Rogue waves on the double periodic background in Hirota equation

TL;DR

This paper develops a method to generate rogue wave solutions on double periodic backgrounds for the Hirota equation using Darboux transformation, identifying eigenfunctions and eigenvalues algebraically.

Contribution

It introduces a novel approach to construct rogue waves on complex backgrounds for the Hirota equation via algebraic Darboux transformation with two eigenvalues.

Findings

Constructed rogue wave solutions on double periodic backgrounds.

Identified eigenfunctions and eigenvalues algebraically.

Presented localized structures for different parameter sets.

Abstract

We construct rogue wave solutions on the double periodic background for the Hirota equation through one fold Darboux transformation formula. We consider two types of double periodic solutions as seed solutions. We identify the squared eigenfunctions and eigenvalues that appear in the one fold Darboux transformation formula through an algebraic method with two eigenvalues. We then construct the desired solution in two steps. In the first step, we create double periodic waves as the background. In the second step, we build rogue wave solution on the top of this double periodic solution. We present the localized structures for two different values of the arbitrary parameter each one with two different sets of eigenvalues.