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

Zhenguo Wang; Yuanxian Hui; Liuyong Pang; Zhenguo Wang; Yuanxian Hui; Liuyong Pang

doi:10.3934/math.20221036

AIMS Mathematics

2022, Volume 7, Issue 10: 18824-18836. doi: 10.3934/math.20221036

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

School of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, China

Received: 29 June 2022 Revised: 31 July 2022 Accepted: 08 August 2022 Published: 25 August 2022
MSC : 65Q10, 39A60

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.
- Schrödinger equations,
- gap solitons,
- critical point theory,
- approximation
Citation: Zhenguo Wang, Yuanxian Hui, Liuyong Pang. Gap solitons in periodic difference equations with sign-changing saturable nonlinearity[J]. AIMS Mathematics, 2022, 7(10): 18824-18836. doi: 10.3934/math.20221036

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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.

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