Rational solitons of wave resonant interaction models

TL;DR

This paper constructs a broad family of rational and mixed rational-exponential soliton solutions for a system of three coupled wave equations, including special cases like vector NLS and three-wave interaction models.

Contribution

It introduces a novel algebraic Darboux-Dressing method using nilpotent matrices to systematically find bounded rational solutions of three-wave resonant interaction equations.

Findings

Discovered a broad family of rational solutions for three-wave interaction models.

First systematic construction of bounded rational solutions in this context.

Provides algebraic framework applicable to integrable wave interaction systems.

Abstract

Integrable models of resonant interaction of two or more waves in 1+1 dimensions are known to be of applicative interest in several areas. Here we consider a system of three coupled wave equations which includes as special cases the vector Nonlinear Schroedinger equations and the equations describing the resonant interaction of three waves. The Darboux-Dressing construction of soliton solutions is applied under the condition that the solutions have rational, or mixed rational-exponential, dependence on coordinates. Our algebraic construction relies on the use of nilpotent matrices and their Jordan form. We systematically search for all bounded rational (mixed rational-exponential) solutions and find, for the first time to our knowledge, a broad family of such solutions of the three wave resonant interaction equations.