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A periodic problem for the Kadomtsev-Petviashvili equation. (English. Russian original) Zbl 0699.35213
Sov. Math., Dokl. 37, No. 1, 157-161 (1988); translation from Dokl. Akad. Nauk SSSR 298, No. 4, 802-807 (1988).
The author discusses the difference between two variants \((\sigma^ 2=1\) or \(\sigma^ 2=-1)\) of the equation \[ (3/4)\sigma^ 2u_{yy}+(u_ t- (3/2)uu_ x+u_{xxx})_ x=0 \] as the explicit solution of the periodic Cauchy problem is concerned. While the case \(\sigma^ 2=-1\) is proved to be formally non-integrable, the Cauchy problem for the case \(\sigma^ 1=1\) can be locally resolved in the following sense: for any smooth, real, periodic and finite-zone solution \(u_ 0(x,y,t)\) there exists a positive constant \(\epsilon\) such that for any smooth, real, periodic function v(x,y) with \(| u_ 0(x,y,0)-v(x,y)| <\epsilon\) a unique solution u(x,y,t) exists satisfying the initial condition \(u(x,y,0)=v(x,y)\).
Reviewer: J.Chrastina

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35B10 Periodic solutions to PDEs
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