PERIODIC AND CONDITIONALLY PERIODIC ANALOGS OF THE MANY-SOLITON SOLUTIONS OF THE KORTEWEG-DE VRIES EQUATION

Abstract

A method of connecting the Korteweg-de Vries (KdV) equation, known from the theory of nonlinear waves, with the Schrodinger equation was discovered in 1967. (1) This method is applied in the present paper to a study of a periodic problem. We find exact analytical formulae for a class of solutions u(x,t) such that at any moment in time t the potential u(x,t) of the Schrodinger operator has only a finite number of forbidden bands in the Bloch spectrum. We find in this connection all potentials with a finite number of bands. This class of solutions contains as a degenerate limiting case the well known N-soliton solutions of the KdV equation, which decrease rapidly as |x| ! 1.

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