# Solitary Waves of the Schrödinger Lattice System with Nonlinear Hopping

Authors: Ming Cheng
Journal: Journal of Function Spaces
 (2015)

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## Abstract

This paper is concerned with the nonlinear Schrödinger lattice with nonlinear hopping. Via variation approach and the Nehari manifold argument, we obtain two types of solution: periodic ground state and localized ground state. Moreover, we consider the convergence of periodic solutions to the solitary waves.


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## Body

## 1. Introduction

In the last decades, a great deal of attention has been paid to study the existence of solitary waves for the lattice systems. They play a role in lots of physical models, such as nonlinear waves in crystals and arrays of coupled optical waveguides. The discrete nonlinear Schrödinger lattice is one of the most famous models in mathematics and physics. The existence and properties of discrete breathers (periodic in time and spatially localized) in discrete nonlinear Schrödinger lattice have been considered in a number of studies. One can see [1–9] and references therein.In the present paper, we consider a variant of the discrete nonlinear Schrödinger lattice as follows:(1)iψ˙l+Δdψl+αψl∑j=1dTjψl+βψl2σψl=0,l∈Zd,where α,β∈R, (Δdψ)l=∑m∈Nd‍ψm-ψl, and the nonlinear operator T is defined by (2)Tjψl=ψl1,…,lj-1,lj+1,lj+1,…,ld2+ψl1,…,lj-1,lj-1,lj+1,…,ld2.Here, Nd denotes the set of the nearest neighbors of the point l∈Zd.Note that, forα=0, β≠0, (1) recovers the classical nonlinear Schrödinger lattice. For α≠0, β≠0, (1) denotes the Schrödinger lattice with nonlinear hopping.There has been a lot of interest in this equation as the modeling of waveguide arrays. Also, nonlinear hopping terms appear from Klein-Gordon and Fermi-Pasta-Ulam chains of anharmonic oscillators coupled with anharmonic intersite potentials or mixed FPU/KG chains. The generalized DNLS system with the nonlinear hopping terms has been derived as a perturbation of the integrable Ablowitz-Ladik system, by the rotating wave approximation on the FPU chain.Karachalios et al. discuss the energy thresholds in the setting of DNLS lattice with nonlinear hopping terms by using fixed point method. The numerical results have also been obtained in their paper [10]. However, the Dirichlet boundary condition in Section 2 of their paper is not suitable. Since ψk+1=ψk+2=0, we have ψk=0 from the equation. With similar argument, the solution of lattice system under the Dirichlet boundary conditions is trivial. Our aim is to investigate the existence of nontrivial solitary waves for the infinite dimensional lattice (1). We firstly consider the k-periodic problem. In the paper of Karachalios et al., they “expect that the variational approach can be applied in the case of periodic boundary conditions, but the details have to be checked.” We obtain the nontrivial periodic solution by Nehari manifolds argument [11]. In Section 3, we obtain the solitary waves for the infinite dimensional lattice (1) by the concentration compactness method [12].Here, we only consider the case of one dimension. That is,d=1. The case of d>1 is similar. It notes that, for classical nonlinear Schrödinger lattice, Weinstein discusses a connection among the dimensionality, the degree of the nonlinearity, and the existence of the excitation threshold [6]. Weinstein prove that if the degree of the nonlinearity σ satisfies σ≥2/d where d is the dimension, then there exists a ground state for the total power which is greater than the excitation threshold and there is no ground state for the total power which is less than the excitation threshold. It is interesting that the power of solitary waves for the infinite dimensional lattice (1) always has a lower bound σ≥1 from our arguments.The paper is organized as follows. In Section2, we firstly consider the k-periodic problem. Note that the dimension in space variable is finite. We obtain the nontrivial periodic solution by Nehari manifolds argument. The existence of solitary waves is more complex. In Section 3, we follow the idea of [13–16] to obtain the solitary waves. The key point is to show the norms of periodic ground state are bounded. It is based on the concentration compactness. In Section 4, we concern the convergence of periodic ground states to solitary waves.


