A higher order system of some coupled nonlinear Schrödinger and Korteweg-de Vries equations

<abstract>In this paper we consider the following system of coupled biharmonic Schrödinger equations <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \ \left\{ \begin{aligned} \Delta^{2}u+\lambda_{1}u = u^{3}+\beta u v^{2}, \\ \Delta^{2}v+\lambda_{2}v = v^{3}+\beta u^{2}v, \end{aligned} \right. \end{equation*} $\end{document} </tex-math></disp-formula> where $ (u, v)\in H^{2}({\mathbb{R}}^{N})\times H^2(\mathbb R^N) $, $ 1\leq N\leq7 $, $ \lambda_{i} > 0 (i = 1, 2) $ and $ \beta $ denotes a real coupling parameter. By Nehari manifold method and concentration compactness theorem, we prove the existence of ground state solution for the coupled system of Schrödinger equations. Previous results on ground state solutions are obtained in radially symmetric Sobolev space $ H_r^2(\mathbb R^N)\times H_r^2(\mathbb R^N) $. When $ \beta $ satisfies some conditions, we prove the existence of ground state solution in the whole space $ H^2(\mathbb R^N)\times H^2(\mathbb R^N) $.</abstract>

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Existence for a Higher Order Coupled System of Korteweg-de Vries Equations

Liu¹

2021

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A higher order system of some coupled nonlinear Schrödinger and Korteweg-de Vries equations

Abstract: We prove existence and multiplicity of bound and ground state solutions, under appropriate conditions on the parameters, for a bi-harmonic stationary system of coupled nonlinear Schrödinger-Korteweg-de Vries equations.

Cited by 3 publications

References 26 publications

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Exploration of some novel solutions to a coupled Schrödinger–KdV equations in the interactions of capillary-gravity waves

Existence of ground state for coupled system of biharmonic Schrödinger equations

Existence for a Higher Order Coupled System of Korteweg-de Vries Equations

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