@MISC{Bidégaray96ona, author = {B. Bidégaray}, title = {On a nonlocal Zakharov equation}, year = {1996} }

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Abstract

We study the Cauchy problem for a nonlocal Zakharov equation, namely ae i' t + \Delta' = r(\Gamma\Delta) \Gamma1 r:(n'); \Gamma2 n tt \Gamma \Deltan = \Deltaj'j 2 : We first study the Cauchy problem for a fixed and then, in a smaller functional space, the limit of the solutions when tends to 1. 1 Introduction Our aim is to prove some results about a nonlocal Zakharov equation introduced by Zakharov (see [14],[15]). The derivation of this system is carried out for x 2 R 3 , however we will suppose that x 2 R N ; N = 1; 2; 3: We study the following system ae i OE + \DeltaOE = \GammaB(nOE); \Gamma2 n \Gamma \Deltan = \DeltajOEj 2 ; where B = r\Delta \Gamma1 r: . We consider the initial value problem, that is 8 ! : OE(x; 0) = OE 0 (x); n(x; 0) = n 0 (x); n t (x; 0) = n 1 (x): This article is divided into three parts. In Part 2, we derive the equation from the physical equations according to Zakharov (see [14]). In Part 3, we study the local Cauchy pro...