A steepest descent algorithm for the computation of traveling dissipative solitons

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. In the literature, there is a conjecture that no stable traveling pulse exists for a 2-component system in 2D unbounded domains. In many instances, such numerical studies investigate only solutions with a small speed, as they rely on good initial guesses based on stable standing pulse profiles. Studying a modified problem with a 2D strip domain R×[-L,L]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}} \times [-L,L]$$\end{document} with zero Dirichlet boundary conditions at y=±L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=\pm L$$\end{document}, by using our algorithm we establish the existence of fast-moving solitons. With an appropriate set of physical parameters in this unbounded rectangular strip domain, 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, we anticipate that this approach could be used to augment or initiate continuation algorithms. We also note that the rectangular strip domain can serve as a first step to investigating waves in the whole of R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^2$$\end{document}.