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Decomposition of initial data of steplike type in the modified Korteweg- de Vries equation. (English. Russian original) Zbl 0723.35070

Sov. Math., Dokl. 41, No. 3, 510-513 (1990); translation from Dokl. Akad. Nauk SSSR 312, No. 5, 1041-1044 (1990).
The initial value problem for the modified Korteweg-de Vries equation \[ U_ t-6U^ 2U_ x+U_{xxx}=0,\quad t>0;\quad U(x,0)=f(x),\quad x\in {\mathbb{R}} \] is considered. The function f(x) satisfies the following boundary conditions: f(x)\(\to a\) (x\(\to -\infty)\), f(x)\(\to b\) \((x\to +\infty)\) with a,b arbitrary constants. This problem is connected with analogous problem for the KdV equation by Miura transformation: \(U^ 2(x,t)-U_ x(x,t)=V(x,t)\). The reconstruction of the solution U(x,t) is given by theorem 1. Its proof may be found in the author’s paper [Math. Notes 46, No.4, 762-769 (1989); translation from Mat. Zametki 46, No.4, 14-24 (1989; Zbl 0702.35222)]. Some results about the decomposition of such solution into solitons for large time are obtained.
Reviewer: V.P.Kotlyarov

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35Q51 Soliton equations

Citations:

Zbl 0702.35222
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