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Local solutions of the Kadomtsev-Petviashvili equation. (English) Zbl 0703.35155

The author considers a Cauchy problem of the Kadomtsev-Petviashvili equation \[ (U_ t+\alpha UU_ x+\beta U_{xxx})_ x+\gamma U_{yy}=0, \] where \(U=U(t,x,y)\) is a scalar unknown function and \(\alpha\), \(\beta\), \(\gamma\) are real constants. The main purpose of this paper is to construct a local (in time) solution of KP equation for initials small in the Sobolev space \(W^{2,s}(\Omega)\) with \(s\geq 3\) and a certain restriction. As for domain \(\Omega\), the author deals with four cases \({\mathbb{R}}^ 2\), \({\mathbb{R}}\times {\mathbb{T}}\), \({\mathbb{T}}\times {\mathbb{R}}\) and \({\mathbb{T}}^ 2\) where \({\mathbb{T}}\) is a one-dimensional torus. The existence of solutions is established by using general local theory for the quasilinear symmetric hyperbolic system. The proofs are given in detail.
Reviewer: N.Kostov

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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