JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2015/385649 385649 Research Article Solitary Waves of the Schrödinger Lattice System with Nonlinear Hopping http://orcid.org/0000-0001-7035-2747 Cheng Ming Motreanu Dumitru College of Mathematics Jilin University Changchun 130012 China jlu.edu.cn 2015 2942015 2015 02 03 2015 15 04 2015 2942015 2015 Copyright © 2015 Ming Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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  and references therein.

In the present paper, we consider a variant of the discrete nonlinear Schrödinger lattice as follows: (1)iψ˙l+Δdψl+αψlj=1dTjψl+βψl2σψl=0,lZd,where α,βR, (Δdψ)l=mNdψm-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 lZd.

Note that, for α=0, β0, (1) recovers the classical nonlinear Schrödinger lattice. For α0, β0, (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 . 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 . In Section 3, we obtain the solitary waves for the infinite dimensional lattice (1) by the concentration compactness method .

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 . Weinstein prove that if the degree of the nonlinearity σ satisfies σ2/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 σ1 from our arguments.

The paper is organized as follows. In Section 2, 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  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,lZ,where σ1.

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,lZ.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 of k-periodic solution which satisfies (6)ul+k=ul,forlZ,where k>2 is an integer.

Let (7)Pk=lZ-k2lk-k2-1.Consider the Banach space lkp with norm: (8)ulkpp=lPkulp.We mention that (9)ulkqulkp,1pq.Denote that ·,·k is natural inner product in lk2.

Define the functional (10)Jku=-Δdu,uk-ωu,uk-αlPkul2ul+12-βσ+1lPkul2σ+2and Nehari manifold (11)Nk=ulk2Iku=-Δdu,uk-ωu,uk-αlPkul2ul+12+ul-12-βlPkul2σ+2=0,u0.Then, the minimizer of the constrained variational problem (12)mk=infuNkJkuis the nontrivial periodic solution of (5). We mention that the minimizer is called a periodic ground state.

Note that (13)0Δdu,uk4ulk22,forulk2.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 frequency ω<0 and α,β>0. There exists a positive k-periodic ground state uk for (5).

Lemma 2.

Under the assumptions of Theorem 1, the Nehari manifold Nk is nonempty.

Proof.

For t0 and u0, define (14)ρt=Iktu=t-Δdu,uk-ωu,uk-2t2αlPkul2ul+12-tσ+1βlPkul2σ+2.Then, (15)ρt=-Δdu,uk-ωu,uk-4tαlPkul2ul+12-σ+1tσβlPkul2σ+2.It holds that ρ(t)>0 for t>0 small enough.

Observe that (16)ρ′′t=-4αlPkul2ul+12-σ+1σtσ-1βlPkul2σ+2<0.Therefore, ρ(t) admits a unique zero point t(0,+). This implies tuNk. It completes the proof.

Lemma 3.

Under the assumptions of Theorem 1, for uNk, the function Jk(tu) has a unique critical point at t=1, which is a global maximum.

Proof.

For t>0 and uNk, we get (17)θt=Jktu=t-Δdu,uk-ωu,uk-t2αlPkul2ul+12-tσ+1βσ+1lPkul2σ+2.Then, (18)θt=-Δdu,uk-ωu,uk-2tαlPkul2ul+12-tσβlPkul2σ+2.It holds that θ(t)>0 for t>0 small enough.

Note that (19)θ′′t=-2αlPkul2ul+12-σtσ-1βlPkul2σ+2<0.We can see that t=1 is the unique maximum point of θ(t). This implies the proof.

Assume that uk is the k-periodic solution of (5); we have (20)ωuklk22-Δduk,ukk-ωuk,ukk=2αlPkulk2ul+1k2+βlPkulk2σ+2uklk22βuklk22σ+2αuklk22.Therefore (21)uklk2C1>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 than C1.

Lemma 5.

Under the assumptions of Theorem 1, Jk(u) is bounded below for all uNk.

Proof.

Let uNk. From the argument in (20) and (21), there exist l0Pk and a positive constant C2 such that (23)ul0>C2>0.Therefore, (24)Jku=αPkul2ul-12+σβσ+1Pkul2σ+2>σβσ+1C22σ+2.It completes the proof.

Lemma 6.

