Solitonic Combinations, Commuting Nonselfadjoint Operators, and Applications

Abstract

In this paper, we consider applications of the connection between the soliton theory and the commuting nonselfadjoint operator theory, established by Livšic and Avishai. An approach to the inverse scattering problem and to the wave equations is presented, based on the Livšic 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 Schrödinger 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 Schrödinger 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.