatω1<λ1,ω2<λ2, and (10) holds. Then the Nehari manifold N is nonempty in E1×E2. Furthermore, for (ϕ,ψ)∈N, J(tϕ,tψ) attains a unique maximum point at t=1.Proof.
First we show thatN≠∅.
From (15) and (16), we rewrite
(18)J(ϕ,ψ)=12(∥ϕ∥E12-ω1∥ϕ∥l22)+12(∥ψ∥E22-ω2∥ψ∥l22)-14∑n∈ℤ‍(a1ϕn4+a2ψn4+2a3ϕn2ψn2),(19)I(ϕ,ψ)=∥ϕ∥E12-ω1∥ϕ∥l22+∥ψ∥E22-ω2∥ψ∥l22-∑n∈ℤ‍(a1ϕn4+a2ψn4+2a3ϕn2ψn2).
Let (ϕ,ψ)∈(E1-{0})×(E2-{0}); then by (19)
(20)I(tϕ,tψ)=t2(∥ϕ∥E12-ω1∥ϕ∥l22)+t2(∥ψ∥E22-ω2∥ψ∥l22)-t4∑n∈ℤ‍(a1ϕn4+a2ψn4+2a3ϕn2ψn2)=t2(∑n∈ℤ‍∥ϕ∥E12-ω1∥ϕ∥l22+∥ψ∥E22-ω2∥ψ∥l22-t2∑n∈ℤ‍(a1ϕn4+a2ψn4+2a3ϕn2ψn2)).
Notice that ∥ϕ∥E12-ω1∥ϕ∥l22≥(λ1-ω1)∥ϕ∥l22>0 and ∥ψ∥E22-ω2∥ψ∥l22≥(λ2-ω2)∥ψ∥l22>0; by (20), we see that I(tϕ,tψ)>0 for t>0 small enough and I(tϕ,tψ)<0 for t>0 large enough. As a consequence, there exists t0>0 such that I(t0ϕ,t0ψ)=0; that is, (t0ϕ,t0ψ)∈N.
LetF(t)=J(tϕ,tψ), (ϕ,ψ)∈N. Computing the derivative of F, we have
(21)F′(t)=t(1-t2)∑n∈ℤ‍(a1ϕn4+a2ψn4+2a3ϕn2ψn2).
This shows that t=1 is a unique maximum point. The proof is completed.Lemma 3.
Assume thatω1<λ1,ω2<λ2, and (10) holds. Then there exists η>0 such that J(ϕ,ψ)≥η, for all (ϕ,ψ)∈N.Proof.
Sinceλ1 is the smallest eigenvalue of E1 and λ2 is the smallest eigenvalue of E2, from the definition of the constant αp and βp, we get λ1=1/α22 and λ2=1/β22. For any (ϕ,ψ)∈N, we have
(22)∥ϕ∥E12-ω1∥ϕ∥l22+∥ψ∥E22-ω2∥ψ∥l22=∑n∈ℤ‍(a1ϕn4+a2ψn4+2a3ϕn2ψn2)≤a*(∥ϕ∥l42+∥ψ∥l42)2≤a*(α42∥ϕ∥E12+β42∥ψ∥E22)2≤a*γ22(∥ϕ∥E12+∥ψ∥E22)2,
where a*=max{a1,a2,a3} and γ2=max{α42,β42}.
Let(23)γ1=min{1,1-ω1λ1,1-ω2λ2}.
By (22), it is easy to see that
(24)γ1(∥ϕ∥E12+∥ψ∥E22)≤a*γ22(∥ϕ∥E12+∥ψ∥E22)2,
and this implies that
(25)∥ϕ∥E12+∥ψ∥E22≥γ1a*γ22.
Moreover, we have
(26)J(ϕ,ψ)=J(ϕ,ψ)-14I(ϕ,ψ)=14(∥ϕ∥E12-ω1∥ϕ∥l22+∥ψ∥E22-ω2∥ψ∥l22)≥γ14(∥ϕ∥E12+∥ψ∥E22)≥γ124a*γ22.
