General rogue waves in the three-wave resonant interaction systems

TL;DR

This paper derives and classifies new rogue wave solutions in three-wave resonant interaction systems using bilinear methods, revealing their existence conditions, dynamics, and novel patterns, and connecting them with previous Darboux transformation results.

Contribution

It introduces a generalized bilinear method to derive rogue waves of arbitrary root multiplicities in three-wave systems, including new solution families and their dynamics.

Findings

New rogue wave solutions associated with two simple roots and higher-order solutions.

Existence of solutions depends on signs of nonlinear coefficients in the system.

Identification of new rogue-wave patterns and their dynamics.

Abstract

General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction respectively. It is shown that while the first family of solutions associated with a simple root exist for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue-wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained.