Multi-pulse solitary waves in a fourth-order nonlinear Schr\"odinger equation

TL;DR

This paper studies the existence and spectral stability of multi-pulse solitary waves in a nonlinear Schrödinger equation with higher-order dispersion, revealing conditions for their existence and demonstrating their instability through analytical and numerical methods.

Contribution

It provides a new criterion for the existence of multi-pulse solutions and analyzes their spectral stability, extending understanding of complex solitary wave structures.

Findings

Multi-pulse solutions exist under specific dispersion coefficient conditions.

All multi-pulses are spectrally unstable under certain assumptions.

Numerical results confirm analytical predictions for double pulses.

Abstract

In the present work, we consider the existence and spectral stability of multi-pulse solitary wave solutions to a nonlinear Schr\"odinger equation with both fourth and second order dispersion terms. We first give a criterion for the existence of a single solitary wave solution in terms of the coefficients of the dispersion terms, and then show that a discrete family of multi-pulse solutions exists which is characterized by the distances between the individual pulses. We then reduce the spectral stability problem for these multi-pulses to computing the determinant of a matrix which is, to leading order, block diagonal. Under an additional assumption, which can be verified numerically, we show that all multi-pulses are spectrally unstable. For double pulses, numerical computations are presented which are in good agreement with our analytical results.