## 2. Periodic Ground State

In this paper, we consider the following equation:(3)iψ˙l+Δdψl+αψlψl+12+ψl-12+βψl2σψl=0,l∈Z,where σ≥1.To obtain breather, we seek the solution(4)ψl=e-iωtul.The equation of ul is (5)ωul+Δdul+αulul+12+ul-12+βul2σul=0,l∈Z.Actually, we give the proofs only in the focusing case with α,β>0 and ω<0. For the defocusing case with α,β<0 and ω>4, the argument is similar. Here, we omit the details.In this section, we prove the existence ofk-periodic solution which satisfies (6)ul+k=ul,forl∈Z,where k>2 is an integer.Let(7)Pk=l∈Z∣-k2≤l≤k-k2-1.Consider the Banach space lkp with norm: (8)ulkpp=∑l∈Pkulp.We mention that (9)ulkq≤ulkp,1≤p≤q≤∞.Denote that 〈·,·〉k is natural inner product in lk2.Define the functional(10)Jku=-Δdu,uk-ωu,uk-α∑l∈Pkul2ul+12-βσ+1∑l∈Pkul2σ+2and Nehari manifold (11)Nk=u∈lk2∣Iku=-Δdu,uk-ωu,uk-α∑l∈Pkul2ul+12+ul-12-β∑l∈Pkul2σ+2=0,u≠0.Then, the minimizer of the constrained variational problem (12)mk=inf⁡u∈Nk⁡Jkuis the nontrivial periodic solution of (5). We mention that the minimizer is called a periodic ground state.Note that(13)0≤Δdu,uk≤4ulk22,foru∈lk2.We want to obtain the periodic solution with prescribed frequency ω<0. With the Nehari manifold approach, we have one of the main results.Theorem 1.
Assume that the frequencyω<0 and α,β>0. There exists a positive k-periodic ground state uk for (5).Lemma 2.
Under the assumptions of Theorem1, the Nehari manifold Nk is nonempty.Proof.
Fort≥0 and u≠0, define (14)ρt=Iktu=t-Δdu,uk-ωu,uk-2t2α∑l∈Pkul2ul+12-tσ+1β∑l∈Pkul2σ+2.Then, (15)ρ′t=-Δdu,uk-ωu,uk-4tα∑l∈Pkul2ul+12-σ+1tσβ∑l∈Pkul2σ+2.It holds that ρ′(t)>0 for t>0 small enough.
Observe that(16)ρ′′t=-4α∑l∈Pkul2ul+12-σ+1σtσ-1β∑l∈Pkul2σ+2<0.Therefore, ρ(t) admits a unique zero point t∗∈(0,+∞). This implies t∗u∈Nk. It completes the proof.Lemma 3.
Under the assumptions of Theorem1, for u∈Nk, the function Jk(tu) has a unique critical point at t=1, which is a global maximum.Proof.
Fort>0 and u∈Nk, we get (17)θt=Jktu=t-Δdu,uk-ωu,uk-t2α∑l∈Pkul2ul+12-tσ+1βσ+1∑l∈Pkul2σ+2.Then, (18)θ′t=-Δdu,uk-ωu,uk-2tα∑l∈Pkul2ul+12-tσβ∑l∈Pkul2σ+2.It holds that θ′(t)>0 for t>0 small enough.
Note that(19)θ′′t=-2α∑l∈Pkul2ul+12-σtσ-1β∑l∈Pkul2σ+2<0.We can see that t=1 is the unique maximum point of θ(t). This implies the proof.Assume thatuk is the k-periodic solution of (5); we have (20)ωuklk22≤-Δduk,ukk-ωuk,ukk=2α∑l∈Pkulk2ul+1k2+β∑l∈Pkulk2σ+2≤uklk22βuklk22σ+2αuklk22.Therefore (21)uklk2≥C1>0,where C1 is the unique positive solution of equation (22)βx2σ+2αx2+ω=0.Observe that C1 is independent of k.Thus, we get a lower bound of the power of the periodic solutions.Theorem 4.
The power of the periodic solution must be greater thanC1.Lemma 5.
Under the assumptions of Theorem1, Jk(u) is bounded below for all u∈Nk.Proof.
Letu∈Nk. From the argument in (20) and (21), there exist l0∈Pk and a positive constant C2 such that (23)ul0>C2>0.Therefore, (24)Jku=α∑Pkul2ul-12+σβσ+1∑Pkul2σ+2>σβσ+1C22σ+2.It completes the proof.Lemma 6.
Under the assumptions of Theorem1, the minimizer of the constrained variational problem mk=inf⁡u∈Nk{Jk(u)} could be attained.Proof.
Assume that{un} is a minimizing sequence. We can see that there exists a constant M>0 such that (25)max⁡⁡Jkun≤M.Thus, (26)ωunlk2≤-Δun,unk-ωun,unk≲M.It holds that unlk∞ is bounded.
Note thatPk is finite dimensional space. Passing to a subsequence, there exists uk such that unj→uk in lk2. Since the set lk2 is closed and the functional Jk is continuous, we obtain that uk∈Nk and Jk(uk)=mk.By Lagrange multiplier method, there exists some constantλ such that (27)λ2-Δduk,vk-2ωuk,vk-4α∑l∈Pkulkvlul+1k2+ul-1k2-2σ+2β·∑l∈Pkulk2σulkvl+2-Δduk,vk-2ωuk,vk-2α∑l∈Pkulkvlul+1k2+ul-1k2-2β∑l∈Pkulk2σulkvl=0.Choose v=uk. Note that uk∈Nk; it holds that (28)λ-2α∑l∈Pkulk2ul+1k2+ul-1k2-σβ∑l∈Pkulk2σ+2=0.We have λ=0. It implies that uk is a nontrivial solution of (3).Now, we prove thatuk is positive. Observe that (29)-Δdu,u-ωu,u≤-Δdu,u-ωu,u.Since that uk is the nontrivial solution, then, there exists t∗∗∈(0,1] such that t∗∗|uk|∈Nk. It is obvious that(30)Jkt∗∗uk≤mk.Hence Jk(t∗∗|uk|)=mk. We can assume that uk=t∗∗|uk|.LetG(n,m) be the Green function of -Δd-ω. From [17], we have G(n,m)>0 for ω<0. It obtains that (31)unk=∑l∈ZGn,lαulkul+1k2+ul-1k2+βulk2σulk,n∈Z.Since uk is nonnegative, this holds unk>0 for all n∈Z. It completes the proof of Theorem 1.


