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.

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 [

In the present paper, we consider a variant of the discrete nonlinear Schrödinger lattice as follows:

Note that, for

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 [

Here, we only consider the case of one dimension. That is,

The paper is organized as follows. In Section

In this paper, we consider the following equation:

To obtain breather, we seek the solution

In this section, we prove the existence of

Let

Define the functional

Note that

Assume that the frequency

Under the assumptions of Theorem

For

Observe that

Under the assumptions of Theorem

For

Note that

Assume that

Thus, we get a lower bound of the power of the periodic solutions.

The power of the periodic solution must be greater than

Under the assumptions of Theorem

Let

Under the assumptions of Theorem

Assume that

Note that

By Lagrange multiplier method, there exists some constant

Now, we prove that

Let

Here, we give some notations. Define the functional

In this section, to obtain the localized ground state

Under the assumptions of Theorem

First, we concern the sequences

Second, we prove that

Now, we rule out case (i). This holds that

Let

By Hölder’s inequality, we have

Let us rule out the nonvanishing case. By the discrete translation invariance, we can assume that

Since

Assume that the frequency

Let

Thus, the sequence

Let

Choose

Combining with (

With similar argument in (

Let

Let

Similar to the argument in [

Since

Also, we have

On the other hand, from the point limits, we have that

Combine with Hölder inequality,

With similar argument, we obtain

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

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.