Let η=γ12/(4a*γ22); then we get J(ϕ,ψ)≥η, for all (ϕ,ψ)∈N. The proof is completed.


## 4. Main Results

Now we state our main results in this paper as follows.Theorem 4.
Assume thatω1<λ1,ω2<λ2, and (10) holds. Then system (13) has a nontrivial solution in E1×E2; that is, system (1) has a nontrivial standing wave solution.In order to prove Theorem4, we consider the following constrained minimization problem:
(27)d≡inf(ϕ,ψ)∈NJ(ϕ,ψ).
From the standard variational method, the proof of Theorem 4 is changed into finding a solution to the minimization problem (27). Now we are ready to prove Theorem 4.Proof.
Letd be given by (27). By Lemma 2, N is nonempty and there exists a sequence {(ϕ(k),ψ(k))}⊂N such that
(28)d=limk→∞J(ϕ(k),ψ(k)).
By Lemma3, d>0 and d≤D=maxk{J(ϕ(k),ψ(k))}<∞. By virtue of (26), we have
(29)∥ϕ(k)∥E12+∥ψ(k)∥E22≤4γ1J(ϕ(k),ψ(k))≤4Dγ1<∞.
Thus, sequences {ϕ(k)} and {ψ(k)} are bounded in Hilbert spaces E1 and E2, respectively. Therefore, there exist subsequences of {ϕ(k)} and {ψ(k)} (denoted by itself) that weakly converge to some ϕ*∈E1 and ψ*∈E2, respectively. By Lemma 1, we get, for any 2≤p≤∞,
(30)limk→∞ϕ(k)=ϕ*,limk→∞ψ(k)=ψ*,inlp.
By virtue of (15) and (16), we have
(31)J(ϕ(k),ψ(k))=12(∥ϕ(k)∥E12-ω1∥ϕ(k)∥l22)+12(∥ψ(k)∥E22-ω2∥ψ(k)∥l22)-14∑n∈ℤ(a1(ϕn(k))4+a2(ψn(k))4+2a3(ϕn(k))2(ψn(k))2)‍=14∑n∈ℤ‍(a1(ϕn(k))4+a2(ψn(k))4+2a3(ϕn(k))2(ψn(k))2).
First, we claim that
(32)limk→∞∑n∈ℤ‍(a1(ϕn(k))4+a2(ψn(k))4+2a3(ϕn(k))2(ψn(k))2)=∑n∈ℤ‍(a1(ϕn*)4+a2(ψn*)4+2a3(ϕn*)2(ψn*)2).
According to (30), it suffices to show that
(33)limk→∞∑n∈ℤ‍(ϕn(k))2(ψn(k))2=∑n∈ℤ‍(ϕn*)2(ψn*)2.
In fact,(34)|∑n∈ℤ‍(ϕn(k))2(ψn(k))2-∑n∈ℤ‍(ϕn*)2(ψn*)2|≤∑n∈ℤ‍|ϕn(k)-ϕn*||ϕn(k)+ϕn*|(ψn(k))2+∑n∈ℤ‍|ψn(k)-ψn*||ψn(k)+ψn*|(ϕn*)2.
Thus Hölder inequality and (30) imply the (33) holds.
Next, we show that(ϕ*,ψ*)∈N and J(ϕ*,ψ*)=d.