Under the assumptions of Theorem 1, the minimizer of the constrained variational problem mk=infuNk{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)maxJkunM.Thus, (26)ωunlk2-Δun,unk-ωun,unkM.It holds that unlk is bounded.

Note that Pk is finite dimensional space. Passing to a subsequence, there exists uk such that unjuk in lk2. Since the set lk2 is closed and the functional Jk is continuous, we obtain that ukNk and Jk(uk)=mk.

By Lagrange multiplier method, there exists some constant λ such that (27)λ2-Δduk,vk-2ωuk,vk-4αlPkulkvlul+1k2+ul-1k2-2σ+2β·lPkulk2σulkvl+2-Δduk,vk-2ωuk,vk-2αlPkulkvlul+1k2+ul-1k2-2βlPkulk2σulkvl=0.Choose v=uk. Note that ukNk; it holds that (28)λ-2αlPkulk2ul+1k2+ul-1k2-σβlPkulk2σ+2=0.We have λ=0. It implies that uk is a nontrivial solution of (3).

Now, we prove that uk 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)Jktukmk.Hence Jk(t|uk|)=mk. We can assume that uk=t|uk|.

Let G(n,m) be the Green function of -Δd-ω. From , we have G(n,m)>0 for ω<0. It obtains that (31)unk=lZGn,lαulkul+1k2+ul-1k2+βulk2σulk,nZ.Since uk is nonnegative, this holds unk>0 for all nZ. 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-αlZul2ul+12-βσ+1lZul2σ+2and Nehari manifold (33)N=ul2Iu=-Δdu,u-ωu,u-αlZul2ul+12+ul-12-βlZul2σ+2=0,u0,where ·,· is natural inner product in l2. Thus, we can see that the minimizer of the constrained variational problem (34)m=infuNJuis 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 state u satisfying (35)limlul=0,we follow the idea of . We want to pass to the limit as k. The key point is the following result.

Lemma 7.

Under the assumptions of Theorem 1, let uk be the k-periodic solution. Therefore, the sequences mk and uklk2 are bounded.

Proof.

First, we concern the sequences mk which are bounded. From similar argument of Lemma 2, this holds that, for any given ul2, there exists t such I(tu)<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~klk2 such that v~lk=t′′u~l for lPk. This holds that Ik(v~k)=I(t′′u~l)=0. And mkJk(v~k)=J(t′′u~) is bounded.

Second, we prove that uklk2 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:

vk is vanishing; that is, vkl0;

vk is not vanishing; passing to a subsequence which is still denoted by itself, there exists δ>0 and bkZ such that |vbkk|>δ for all k.

Now, we rule out case (i). This holds that (36)0=Ikukuklk2=-Δd-ωvk,vkk-2αlPkvlk2ul-1k2-βlPkvlk2ulk2σ.Hence, (37)ω=ωvklk2-Δd-ωvk,vkk=2αlPkvlk2ul-1k2+βlPkvlk2ulk2σ.Assume that (38)Ak=lPkulk<M0,Bk=PkAk,where M0>0 is a constant which is defined below.

Let M0 be small enough such that (39)2αlAkvlk2ul-1k2+βlAkvlk2ulk2σ<2αM02lAkvl+1k2+βM02σlAkvlk2ω2.Combine with (37), we have (40)ω2liminfk2αlBkvl+1k2ulk2+βlBkvlk2ulk2σ.From the argument above, there exists a constant M>0 such that (41)mk=lPkαulk2ul+1k2+σβσ+1lPkulk2σ+2<M.Hence, uklk2σ+2 is uniformly bounded.

By Hölder’s inequality, we have (42)nvn2un2ul2σ+22vl2σ+2/σ2,nvn2un2σul2σ+22σvl2σ+22,vlpvlp-2/pvl22/p,forp>2.Since vk is vanishing, we can see that (43)limkvklkp=0,forp>2.It concludes that (44)liminfk2αBkvl+1k2ulk2+βBkvlk2ulk2σ0,ask.It contradicts with (40).

Let us rule out the nonvanishing case. By the discrete translation invariance, we can assume that bk=0. Since vklk2=1, there exists v={vl} such that vlkvl for all lZ. It is obvious that vl2, vl21, and |v0|δ.

