Multi-pulse solitary waves in a fourth-order nonlinear Schrödinger equation

Abstract

In the present work, we consider the existence and spectral stability of multi-pulse solitary wave solutions to a nonlinear Schrödinger 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 additional assumptions, which can be verified numerically and are sufficient to prove orbital stability of the primary solitary wave, we show that all multi-pulses are spectrally unstable. Numerical computations for the spectra of double pulses are presented which are in good agreement with our analytical results. Results of timestepping simulations are also provided to characterize the nature of these instabilities.