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The existence of nonzero periodic travelling wave solutions for a general discrete nonlinear Schrödinger equation (DNLS) on one-dimensional lattices is proved. The DNLS features a general nonlinear term and variable range of interactions going beyond the usual nearest-neighbour interaction. The problem of the existence of travelling wave solutions is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder’s Fixed Point Theorem.

Coherent structures arising in the form of travelling waves, solitons, and breathers in systems of coupled oscillators have attracted considerable interest not least due to the important role they play for applications in physics, biology, and chemistry (for reviews see [

In particular with regard to the existence of periodic travelling waves (TWs) in nonlinear lattice systems various methods have been used. For instance, the existence of small amplitude waves in nonlinear discrete Klein-Gordon systems was proved with the usage of spatial dynamics and centre manifold reduction [

In [

In the current study we are interested in the existence of periodic TW solutions of the following general discrete nonlinear Schrödinger equation on finite one-dimensional lattices:

The solutions satisfy periodicity conditions:

Assume the following condition on

The standard DNLS, arising for

System (

We consider travelling wave solutions of the form

In order for a travelling wave solution to satisfy the periodicity conditions in (

Regarding the existence of periodic travelling wave solutions we state the following.

Let

In the following we reformulate the original problem as a fixed point problem in a Banach space in a similar vein to the approach in [

To prove the assertions of the theorem we utilise Schauder’s Fixed Point Theorem (see, e.g., in [

Travelling wave solutions

Substituting (

Thus, the task amounts to finding

For the forthcoming discussion (

Let

We decompose functions

Related to the l.h.s. of (

Applying the operator

By the assumption (

For periodic travelling wave solutions

We consider then the closed and convex subsets of

Furthermore associated with the r.h.s. of (

Clearly, the operator

Finally, we express problem (

We get

It remains to prove that

Furthermore, the spectrum of linear plane wave solutions (phonons) arising for zero nonlinear term, determined by the r.h.s. of system (

To summarise, we have proven the existence of nonzero periodic travelling wave solutions for a general DNLS (including as a special case the standard DNLS) on finite one-dimensional lattices. To this end the existence problem has been reformulated as a fixed point problem for an operator on a function space which is solved with the help of Schauder’s Fixed Point Theorem. Our method can be straightforwardly extended not only to treating the general DNLS on lattices of higher dimension but also to other types of nonlinear lattice systems such as nonlinear discrete Klein-Gordon systems.

The author declares that there are no conflicts of interest.