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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
##### Keywords:
formal integrability; Cauchy problem; finite-zone solution