Orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear term
Abstract In this paper, we investigate the orbital stability of solitary waves for the following generalized long-short wave resonance equations of Hamiltonian form: 0.1 { i u t + u x x = α u v + γ | u | 2 u + δ | u | 4 u , v t + β | u | x 2 = 0 . $$ \textstyle\begin{cases} iu_{t}+u_{{xx}}=\alpha uv...
Main Authors: | , , |
---|---|
Format: | Article |
Language: | English |
Published: |
SpringerOpen
2020-11-01
|
Series: | Journal of Inequalities and Applications |
Subjects: | |
Online Access: | http://link.springer.com/article/10.1186/s13660-020-02505-7 |
id |
doaj-44a959dd565747f4b1c0ae10e0806b7a |
---|---|
record_format |
Article |
spelling |
doaj-44a959dd565747f4b1c0ae10e0806b7a2020-11-25T04:05:57ZengSpringerOpenJournal of Inequalities and Applications1029-242X2020-11-012020111910.1186/s13660-020-02505-7Orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear termXiaoxiao Zheng0Huafei Di1Xiaoming Peng2School of Mathematical Sciences, Qufu Normal UniversitySchool of Mathematics and Information Science, Guangzhou UniversitySchool of Statistics and Mathematics, Guangdong University of Finance and EconomicsAbstract In this paper, we investigate the orbital stability of solitary waves for the following generalized long-short wave resonance equations of Hamiltonian form: 0.1 { i u t + u x x = α u v + γ | u | 2 u + δ | u | 4 u , v t + β | u | x 2 = 0 . $$ \textstyle\begin{cases} iu_{t}+u_{{xx}}=\alpha uv+\gamma \vert u \vert ^{2}u+\delta \vert u \vert ^{4}u, \\ v_{t}+\beta \vert u \vert ^{2}_{x}=0. \end{cases} $$ We first obtain explicit exact solitary waves for Eqs. (0.1). Second, by applying the extended version of the classical orbital stability theory presented by Grillakis et al., the approach proposed by Bona et al., and spectral analysis, we obtain general results to judge orbital stability of solitary waves. We finally discuss the explicit expression of det ( d ″ ) $\det (d^{\prime \prime })$ in three cases and provide specific orbital stability results for solitary waves. Especially, we can get the results obtained by Guo and Chen with parameters α = 1 $\alpha =1$ , β = − 1 $\beta =-1$ , and δ = 0 $\delta =0$ . Moreover, we can obtain the orbital stability of solitary waves for the classical long-short wave equation with γ = δ = 0 $\gamma =\delta =0$ and the orbital instability results for the nonlinear Schrödinger equation with β = 0 $\beta =0$ .http://link.springer.com/article/10.1186/s13660-020-02505-7Long-short resonance wave equationsCubic-quintic nonlinearitySolitary wavesOrbital stability |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Xiaoxiao Zheng Huafei Di Xiaoming Peng |
spellingShingle |
Xiaoxiao Zheng Huafei Di Xiaoming Peng Orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear term Journal of Inequalities and Applications Long-short resonance wave equations Cubic-quintic nonlinearity Solitary waves Orbital stability |
author_facet |
Xiaoxiao Zheng Huafei Di Xiaoming Peng |
author_sort |
Xiaoxiao Zheng |
title |
Orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear term |
title_short |
Orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear term |
title_full |
Orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear term |
title_fullStr |
Orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear term |
title_full_unstemmed |
Orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear term |
title_sort |
orbital stability of solitary waves for the generalized long-short wave resonance equations with a cubic-quintic strong nonlinear term |
publisher |
SpringerOpen |
series |
Journal of Inequalities and Applications |
issn |
1029-242X |
publishDate |
2020-11-01 |
description |
Abstract In this paper, we investigate the orbital stability of solitary waves for the following generalized long-short wave resonance equations of Hamiltonian form: 0.1 { i u t + u x x = α u v + γ | u | 2 u + δ | u | 4 u , v t + β | u | x 2 = 0 . $$ \textstyle\begin{cases} iu_{t}+u_{{xx}}=\alpha uv+\gamma \vert u \vert ^{2}u+\delta \vert u \vert ^{4}u, \\ v_{t}+\beta \vert u \vert ^{2}_{x}=0. \end{cases} $$ We first obtain explicit exact solitary waves for Eqs. (0.1). Second, by applying the extended version of the classical orbital stability theory presented by Grillakis et al., the approach proposed by Bona et al., and spectral analysis, we obtain general results to judge orbital stability of solitary waves. We finally discuss the explicit expression of det ( d ″ ) $\det (d^{\prime \prime })$ in three cases and provide specific orbital stability results for solitary waves. Especially, we can get the results obtained by Guo and Chen with parameters α = 1 $\alpha =1$ , β = − 1 $\beta =-1$ , and δ = 0 $\delta =0$ . Moreover, we can obtain the orbital stability of solitary waves for the classical long-short wave equation with γ = δ = 0 $\gamma =\delta =0$ and the orbital instability results for the nonlinear Schrödinger equation with β = 0 $\beta =0$ . |
topic |
Long-short resonance wave equations Cubic-quintic nonlinearity Solitary waves Orbital stability |
url |
http://link.springer.com/article/10.1186/s13660-020-02505-7 |
work_keys_str_mv |
AT xiaoxiaozheng orbitalstabilityofsolitarywavesforthegeneralizedlongshortwaveresonanceequationswithacubicquinticstrongnonlinearterm AT huafeidi orbitalstabilityofsolitarywavesforthegeneralizedlongshortwaveresonanceequationswithacubicquinticstrongnonlinearterm AT xiaomingpeng orbitalstabilityofsolitarywavesforthegeneralizedlongshortwaveresonanceequationswithacubicquinticstrongnonlinearterm |
_version_ |
1724433158982598656 |