Exact Jacobi elliptic solutions of some models for the interaction of long and short waves

TL;DR

This paper derives exact Jacobi elliptic solutions for models describing long and short wave interactions, proving the uniqueness of these periodic solutions and extending previous findings on synchronized wave solutions.

Contribution

It provides explicit Jacobi elliptic function solutions for the models and establishes their uniqueness, advancing the understanding of wave interactions in dispersive media.

Findings

Exact periodic solutions in terms of Jacobi elliptic functions

Uniqueness of cnoidal wave solutions

Extension of previous synchronized wave solutions

Abstract

Some systems were recently put forth by Nguyen et. al. as models for studying the interaction of long and short waves in dispersive media. These systems were shown to possess synchronized Jacobi elliptic solutions as well as synchronized solitary wave solutions under certain constraints, i.e., vector solutions where the two components are proportional to one another. In this paper, the exact periodic traveling wave solutions to these systems in general are found to be given by Jacobi elliptic functions. Moreover, these cnoidal wave solutions are unique. Thus, the explicit synchronized solutions under some conditions obtained by Nguyen et. al. are also indeed unique.