SinceE1 and E2 are Hilbert spaces, by (32) we have
(35)∥ϕ*∥E12+∥ψ*∥E22=∥weak-limk→∞ϕ(k)∥E12+∥weak-limk→∞ψ(k)∥E22≤liminfk→∞∥ϕ(k)∥E12+liminfk→∞∥ψ(k)∥E22≤liminfk→∞(∥ϕ(k)∥E12+∥ψ(k)∥E22)=liminfk→∞(∑n∈ℤ‍(a1(ϕn(k))4+a2(ψn(k))4+2a3(ϕn(k))2(ψn(k))2)+ω1∥ϕ(k)∥l22+ω2∥ψ(k)∥l22∑n∈ℤ(ψn(k))4‍)=∑n∈ℤ‍(a1(ϕn*)4+a2(ψn*)4+2a3(ϕn*)2(ψn*)2)+ω1∥ϕ*∥l22+ω2∥ψ*∥l22,
which implies I(ϕ*,ψ*)=∥ϕ*∥E12-ω1∥ϕ*∥l22+∥ψ*∥E22-ω2∥ψ*∥l22-∑n∈ℤ‍(a1(ϕn*)4+a2(ψn*)4+2a3(ϕn*)2(ψn*)2)≤0. Through a similar argument to the proof of Lemma 2, we know that I(tϕ*,tψ*) is positive as t is small enough. Therefore there exists t*∈(0,1] such that I(t*ϕ*,t*ψ*)=0 which implies (t*ϕ*,t*ψ*)∈N. Thus we have J(t*ϕ*,t*ψ*)=(1/4)W(t*) and by (32), W(1)=4d, where
(36)W(t)=t4∑n∈ℤ‍(a1(ϕn*)4+a2(ψn*)4+2a3(ϕn*)2(ψn*)2).
Clearly, W(t) is strictly increasing on 0<t<∞. Therefore by (27),
(37)d≤J(t*ϕ*,t*ψ*)=14W(t*)≤14W(1)=d.
This implies that t*=1 and J(ϕ*,ψ*)=d.
Finally, we will prove(ϕ*,ψ*) is a nontrivial solution to system (13).
Since(ϕ*,ψ*) is an energy minimizer on Nehari manifold N, there exists a Lagrange multiplier Λ such that
(38)(J′(ϕ*,ψ*)+ΛI′(ϕ*,ψ*),(ϕ,ψ))=0,
for any (ϕ,ψ)∈E1×E2. Let (ϕ,ψ)=(ϕ*,ψ*) in (38). (J′(ϕ*,ψ*),(ϕ*,ψ*))=I(ϕ*,ψ*)=0 implies that
(39)Λ(I′(ϕ*,ψ*),(ϕ*,ψ*))=0,
but
(40)(I′(ϕ*,ψ*),(ϕ*,ψ*))=2((L1-ω1)ϕ*,ϕ*)+2((L2-ω2)ψ*,ψ*)-4∑n∈ℤ‍(a1(ϕn*)4+a2(ψn*)4+2a3(ϕn*)2(ψn*)2)=-2∑n∈ℤ‍(a1(ϕn*)4+a2(ψn*)4+2a3(ϕn*)2(ψn*)2)<0.
Thus, Λ=0 and
(41)(J′(ϕ*,ψ*),(ϕ,ψ))=0,
for any (ϕ,ψ)∈E1×E2. Take (ϕ,ψ)=(e(k),0) and (ϕ,ψ)=(0,e(k)) in (41) for k∈ℤ, where
(42)en(k)={1,n=k,0,n≠k.
We see that J′(ϕ*,ψ*)=0. Thus, (ϕ*,ψ*) is a nontrivial solution to system (13). The proof is completed.By Theorem4, the system (1) has a nontrivial solution. However, it is uncertain if two components of this solution are nonzero. Therefore, we want to find solutions of the system (1) which have both of the components not identically zero. In order to achieve this goal, we consider the system (1) with b1n=b2n,n∈ℤ; that is,
(43)idundt=-(𝒜u)n+b1nun-a1|un|2un-a3|vn|2un,idvndt=-(𝒜v)n+b1nvn-a2|vn|2vn-a3|un|2vn.
In system (43), we know that L1=L2, where Li,i=1,2, is given by (11). By the definition of Ei,i=1,2, in Section 2 of this paper, we obtain that E1=E2. Hence, λ1=λ2. For the sake of simplicity, we let L1=L2=L, E1=E2=E, and λ1=λ2=λ. The notations in Section 2, such as J(ϕ,ψ),I(ϕ,ψ),andN are the same.Now, we give the second result of this paper as follows.Theorem 5.
Assume thatω1<λ,ω2<λ, a3>max{a1,a2,((λ-ω2)/(λ-ω1))a1,((λ-ω1)/(λ-ω2))a2}, and (10) holds. Then system (43) has a nontrivial standing wave solution (ϕ~,ψ~) in E×E with ϕ~≠0 and ψ~≠0.Proof.
