Approximation to the global solution of generalized Zakharov equations in mathbf{R}^{2}

Shujun You

doi:10.1186/s13660-018-1813-9

Abstract

We consider the initial value problem for the two-dimensional generalized Zakharov equations which model the propagation of Langmuir waves in plasmas. It is obtained that the solutions of the two-dimensional generalized Zakharov equations converge as alphato0 to a solution of the Zakharov equations. Both weak and strong solutions are considered.

Highlights

1 Introduction Zakharov derived a set of coupled nonlinear wave equations describing the interaction between high-frequency Langmuir waves and low-frequency ion-acoustic waves at the classical level [1]
The usual Zakharov system defined in space time Rd+1 is given by iEt + E = nE, (1)
Tsutsumi, and Velo studied the local Cauchy problem in time for the Zakharov system governing Langmuir turbulence, with initial data (u(0), n(0), ∂tn(0)) ∈ Hk × Hl × Hl–1, in arbitrary space dimension. They proved that the Zakharov system is locally well-posed for a variety of values of (k, l) [6]

Summary

Introduction

Zakharov derived a set of coupled nonlinear wave equations describing the interaction between high-frequency Langmuir waves and low-frequency ion-acoustic waves at the classical level [1]. Tsutsumi, and Velo studied the local Cauchy problem in time for the Zakharov system governing Langmuir turbulence, with initial data (u(0), n(0), ∂tn(0)) ∈ Hk × Hl × Hl–1, in arbitrary space dimension. Guo et al established local in time existence and uniqueness for a generalized Zakharov equation in the case of dimension d = 1, 2, 3.

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