Scattering theory for the Korteweg-de Vries (KdV) equation and its Hamiltonian interpretation. (English) Zbl 0618.35100

Buslaev, V. S.; Faddeev, L. D.; Takhtadzhan, L. A.

Physica D 18, 255-266 (1986).

The paper deals with the Korteweg-de Vries equation of the form \(u_ t- 6uu_ x+u_{xxx}=0\) and shows that the scattering transform for it is not canonical in the naive sense as it produces ”paradoxes”, one of which is also presented. An explanation of this phenomenon is found, a correct Hamiltonian formulation of the scattering theory is proposed.
It is also shown that a necessary condition is the presence of a boundary in the continuous spectrum of the corresponding auxiliary linear problem: the continuous spectrum should not fill the whole real axis (for KdV equation spectrum fills the semi-axis \(\lambda =k^ 2\geq 0\), for the nonlinear Schrödinger equation it fills the real axis with a gap \(| \lambda | <\omega\), for the Toda chain - the interval \(-2\leq \lambda =2\cos k\leq 2)\).

Reviewer: J.Kostarcuk

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
35A30	Geometric theory, characteristics, transformations in context of PDEs
35P25	Scattering theory for PDEs

Keywords:

Korteweg-de Vries equation; scattering transform; Hamiltonian formulation; continuous spectrum; nonlinear Schrödinger equation; Toda chain

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