Solitary wave of the Schrodinger lattice system with nonlinear hopping

This paper is concerned with the nonlinear Schrodinger 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 wave.


Introduction
In the last decades, a great deal of attention has been paid to study the existence of solitary wave for the lattice systems [9,10,11]. 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 [13,14].
Here, N d denotes the set of the nearest neighbors of the point l ∈ Z d . Note that for α = 0, β = 0, it recovers the classical nonlinear Schrödinger lattice. For α = 0, β = 0, it denotes the Schrödinger lattice with nonlinear hopping.
There has been a lot of interests 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 inter-site potentials, or mixed FPU/KG chains. N. I. 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 [1].
Our aim is to investigate the existence of nontrivial solitary wave for the infinite dimensional lattice (1.1). Here, we only consider the case of one dimension. i. e., d = 1. The case of d > 1 is similar. It notes that for classical nonlinear Schrödinger lattice, Weinstein [8] discusses a connection among the dimensionality, the degree of the nonlinearity and the existence of the excitation threshold. They 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 is greater than the excitation threshold and there is no ground state for the total power is less than the excitation threshold. However, we get that the power of solitary wave always has a lower bound for the equation (1.1) with σ ≥ 1.
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 nontrival periodic solution by Nehari manifolds argument [6]. The existence of solitary wave is more complex. In Section 3, we follow the idea of [2,3,4,5] to obtain the solitary wave. 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 a solitary ground state.

Periodic solution
In this paper, we consider the the following equation: where σ ≥ 1.
To obtain breather, we seek the solution: The equation of u l is 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 where k > 2 is an integer.
Consider the Banach space l p k with norm: We mention that and Nehari manifold Then, the minimizer of the constrained variational problem: is the nontrivial periodic solution of (2.3). We mention that the minimizer is called a periodic ground state.
We want to obtain the periodic solution with prescribed frequency ω < 0. With the Nehari manifold approach, we have the following result.  Proof. For t ≥ 0 and u = 0, define Then, There holds that ρ ′ (t) > 0 for t > 0 small enough. Observe that Therefore, ρ(t) admits a unique zero point t * ∈ (0, +∞). This implies √ t * u ∈ N k . It completes the proof. Proof. For t > 0 and u ∈ N k , we get Then, We can see that t = 1 is the unique maximum point of θ(t). This implies the proof.
Assume that u k is the k-periodic solution of (2.3), we have where C 1 is the unique positive solution of equation: Observe that C 1 is independent of k.
Thus, we get a lower bound of the power of the periodic solutions. Proof. Let u ∈ N k . From the argument in (2.4) and (2.5), there exists l 0 ∈ P k and a positive constant C 2 such that It completes the proof. Proof. Assume that {u n } is a minimizing sequence. We can see that there exists a constant M > 0 such that Thus, |ω|||u n || l 2 k ≤ −△u n , u n k − ω u n , u n k ≤ M. There holds that ||u n || l ∞ k is bounded. Note that P k is finite dimensional space. Passing to a subsequence, there exists u k such that u n j → u k in l 2 k . Since the set l 2 k is closed and the functional J k is continuous, we obtain that u k ∈ N k and J k (u k ) = m k .
By Lagrange multiplier method, there exists some constant λ such that Choose v = u k . Note that u k ∈ N k , there holds We have λ = 0. It implies that u k is a nontrival solution of equation (2.2). Now, we prove that u k is positive. Observe that Since that u k is the nontrival solution. Then, there exists t * * ∈ (0, 1] such that √ t * * |u k | ∈ N k . It is obvious that We can assume that u k = √ t * * |u k |. Let G(n, m) be the Green function of −△ d − ω. From [7], we have G(n, m) > 0 for ω < 0. It obtains that Since that u k is nonnegative, there holds u k n > 0 for all n ∈ Z. It completes the proof of Theorem 2.1.