Since |v0|0, then |u0k|, as k. On the other hand, we have (45)σβσ+1u0k2σ+2mkM.It is a contradiction.

Theorem 8.

Assume that the frequency ω<0 and α,β>0. There exists a positive localized ground state u for (5).

Proof.

Let uklk2 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 limkuklkp0, for p>2. This holds that (46)ωuklk22-Δd-ωuk,ukk=2αlPkulk2ul-1k2+βlPkulk2σ+22αlPkulk4+βlPkulk2σ+20.It is a contradiction.

Thus, the sequence uk is nonvanishing. By the discrete translation invariance, we assume that |u0k|δ>0. There exists u={ul} such that ulkul for all lZ. It is obvious that ul2 and u0. 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.

Let L be a positive integer such that (47)liminfkJkukliminfkα-LlLulk2ul-1k2+σβσ+1-LlLulk2σ+2α-LlLul2ul-12+σβσ+1-LlLul2σ+2.Let L; it obtains that (48)liminfkJkukJum,liminfkmkm.For any given ϵ>0, let uN such that (49)Ju=αlZul2ul-12+σβσ+1lZul2σ+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,αlZvl2vl-12+σβσ+1lZvl2σ+2<m+ϵ.Thus, there exists t2(0,1) such that t2vN and J(t2v)<m+ϵ.

Choose k large enough such that Pk contains the support of v. Let vklk2 such that vlk=t2vl for lPk. It concludes that (52)Ikvk=It2v,Jkvk=Jt2v<m+ϵ.It implies (53)limsupkmk<m+ϵ.

Combining with (48), we have limkmk=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.

Let uklk2 be the periodic ground state to (5). Then, there exists a ground state ul2 such that uk is strongly convergent to u in lk2 after some discrete translation.

Proof.

Let uklk2 be the periodic ground state and bkZ. Now, we consider a translation (54)ulk=ul+bkk.From the argument above, we can assume that ulkul for all lZ where u is a ground state. We want to prove that uk-ulk2 is convergent to 0 as k. First, it concludes that (55)Jkuk-u0,Ikuk-u0,ask.Indeed,(56)Jkuk-u=-Δduk-u,uk-u-ωuk-u,uk-u-αlPkulk-ul2ul+1k-ul+12-βσ+1lPkulk-ul2σ+2=Jkuk-Jku-2-Δduk-u,u-2ωuk-u,u-αlPkulk-ul2ul+1k-ul+12-βσ+1lPkulk-ul2σ+2+αlPkulk2ul+1k2+βσ+1lPkulk2σ+2-αlPkul2ul+12-βσ+1lPkul2σ+2.

Similar to the argument in , it obtains that Jk(uk)J(u)=m and -2-Δduk-u,u-2ωuk-u,u0, as k.

Since uklk2 and ul2 are bounded. For any given ϵ>0, there exists M>0 such that |l|Mul2<ϵ. Therefore, we have (57)αlPk,lMulk2ul+1k2-αlPk,lMulk-ul2ul+1k-ul+12αlPk,lMulk2-ulk-ul2ul+12+αlPk,lMulk-ul2ul+1k2-ul+1k-ul+12α2uklk2ul2+ul22lPk,lMul+12+αuklk22+ul222uklk2+ul2·lPk,lMul+121/2M,ul22ϵfor k large enough.

Also, we have(58)-βσ+1lPk,lMulk-ul2σ+2+βσ+1lPk,lMulk2σ+2M,ul22ϵfor k large enough.

On the other hand, from the point limits, we have that (59)-αlPk,l<Mulk-ul2ul+1k-ul+12-βσ+1lPk,l<Mulk-ul2σ+2+αlPk,l<Mulk2ul+1k2+βσ+1lPk,l<Mulk2σ+2-αlPk,l<Mul2ul+12-βσ+1lPk,l<Mul2σ+2<ϵfor k large enough.

Combine with Hölder inequality, Jk(uk-u)0 holds.

