Instability of single- and double-periodic waves in the fourth-order nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the spectral stability and computes the instability rates of single- and double-periodic wave solutions in a fourth-order nonlinear Schrödinger equation, revealing that double-periodic waves exhibit higher instability growth rates.

Contribution

It derives explicit single- and double-periodic solutions using Jacobian elliptic functions and computes their spectral stability spectra and instability rates, highlighting novel features of the system.

Findings

Double-periodic waves have higher instability growth rates than single-periodic waves.

Spectral spectra are computed for various elliptic modulus parameters.

The fourth-order dispersion parameter influences the instability rates significantly.

Abstract

We compute the instability rate for single- and double-periodic wave solutions of a fourth-order nonlinear Schr\"odinger equation. The single- and double-periodic solutions of a fourth-order nonlinear Schr\"odinger equation are derived in terms of Jacobian elliptic functions such as $dn$, $cn$, and $sn$. From the spectral problem, we compute Lax and stability spectrum of single-periodic waves. We then calculate the instability rate of single-periodic waves (periodic in the spatial variable). We also obtain the Lax and stability spectrum of double-periodic wave solutions for different values of the elliptic modulus parameter. We also highlight certain novel features exhibited by the considered system. We then compute instability rate for two families of double-periodic wave solutions of the considered equation for different values of the system parameter. Our results reveal that the instability growth rate is higher for the double-periodic waves due to the fourth-order dispersion parameter when compared to single-periodic waves.