An Initial Boundary Value Problem for the Zakharov Equation

This paper studies an inhomogeneous initial boundary value problem for the one-dimensional Zakharov equation. Existence and uniqueness of the global strong solution are proved by Galerkin’s method and integral estimates. 1. Introduction In this paper, we consider the following inhomogeneous initial boundary value problem for the Zakharov equations in one dimension: Zakharov equations play an important role in the turbulence theory for plasma waves and resemble closely to the nonlinear Schr？dinger equations. There has been extensive study both theoretically and numerically on these equations (e.g., see [1–17]). Most investigation have been focused on the Cauchy problem of the system of Zakharov equations (SZEs), sometimes with a homogeneous boundary condition. It is well known that Zakharov equations possess 1D soliton solutions. Some numerical experiments suggest that the solutions of 2D and 3D SZE may become singular in finite time [7, 18]. Global existence of solutions of the -dimensional SZE ( ) has been only proved for small initial data [5, 6]. It has been shown that solutions with large initial data may blow up in finite time [8]. The system (1)-(2) above describes the interaction of a Langmuir wave and an ion acoustic wave in a plasma (Dendy [19] and Bellan [20]). Here, is an unknown complex vector-valued function that denotes a slowly varying envelope of a highly oscillatory electric field. Meanwhile, is an unknown real function that denotes the fluctuation in the ion density about its equilibrium value (see [19, 21]). We assume that , are given smooth functions. Let be any function on with compact support satisfying , and we define Thus, the problem (1)–(4) is equivalent to In Section 2, we obtain the existence of a local weak solution via Galerkin’s method and the principle of compactness. In Section 3, we derive estimates of higher-order derivatives of Galerkin’s approximate solution to obtain the existence and uniqueness of the local strong solution. In Section 3, we prove the existence and uniqueness of the global strong solution. 2. Existence of a Local Strong Solution We first work on Galerkin’s approximation solution for the problem (6)–(9) by choosing the basic functions as follows: The approximate solution for the problem (6)–(9) can be written as According to Galerkin’s method, these undetermined coefficients must satisfy the following initial value problem for a system of ordinary differential equations: with , , and To obtain existence of a local weak solution, we need the following lemmas. Lemma 1. Assume that , then . Proof. We