With similar argument, we obtain Ik(uk-u)0. Therefore, (60)Jkuk-u-Ikuk-u=αlPkulk-ul2ul-1k-ul-12+σβσ+1·lPkulk-ul2σ+20.Since ·lk·lk2σ+2, we have uk-ulk0. From Lemma 7, it is known that uk-ulkp0 for p>2. Hence, (61)ωuk-ulk22-Δduk-u,uk-u-ωuk-u,uk-u=2αlPkulk-ul2ul+1k-ul+12+β·lPkulk-ul2σ+22αuk-ulk44+βuk-ulk2σ+22σ+20.It completes the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author sincerely thanks Professor Yong Li for many useful suggestions. The author also thanks the referees for their comments that improved this paper. This work was supported by NSF of China (NSFC) Grant no. 11401250.

Johansson M. Aubry S. Existence and stability of quasiperiodic breathers in the discrete nonlinear Schrödinger equation Nonlinearity 1997 10 5 1151 1178 10.1088/0951-7715/10/5/008 MR1473378 2-s2.0-0000576831 Aubry S. Breathers in nonlinear lattices: existence, linear stability and quantization Physica D 1997 103 1–4 201 250 10.1016/s0167-2789(96)00261-8 MR1464249 Zhang G. Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials Journal of Mathematical Physics 2009 50 1 013505 013505 10.1063/1.3036182 MR2492615 2-s2.0-59349090313 Cuevas J. Karachalios N. I. Palmero F. Lower and upper estimates on the excitation threshold for breathers in discrete nonlinear Schrödinger lattices Journal of Mathematical Physics 2009 50 11 10 112705 10.1063/1.3263142 MR2567201 2-s2.0-72249085777 Karachalios N. I. Yannacopoulos A. N. Global existence and compact attractors for the discrete nonlinear Schrodinger equation Journal of Differential Equations 2005 217 1 88 123 10.1016/j.jde.2005.06.002 MR2170529 2-s2.0-24644468626 Weinstein M. I. Excitation thresholds for nonlinear localized modes on lattices Nonlinearity 1999 12 3 673 691 ZBL0984.35147 2-s2.0-0033130275 10.1088/0951-7715/12/3/314 MR1690199 Kevrekidis P. G. Rasmussen K. Ø. Bishop A. R. The discrete nonlinear schrödinger equation: a survey of recent results International Journal of Modern Physics B 2001 15 21 2833 2900 10.1142/s0217979201007105 2-s2.0-0035921187 Zhang G. Pankov A. Standing waves of the discrete nonlinear Schrodinger equations with growing potentials Communications in Mathematical Analysis 2008 5 2 38 49 MR2421490 Zhang G. Pankov A. Standing wave solutions of the discrete non-linear Schrodinger equations with unbounded potentials, II Applicable Analysis 2010 89 9 1541 1557 10.1080/00036810902942234 MR2682118 2-s2.0-77954969016 Karachalios N. I. Sánchez-Rey B. Kevrekidis P. G. Cuevas J. Breathers for the discrete nonlinear Schrödinger equation with nonlinear hopping Journal of Nonlinear Science 2013 23 2 205 239 2-s2.0-84879692231 10.1007/s00332-012-9149-y MR3041624 Nehari Z. On a class of nonlinear second-order differential equations Transactions of the American Mathematical Society 1960 95 101 123 ZBL0097.29501 10.1090/S0002-9947-1960-0111898-8 MR0111898 Lions P. L. The concentration compactness principle in the calculus of variations I: the locally compact case Annales de l'institut Henri Poincaré (C) Analyse non linéaire 1984 1 4 223 283 Pankov A. Rothos V. Periodic and decaying solutions in discrete nonlinear Schrodinger with saturable nonlinearity Proceedings of the Royal Society A 2008 464 3219 3236 Pankov A. Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices 2005 London, UK Imperial College Press 10.1142/9781860947216 MR2156331 Pankov A. Periodic nonlinear Schrödinger equation with application to photonic crystals Milan Journal of Mathematics 2005 73 1 259 287 2-s2.0-27844448598 10.1007/s00032-005-0047-8 MR2175045 Pankov A. Gap solitons in periodic discrete nonlinear Schrödinger equations Nonlinearity 2006 19 1 27 40 2-s2.0-29244484313 10.1088/0951-7715/19/1/002 MR2191617 Teschl G. Jacobi Operators and Completely Integrable Nonlinear Lattices 2000 Providence, RI, USA American Mathematical Society