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Concentrating standing waves for Davey–Stewartson systems

Part of: Elliptic equations and systems Incompressible inviscid fluids Partial differential equations

Published online by Cambridge University Press:  01 October 2021

Yi He  and
Xiao Luo
Yi He
Affiliation:
School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, P. R. China (heyi19870113@163.com)
Xiao Luo
Affiliation:
School of Mathematics, Hefei University of Technology, Hefei 230009, P. R. China (luoxiao@hfut.edu.cn)
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Abstract

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

Keywords

ConcentrationDavey–Stewartson systemmulti-peak solutionssingular integral operator

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35B25: Singular perturbations 35J60: Nonlinear elliptic equations 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 6 , December 2022 , pp. 1411 - 1450
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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