# A global algorithm for the computation of traveling dissipative solitons

## TL;DR

This paper introduces a global variational algorithm based on steepest descent for computing traveling dissipative solitons in FitzHugh-Nagumo equations, capable of finding multiple solutions and handling bifurcations.

## Contribution

It presents a robust, global method for calculating dissipative solitons that can find multiple solutions and is effective even with poor initial guesses.

## Key findings

- Successfully computes single and multi-soliton solutions.
- Handles bifurcations with minimal performance impact.
- Automatically finds various pulse solutions as energy minimizers.

## Abstract

An algorithm is proposed to calculate traveling dissipative solitons for the FitzHugh-Nagumo equations. It is based on the application of the steepest descent method to a certain functional. This approach can be used to find solitons whenever the problem has a variational structure. Since the method seeks the lowest energy configuration, it has robust performance qualities. It is global in nature, so that initial guesses for both the pulse profile and the wave speed can be quite different from the correct solution. Also, bifurcations have a minimal effect on the performance. With an appropriate set of physical parameters in two dimensional domains, we observe the co-existence of single-soliton and 2-soliton solutions together with additional unstable traveling pulses. The algorithm automatically calculates these various pulses as the energy minimizers at different wave speeds. In addition to finding individual solutions, this approach could be used to augment or initiate continuation algorithms.