Convergent Analytic Solutions for Homoclinic Orbits in Reversible and Non-reversible Systems

TL;DR

This paper derives convergent series solutions for homoclinic orbits in a fourth-order ODE system, applicable to nonlinear PDEs, and extends previous analyses to non-reversible cases with potential for broader solution classification.

Contribution

It provides the first convergent series solutions for homoclinic orbits in both reversible and non-reversible systems, enhancing analytical understanding of complex wave phenomena.

Findings

Solutions closely match numerical results

Extends analysis beyond traditional bifurcation theory

Applicable to a wide class of nonlinear PDE reductions

Abstract

In this paper, convergent, multi-infinite, series solutions are derived for the homoclinic orbits of a canonical fourth-order ODE system, in both reversible and non-reversible cases. This ODE includes traveling-wave reductions of many important nonlinear PDEs or PDE systems, for which these analytical solutions would correspond to regular or localized pulses of the PDE. As such, the homoclinic solutions derived here are clearly topical, and they are shown to match closely to earlier results obtained by homoclinic numerical shooting. In addition, the results for the non-reversible case go beyond those that have been typically considered in analyses conducted within bifurcation-theoretic settings. We also comment on generalizing the treatment here to parameter regimes where solutions homoclinic to exponentially small periodic orbits are known to exist, as well as another possible extension placing the solutions derived here within the framework of a comprehensive categorization of ALL possible traveling-wave solutions, both smooth and non-smooth, for our governing ODE.