Construction of the solution of the inverse spectral problem for a system depending rationally on the spectral parameter, Borg-Marchenko-type theorem, and sine-Gordon equation

TL;DR

This paper develops Weyl theory for a non-classical rational spectral system, proves a Borg-Marchenko-type uniqueness theorem, constructs the inverse problem solution, and applies it to the sine-Gordon equation.

Contribution

It introduces a new Weyl theory for rational spectral systems, establishes a uniqueness theorem, and constructs solutions with applications to sine-Gordon equations.

Findings

Proved a Borg-Marchenko-type uniqueness theorem.

Constructed the inverse spectral problem solution.

Applied the theory to the sine-Gordon equation.

Abstract

Weyl theory for a non-classical system depending rationally on the spectral parameter is treated. Borg-Marchenko-type uniqueness theorem is proved. The solution of the inverse problem is constructed. An application to sine-Gordon equation in laboratory coordinates is given.