By Theorem4, we know that system (43) has a nontrivial standing wave solution (ϕ~,ψ~) in E×E.
Now we will prove thatϕ~≠0 and ψ~≠0.
Since(ϕ~,ψ~)∈N, we know that (ϕ~,ψ~)≠(0,0). If one of the components (ϕ~,ψ~), say ψ~=0, then ϕ~≠0. For ϵ small enough, we consider (ϕ~,ϵϕ~)∈(E-{0})×(E-{0}); by a similar argument to the proof of Lemma 2, we know that there exists t* such that I(t*ϕ~,t*ϵϕ~)=0; that is, (t*ϕ~,t*ϵϕ~)∈N.
By (20) and I(t*ϕ~,t*ϵϕ~)=0, we have (t*)2=(H1(ϕ~,ϵϕ~))/(H2(ϕ~,ϵϕ~)), where
(44)H1(ϕ,ψ)=∥ϕ∥E2-ω1∥ϕ∥l22+∥ψ∥E2-ω2∥ψ∥l22,H2(ϕ,ψ)=∑n∈ℤ‍(a1ϕn4+a2ψn4+2a3ϕn2ψn2),
and J(t*ϕ~,t*ϵϕ~)=(H12(ϕ~,ϵϕ~))/(4H2(ϕ~,ϵϕ~)).
We noticed thatJ(ϕ~,0)=(a1/4)∑n∈ℤ‍ϕ~4=inf(ϕ,ψ)∈NJ(ϕ,ψ) and
(45)H2(ϕ~,0)=H1(ϕ~,0)=∥ϕ~∥E2-ω1∥ϕ~∥l22≥(λ-ω1)∥ϕ~∥l22.
For the sake of simplicity, we let(46)B=∑n∈ℤ‍ϕ~n4,D=∥ϕ~∥E2-ω2∥ϕ~∥l22.
If ω1≤ω2<λ, then
(47)D=∥ϕ~∥E2-ω2∥ϕ~∥l22=∥ϕ~∥E2-ω1∥ϕ~∥l22+(ω1-ω2)∥ϕ~∥l22=a1B+(ω1-ω2)∥ϕ~∥l22≤a1B.
Thus, a3>a1 and (47) yields a1BD<a1a3B2.
Ifω2<ω1<λ, then by (45),
(48)D=∥ϕ~∥E2-ω2∥ϕ~∥l22=∥ϕ~∥E2-ω1∥ϕ~∥l22+(ω1-ω2)∥ϕ~∥l22=a1B+(ω1-ω2)∥ϕ~∥l22≤a1B+ω1-ω2λ-ω1a1B=λ-ω2λ-ω1a1B.
Thus, a3>((λ-ω2)/(λ-ω1))a1 and (48) yields a1BD<a1a3B2.
From the above arguments, ifa3>max{a1,((λ-ω2)/(λ-ω1))a1}, then a1BD<a1a3B2.
Forϵ small enough, we have
(49)H12(ϕ~,ϵϕ~)=(∥ϕ~∥E2-ω1∥ϕ~∥l22+∥ϵϕ~∥E2-ω2∥ϵϕ~∥l22)2=(a1B+ϵ2D)2=a12B2+2a1BDϵ2+D2ϵ4<a12B2+2a1a3B2ϵ2+a1a2B2ϵ4=a1B(a1B+a2Bϵ4+2a3Bϵ2)=H2(ϕ~,0)H2(ϕ~,ϵϕ~).
Hence, by (49), we have
(50)J(t*ϕ~,t*ϵϕ~)=H12(ϕ~,ϵϕ~)4H2(ϕ~,ϵϕ~)<14H2(ϕ~,0)=J(ϕ~,0)=inf(ϕ,ψ)∈NJ(ϕ,ψ).
This is a contradiction. So, ψ~≠0.
Similarly, ifψ~≠0 and a3>max{a2,((λ-ω1)/(λ-ω2))a2}, then ϕ~≠0. The proof is completed.



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