Gap solitons in periodic difference equations with sign-changing saturable nonlinearity

Abstract

In this paper, we consider the existence of gap solitons for a class of difference equations: $ \begin{equation*} Lu_{n}-\omega u_{n} = f_{n}(u_{n}), n\in\mathbb{Z}, \end{equation*} $ where $ Lu_{n} = a_{n}u_{n+1}+a_{n-1}u_{n-1}+b_{n}u_{n} $ is the discrete difference operator in one spatial dimension, $ \{a_{n}\} $ and $ \{b_{n}\} $ are real valued T-periodic sequences, $ \omega\in \mathbb{R} $, $ f_{n}(\cdot)\in C(\mathbb{R}, \mathbb{R}) $ and $ f_{n+T}(\cdot) = f_{n}(\cdot) $ for each $ n\in\mathbb{Z} $. Under general asymptotically linear conditions on the nonlinearity $ f_{n}(\cdot) $, we establish the existence of gap solitons for the above equation via variational methods when $ t f_{n}(t) $ is allowed to be sign-changing. Our methods further extend and improve the existing results.

Keywords

• schrödinger equations
• gap solitons
• critical point theory
• approximation