Existence of Standing Waves for a Generalized Davey-Stewartson System

Xiaoxiao Hu,Min-Bo Yang,Xiao-Xun Zhou,Wu Tunhua

doi:10.1155/2013/848690

Abstract

The purpose of this paper is to investigate the existence of standing waves for a generalized Davey-Stewartson system. By reducing the system to a single Schrödinger equation problem, we are able to establish some existence results for the system by variational methods.

Highlights

Introduction and Main ResultsIn this paper, we are going to consider the existence of standing waves for a generalized Davey-Stewartson system in R3 iψt + Δψ = b (x) ψφx1 − a (x) 󵄨󵄨󵄨󵄨ψ󵄨󵄨󵄨󵄨p−2ψ, −Δφ = (b(x)󵄨󵄨󵄨󵄨ψ󵄨󵄨󵄨󵄨2 ) x1 (1)Here Δ is the Laplacian operator in R3 and i is the imaginary unit, a(x), b(x), and p satisfy some additional assumptions.The Davey-Stewartson system is a model for the evolution of weakly nonlinear packets of water waves that travel predominantly in one direction, but in which the amplitude of waves is modulated in two spatial directions
The function U is a critical point of C2 functional I0 : H1(R3) → R defined by I0 possesses a 3-dimensional manifold of critical points
(5) let TθZ denote the tangent space to Z at zθ, the manifold Z is nondegenerate in the following sense: Ker(I0󸀠󸀠(z)) = TθZ and I0󸀠󸀠(zθ) is an index-0 Fredholm operator for any zθ ∈ Z

Summary

Introduction and Main Results

We are going to consider the existence of standing waves for a generalized Davey-Stewartson system in R3 iψt + Δψ = b (x) ψφx1 − a (x) 󵄨󵄨󵄨󵄨ψ󵄨󵄨󵄨󵄨p−2ψ,. In [4, 5], Ambrosetti and Badiale established an abstract theory to reduce a class of perturbation problems to a finite dimensional one by some careful observation on the unperturbed problems and the Lyapunov-Schmit reduction procedure. This method has been successfully applied to many different problems, see [6] for examples. In this paper we are going to consider the following two types of perturbed problems for generalized Davey-Stewartson system.

The Abstract

Proof of the Main Results