Symbolic Computation of Solitary Wave Solutions and Solitons Through Homogenization of Degree

A simplified Hirota method for the computation of solitary waves and solitons of nonlinear partial differential equations (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 fifth-order Korteweg-de Vries (KdV) equations due to Lax, Sawada-Kotera, and Kaup-Kupershmidt. The method works for non-quadratic homogeneous equations for which the bilinear form might be unknown. 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 implemented in Mathematica.