Symbolic computation of solitary wave solutions and solitons through homogenization of degree

TL;DR

This paper introduces a simplified, homogenization-based method for computing solitary wave solutions and solitons of nonlinear PDEs, applicable to equations lacking known bilinear forms and extendable to non-solitonic equations.

Contribution

It presents a novel, algorithmic approach using homogenization of degree to compute solitary waves, including for equations without established bilinear forms.

Findings

Successfully applied to fifth-order KdV equations.

Extended to equations without solitons like Fisher and FitzHugh-Nagumo.

Implemented in Mathematica for practical computation.

Abstract

A simplified version of Hirota's method for the computation of solitary waves and solitons of nonlinear PDEs is presented. A change of dependent variable transforms the PDE into an equation that is homogeneous of degree. Solitons are then computed using a perturbation-like scheme involving linear and nonlinear operators in a finite number of steps. The method is applied to a class of fifth-order KdV equations due to Lax, Sawada-Kotera, and Kaup-Kupershmidt. The method works for non-quadratic homogeneous equations for which the bilinear form might not be known. Furthermore, homogenization of degree allows one to compute solitary wave solutions of nonlinear PDEs that do not have solitons. Examples include the Fisher and FitzHugh-Nagumo equations, and a combined KdV-Burgers equation. When applied to a wave equation with a cubic source term, one gets a bi-soliton solution describing the coalescence of two wavefronts. The method is largely algorithmic and is implemented in Mathematica.