## 3. Localized Ground State

Here, we give some notations. Define the functional(32)Ju=-Δdu,u-ωu,u-α∑l∈Zul2ul+12-βσ+1∑l∈Zul2σ+2and Nehari manifold (33)N=u∈l2∣Iu=-Δdu,u-ωu,u-α∑l∈Zul2ul+12+ul-12-β∑l∈Zul2σ+2=0,u≠0,where 〈·,·〉 is natural inner product in l2. Thus, we can see that the minimizer of the constrained variational problem (34)m=inf⁡u∈N⁡Juis the nontrivial solitary waves of (5). We call this minimizer a localized ground state. Similar to Lemmas 2 and 3, the results are obtained by replacing Jk(u), Ik(u) with J(u), I(u).In this section, to obtain the localized ground stateu satisfying (35)liml→∞⁡ul=0,we follow the idea of [13]. We want to pass to the limit as k→∞. The key point is the following result.Lemma 7.
Under the assumptions of Theorem1, let uk be the k-periodic solution. Therefore, the sequences mk and uklk2 are bounded.Proof.
First, we concern the sequencesmk which are bounded. From similar argument of Lemma 2, this holds that, for any given u∈l2, there exists t′ such I(t′u)<0. Since the sequences with finite support are dense in l2, therefore, there exists u~ with finite support such that I(u~)<0. It obtains that there exists t′′ such that I(t′′u~)=0. For k large enough, we have suppt′′u~⊂Pk. We can get v~k∈lk2 such that v~lk=t′′u~l for l∈Pk. This holds that Ik(v~k)=I(t′′u~l)=0. And mk≤Jk(v~k)=J(t′′u~) is bounded.
Second, we prove thatuklk2 is uniformly bounded. Assume that uklk2 is unbounded. Passing to a subsequence which is still denoted by itself, we have uklk2→∞ for k→∞. Let vk=uk/uklk2. One of the following should hold:(i)
vk is vanishing; that is, vkl∞→0;(ii)
vk is not vanishing; passing to a subsequence which is still denoted by itself, there exists δ>0 and bk∈Z such that |vbkk|>δ for all k.
Now, we rule out case (i). This holds that(36)0=Ikukuklk2=-Δd-ωvk,vkk-2α∑l∈Pkvlk2ul-1k2-β∑l∈Pkvlk2ulk2σ.Hence, (37)ω=ωvklk2≤-Δd-ωvk,vkk=2α∑l∈Pkvlk2ul-1k2+β∑l∈Pkvlk2ulk2σ.Assume that (38)Ak=l∈Pk∣ulk<M0,Bk=Pk∖Ak,where M0>0 is a constant which is defined below.
LetM0 be small enough such that (39)2α∑l∈Akvlk2ul-1k2+β∑l∈Akvlk2ulk2σ<2αM02∑l∈Akvl+1k2+βM02σ∑l∈Akvlk2≤ω2.Combine with (37), we have (40)ω2≤liminf⁡k→∞⁡2α∑l∈Bkvl+1k2ulk2+β∑l∈Bkvlk2ulk2σ.From the argument above, there exists a constant M′>0 such that (41)mk=∑l∈Pkαulk2ul+1k2+σβσ+1∑l∈Pkulk2σ+2<M′.Hence, uklk2σ+2 is uniformly bounded.
