Existence and spectral stability of multi-pulses in discrete Hamiltonian lattice systems

TL;DR

This paper develops a framework for analyzing the existence and spectral stability of multi-pulse solutions in discrete Hamiltonian lattice systems, exemplified by the discrete nonlinear Schrödinger equation, with analytical and numerical validation.

Contribution

It introduces a discrete analogue of Lin's method for multi-pulse analysis and provides explicit conditions and stability eigenvalues for such solutions.

Findings

Explicit existence conditions for multi-pulse solutions.

Analytical expressions for stability eigenvalues.

Good agreement between analytical and numerical results.

Abstract

In the present work, we consider the existence and spectral stability of multi-pulse solutions in Hamiltonian lattice systems. We provide a general framework for the study of such wave patterns based on a discrete analogue of Lin's method, previously used in the continuum realm. We develop explicit conditions for the existence of $m$-pulse structures and subsequently develop a reduced matrix allowing us to address their spectral stability. As a prototypical example for the manifestation of the details of the formulation, we consider the discrete nonlinear Schr\"{o}dinger equation. Different families of $2$- and $3$-pulse solitary waves are discussed, and analytical expressions for the corresponding stability eigenvalues are obtained which are in very good agreement with numerical results.