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The Korteweg-de Vries equation in classes of increasing functions with prescribed asymptotics as \(| x| \to \infty\). (English. Russian original) Zbl 0568.35083

Math. USSR, Sb. 50, 125-135 (1985); translation from Mat. Sb., Nov. Ser. 122(164), No. 2, 131-141 (1983).
The initial-value problem for the KdV equation \(u_ t+uu_ x+u_{xxx}=0\), \(u|_{t=0}=u_ 0(x)\) is considered. The initial function \(u_ 0(x)\) is assumed to be asymptotically equivalent to \(\sum^{\infty}_{j=0}c_ j^{\pm}(\pm x)^{\alpha_ j}\), \(\alpha_ 0<1\), \(\alpha_ j>\alpha_{j+1}\), \(\alpha_ j\to -\infty\) as \(j\to \infty\). Asymptotic solutions of the problem with leading term \(c_ 0^{\pm}(\pm x)^{\alpha_ 0}\) have been constructed in a preceding paper [Sov. Math., Dokl. 26, 716-719 (1982); translation from Dokl. Akad. Nauk SSSR 267, 1035-1038 (1982; Zbl 0522.35076)]. Now it is demonstrated that the asymptotic solution differs from the rigorous one by a function w(x,t) belonging to the Schwartz space \(S({\mathbb{R}}_ x)\) for any \(t\in {\mathbb{R}}\).
Reviewer: I.Ya.Dorfman

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35C20 Asymptotic expansions of solutions to PDEs

Citations:

Zbl 0522.35076
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