By Hölder’s inequality, we have(42)∑nvn2un2≤ul2σ+22vl2σ+2/σ2,∑nvn2un2σ≤ul2σ+22σvl2σ+22,vlp≤vl∞p-2/pvl22/p,forp>2.Since vk is vanishing, we can see that (43)limk→∞⁡vklkp=0,forp>2.It concludes that (44)liminf⁡k→∞⁡2α∑Bkvl+1k2ulk2+β∑Bkvlk2ulk2σ⟶0,ask⟶∞.It contradicts with (40).
Let us rule out the nonvanishing case. By the discrete translation invariance, we can assume thatbk=0. Since vklk2=1, there exists v={vl} such that vlk→vl for all l∈Z. It is obvious that v∈l2, vl2≤1, and |v0|≥δ.
Since|v0|≠0, then |u0k|→∞, as k→∞. On the other hand, we have (45)σβσ+1u0k2σ+2≤mk≤M′.It is a contradiction.Theorem 8.
Assume that the frequencyω<0 and α,β>0. There exists a positive localized ground state u for (5).Proof.
Letuk∈lk2 be a periodic ground state. From Lemma 7, the sequence uklk2 is bounded. Therefore, uk is either vanishing or nonvanishing. In the case of vanishing, we have limk→∞uklkp→0, for p>2. This holds that (46)ωuklk22≤-Δd-ωuk,ukk=2α∑l∈Pkulk2ul-1k2+β∑l∈Pkulk2σ+2≤2α∑l∈Pkulk4+β∑l∈Pkulk2σ+2⟶0.It is a contradiction.
Thus, the sequenceuk is nonvanishing. By the discrete translation invariance, we assume that |u0k|≥δ>0. There exists u={ul} such that ulk→ul for all l∈Z. It is obvious that u∈l2 and u≠0. Also, we obtain that u is a nontrivial solution for (5) by pointwise limits. Now, we want to prove that u is a localized ground state.
LetL be a positive integer such that (47)liminf⁡k→∞⁡Jkuk≥liminf⁡k→∞⁡α∑-L≤l≤Lulk2ul-1k2+σβσ+1∑-L≤l≤Lulk2σ+2≥α∑-L≤l≤Lul2ul-12+σβσ+1∑-L≤l≤Lul2σ+2.Let L→∞; it obtains that (48)liminf⁡k→∞⁡Jkuk≥Ju≥m,liminf⁡k→∞⁡mk≥m.For any given ϵ>0, let u′∈N such that (49)Ju′=α∑l∈Zul′2ul-1′2+σβσ+1∑l∈Zul′2σ+2<m+ϵ.Choose t1>1 such that (50)Jt1u′<m+ϵ,It1u′<0.From density argument, there exists a finite supported sequence v={vl} sufficiently close to t1u′ in l2 such that(51)Iv<0,α∑l∈Zvl2vl-12+σβσ+1∑l∈Zvl2σ+2<m+ϵ.Thus, there exists t2∈(0,1) such that t2v∈N and J(t2v)<m+ϵ.
Choosek large enough such that Pk contains the support of v. Let vk∈lk2 such that vlk=t2vl for l∈Pk. It concludes that (52)Ikvk=It2v,Jkvk=Jt2v<m+ϵ.It implies (53)limsup⁡k→∞⁡mk<m+ϵ.
Combining with (48), we have limk→∞mk=m. It completes the proof.Remark 9.
With similar argument in (20) and (21), the power of the localized ground state has a lower bound C1>0.