Localized ground state
Here, we give some notations. Define the functional and Nehari manifold where ·, · is natural inner product in l 2 . Thus, we can see that the minimizer of the constrained variational problem: is the nontrivial solitary wave of (2.3). We call this minimizer a localized ground state. Similar with Lemma 2.1 and 2.2, the results are obtained by replacing J k (u), I k (u) to J(u), I(u).
In this section, to obtain the localized ground state u satisfying lim l→∞ |u l | = 0, we follow the idea of [2]. We want to pass to the limit as k → ∞. The key point is the following result.
Lemma 3.1. Under the assumptions of Theorem 2.1. Let u k be the kperiodic solution. Therefore, the sequences m k and ||u k || l 2 k are bounded. Proof. First, we concern the sequences m k are bounded. From the similar argument of Lemma 2.1, there holds that for any given u ∈ l 2 , there exists t ′ such I( √ t ′ u) < 0. Since the sequences with finite support are dense in l 2 . Therefore, there existsũ with finite support such that I(ũ) < 0. It obtains that there exists t ′′ such that I( √ t ′′ũ ) = 0. For k large enough, we have supp √ t ′′ũ ⊂ P k . We can getṽ k ∈ l 2 k such thatṽ k l = √ t ′′ũ l for l ∈ P k . There holds that I k (ṽ k ) = I( √ t ′′ũ l ) = 0. And m k ≤ J k (ṽ k ) = J( √ t ′′ũ ) is bounded. Second, we prove that ||u k || l 2 k is uniformly bounded. Assume that ||u k || l 2 k is unbounded. Passing to a subsequence which is still denoted by itself, we . One of the following should holds: (i) v k is vanishing, i.e. ||v k || l ∞ → 0.
(ii) v k is not vanishing. Passing to a subsequence which is still denoted by itself, there exists δ > 0 and b k ∈ Z such that |v k b k | > δ for all k. Now, we rule out the case (i). There holds that Hence, (3.6) Assume that where M 0 > 0 is a constant which is defined below.
It concludes that It contradicts with (3.7).
Let's rule out the non-vanishing case. By the discrete translation invariance, we can assume that b k = 0. Since ||v k || l 2 It is obvious that v ∈ l 2 , ||v|| l 2 ≤ 1 and |v 0 | ≥ δ.
Since |v 0 | = 0, then |u k 0 | → ∞, as k → ∞. On the other hand, we have It is a contradiction. Proof. Let u k ∈ l 2 k be a periodic ground state. From Lemma 3.1, the sequence ||u k || l 2 k is bounded. Therefore, u k is either vanishing or non-vanishing. In the case of vanishing, we have lim k→∞ ||u k || l p k → 0, for p > 2. There holds that |ω|||u k || 2 It is a contradiction. Thus, the sequence u k is non-vanishing. By the discrete translation invariance, we assume that |u k 0 | ≥ δ > 0. There exists u = {u l } such that u k l → u l for all l ∈ Z. It is obvious that u ∈ l 2 and u = 0. Also, we obtains that u is a nontrival solution for (2.3) by point-wise limits. Now, we want to prove that u is a localized ground state.
Let L be a positive integar such that For any given ǫ > 0, let u ′ ∈ N such that Choose t 1 > 1 such that From density argument, there exists a finite supported sequence v = {v l } sufficiently close to t 1 u ′ in l 2 such that I(v) < 0 and α l∈Z |v l | 2 |v l−1 | 2 + σβ σ + 1 l∈Z |v l | 2σ+2 < m + ǫ.
Thus, there exists t 2 ∈ (0, 1) such that t 2 v ∈ N and J(t 2 v) < m + ǫ. Choose k large enough such that P k contains the support of v. Let v k ∈ l 2 k such that v k l = t 2 v l for l ∈ P k . It concludes that I k (v k ) = I(t 2 v), J k (v k ) = J(t 2 v) < m + ǫ. It implies lim sup k→∞ m k < m + ǫ.
Combining with (3.8), we have lim k→∞ m k = m. It completes the proof.