Computational aspects of the Maslov index of solitary waves

TL;DR

This paper develops a robust numerical algorithm to compute the Maslov index of solitary waves, providing new insights into their stability across various equations and phase space dimensions.

Contribution

It introduces a novel exterior algebra-based algorithm for calculating the Maslov index and applies it to different types of solitary wave solutions.

Findings

New algorithms for Maslov index computation are robust and fast.

Applications to reaction-diffusion, KdV, and resonance equations reveal stability properties.

Part 1 and 2 address different phase space dimensions.

Abstract

When solitary waves are characterized as homoclinic orbits of a finite-dimensional Hamiltonian system, they have an integer-valued topological invariant, the Maslov index. We are interested in developing a robust numerical algorithm to compute the Maslov index, to understand its properties, and to study the implications for the stability of solitary waves. The algorithms reported here are developed in the exterior algebra representation, which leads to a robust and fast algorithm with some novel properties. We use two different representations for the Maslov index, one based on an intersection index and one based on approximating the homoclinic orbit by a sequence of periodic orbits. New results on the Maslov index for solitary wave solutions of reaction-diffusion equations, the fifth-order Korteweg-De Vries equation, and the longwave-shortwave resonance equations are presented. Part 1 considers the case of four-dimensional phase space, and Part 2 considers the case of $2n-$dimensional phase space with $n>2$.