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Commutation methods applied to the mKdV-equation. (English) Zbl 0728.35106

An explicit construction of solutions of the modified Korteweg-de Vries (mKdV) equation given a solution of the Korteweg-de Vries (KdV) equation is provided. There are considered (i) soliton-like; (ii) singular; (iii) periodic two-zone solutions and (iv) solitons relative to periodic two- zone background solutions of the KdV-equation.
The method is based on the fact that the connection between the KdV- and mKdV-equation, effected by the Miura transformation reflects itself in the connection between the corresponding two Lax pairs; and on inversion of the Miura transformation. The realization of these ideas requires the study of
(i) spectral and scattering properties of Dirac and one-dimensional (1d) Schrödinger operators (including the case where the potential has a nontrivial spatial asymptotics as \(x\to \pm \infty);\)
(ii) commutation methods (i.e. \(N=1\) supersymmetry or equivalently Darboux (Crum-Krein) transformation) and
(iii) zero-energy spectral properties of 1d Schrödinger operators.
The methods developed in the present paper are expected to be generalized in the \((1+2)d\) case (e.g. Kadomtsev-Petviashvili equation) as well as in the context of discrete nonlinear systems.
Reviewer: Y.P.Mishev (Sofia)

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
76B25 Solitary waves for incompressible inviscid fluids
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