Weyl functions, inverse problem and special solutions for the system auxiliary to the nonlinear optics equation

TL;DR

This paper establishes a uniqueness theorem for a system related to the N-wave equation using Weyl functions, introduces an inverse problem solution procedure, and provides explicit solutions and evolution analysis.

Contribution

It presents a new uniqueness theorem based on Weyl functions, along with a simple sufficient condition for the inverse problem's unique solvability.

Findings

Proved a Borg-Marchenko type uniqueness theorem.

Developed a procedure to solve the inverse problem.

Obtained explicit solutions and analyzed the evolution of the Weyl function.

Abstract

A Borg-Marchenko type uniqueness theorem (in terms of the Weyl function) is obtained here for the system auxiliary to the N-wave equation. A procedure to solve inverse problem is used for this purpose. The asymptotic condition on the Weyl function, under which the inverse problem is uniquely solvable, is completed by the new and simple sufficient condition on the potential, granting the fulfillment of this asymptotic condition. The evolution of the Weyl function is discussed and the solution of an initial-boundary value problem for the N-wave equation follows. Explicit solutions of the system are obtained. System with a shifted argument is treated.