## 4. Global Convergence

Theorem 10.
Letuk∈lk2 be the periodic ground state to (5). Then, there exists a ground state u∈l2 such that uk is strongly convergent to u in lk2 after some discrete translation.Proof.
Letuk∈lk2 be the periodic ground state and bk∈Z. Now, we consider a translation (54)ul′k=ul+bkk.From the argument above, we can assume that ul′k→ul for all l∈Z where u is a ground state. We want to prove that u′k-ulk2 is convergent to 0 as k→∞. First, it concludes that (55)Jku′k-u⟶0,Iku′k-u⟶0,ask⟶∞.Indeed,(56)Jku′k-u=-Δdu′k-u,u′k-u-ωu′k-u,u′k-u-α∑l∈Pkul′k-ul2ul+1′k-ul+12-βσ+1∑l∈Pkul′k-ul2σ+2=Jku′k-Jku-2-Δdu′k-u,u-2ωu′k-u,u-α∑l∈Pkul′k-ul2ul+1′k-ul+12-βσ+1∑l∈Pkul′k-ul2σ+2+α∑l∈Pkul′k2ul+1′k2+βσ+1∑l∈Pkul′k2σ+2-α∑l∈Pkul2ul+12-βσ+1∑l∈Pkul2σ+2.
Similar to the argument in [13], it obtains that Jk(u′k)→J(u)=m and -2-Δdu′k-u,u-2ωu′k-u,u→0, as k→∞.
Sinceu′klk2 and ul2 are bounded. For any given ϵ>0, there exists M>0 such that ∑|l|≥Mul2<ϵ. Therefore, we have (57)α∑l∈Pk,l≥Mul′k2ul+1′k2-α∑l∈Pk,l≥Mul′k-ul2ul+1′k-ul+12≤α∑l∈Pk,l≥Mul′k2-ul′k-ul2ul+12+α∑l∈Pk,l≥Mul′k-ul2ul+1′k2-ul+1′k-ul+12≤α2u′klk2ul2+ul22∑l∈Pk,l≥Mul+12+αu′klk22+ul222u′klk2+ul2·∑l∈Pk,l≥Mul+121/2≲M′,ul22ϵfor k large enough.
Also, we have(58)-βσ+1∑l∈Pk,l≥Mul′k-ul2σ+2+βσ+1∑l∈Pk,l≥Mul′k2σ+2≲M′,ul22ϵfor k large enough.
On the other hand, from the point limits, we have that(59)-α∑l∈Pk,l<Mul′k-ul2ul+1′k-ul+12-βσ+1∑l∈Pk,l<Mul′k-ul2σ+2+α∑l∈Pk,l<Mul′k2u′l+1k2+βσ+1∑l∈Pk,l<Mul′k2σ+2-α∑l∈Pk,l<Mul2ul+12-βσ+1∑l∈Pk,l<Mul2σ+2<ϵfor k large enough.
Combine with Hölder inequality,Jk(u′k-u)→0 holds.
With similar argument, we obtainIk(u′k-u)→0. Therefore, (60)Jku′k-u-Iku′k-u=α∑l∈Pkul′k-ul2ul-1′k-ul-12+σβσ+1·∑l∈Pkul′k-ul2σ+2⟶0.Since ·lk∞≤·lk2σ+2, we have u′k-ulk∞→0. From Lemma 7, it is known that u′k-ulkp→0 for p>2. Hence, (61)ωu′k-ulk22≤-Δdu′k-u,u′k-u-ωu′k-u,u′k-u=2α∑l∈Pkul′k-ul2ul+1′k-ul+12+β·∑l∈Pkul′k-ul2σ+2≤2αu′k-ulk44+βu′k-ulk2σ+22σ+2⟶0.It completes the proof.



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