Initial boundary value problem for generalized Zakharov equations with nonlinear function terms

Xueqin Wang,Chunlin Lei,Yadong Shang

doi:10.1186/s13661-020-01383-8

Xueqin Wang, Chunlin Lei + Show 1 more

Open Access

https://doi.org/10.1186/s13661-020-01383-8

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Abstract

In this paper, we consider the initial boundary value problem for generalized Zakharov equations. Firstly, we prove the existence and uniqueness of the global smooth solution to the problem by a priori integral estimates, the Galerkin method, and compactness theory. Furthermore, we discuss the approximation limit of the global solution when the coefficient of the strong nonlinear term tends to zero.

Highlights

The Zakharov system, derived by Zakharov in 1972 [1], describes the interaction between Langmuir and ion acoustic waves in an unmagnetized plasma
The usual Zakharov system defined in the space Rd+1 is given by iεt + ε = nε, ntt – n = |ε|2, where the wave fields ε(x, t) and n(x, t) are complex and real, respectively
It has become commonly accepted that the Zakharov system is a general model to govern interaction of dispersive and nondispersive waves

Summary

Introduction

The Zakharov system, derived by Zakharov in 1972 [1], describes the interaction between Langmuir (dispersive) and ion acoustic (approximately nondispersive) waves in an unmagnetized plasma. Gajewski and Zacharias [14] studied the following generalized Zakharov system and established the global existence for initial value problem: iεt + εxx + (α – n)ε = 0,. You and Ning [15] considered the existence and uniqueness of the global smooth solution for the initial value problem of the following generalized Zakharov equations in dimension two: iεt + ε – nε = 0, 2∂ vt + i=1 ∂xi grad φ(v) –. We study the following initial boundary value problem for generalized Zakharov equations: iεt + εxx + (α – n)ε + δ|ε|pε = 0,. Proof Taking the inner product of (2.7) and –εtxx, it follows that iεtt + εxxt + αεt – ntε – nεt + δ|ε|pε t, –εtxx = 0.

L2 εxx

G L2 Gx L2