Formation of rogue waves on the periodic background in a fifth-order nonlinear Schr\"odinger equation

TL;DR

This paper constructs rogue wave solutions on periodic backgrounds in a fifth-order nonlinear Schrödinger equation, revealing how their stability varies with system parameters and background types.

Contribution

It introduces a method combining Darboux transformation and spectral nonlinearization to generate rogue waves on elliptic function backgrounds in a fifth-order NLS system.

Findings

Rogue wave solutions are constructed on elliptic function backgrounds.

Instability growth rates depend on the type of periodic background and elliptic modulus.

Novel features of rogue waves on different periodic backgrounds are analyzed.

Abstract

We construct rogue wave solutions of a fifth-order nonlinear Schr\"odinger equation on the Jacobian elliptic function background. By combining Darboux transformation and the nonlinearization of spectral problem, we generate rogue wave solution on two different periodic wave backgrounds. We analyze the obtained solutions for different values of system parameter and point out certain novel features of our results. We also compute instability growth rate of both $dn$ and $cn$ periodic background waves for the considered system through spectral stability problem. We show that instability growth rate decreases (increases) for $dn$-$(cn)$ periodic waves when we vary the value of the elliptic modulus parameter.