Solitonic combinations, commuting nonselfadjoint operators, and applications

TL;DR

This paper explores the link between soliton theory and commuting nonselfadjoint operator theory, presenting new methods for inverse scattering problems and deriving generalized equations for various differential equations.

Contribution

It introduces an approach based on Livšic's operator colligation theory for commuting nonselfadjoint operators, extending to nondissipative operators and deriving generalized Gelfand-Levitan-Marchenko equations.

Findings

Derived generalized Gelfand-Levitan-Marchenko equations for multiple differential equations.

Established relations between input and output wave equations in open systems.

Identified differential equations satisfied by components of input and output in specific cases.

Abstract

In this paper, applications of the connection between the soliton theory and the commuting nonselfadjoint operator theory, established by M.S. Liv\v{s}ic and Y. Avishai, are considered. An approach to the inverse scattering problem and to the wave equations is presented, based on the Liv\v{s}ic operator colligation theory (or vessel theory) in the case of commuting bounded nonselfadjoint operators in a Hilbert space, when one of the operators belongs to a larger class of nondissipative operators with asymptotics of the corresponding nondissipative curves. The generalized Gelfand-Levitan-Marchenko equation of the cases of different differential equations (the Korteweg-de Vries equation, the Schr\"{o}dinger equation, the Sine-Gordon equation, the Davey-Stewartson equation) are derived. Relations between the wave equations of the input and the output of the generalized open systems, corresponding to the Schr\"{o}dinger equation and the Korteweg-de Vries equation, are obtained. In these two cases, differential equations (the Sturm-Liouville equation and the 3-dimensional differential equation), satisfied by the components of the input and the output of the corresponding generalized open systems, are derived.