Factorization of damped wave equations with cubic nonlinearities

TL;DR

This paper applies a novel factorization method to damped wave equations with cubic nonlinearities, revealing that traveling kink solutions exist only along specific hyperbolic curves in the velocity-damping plane.

Contribution

It extends a previous factorization scheme to a new class of nonlinear PDEs, providing a geometric characterization of kink solutions.

Findings

Traveling kink solutions occur only along hyperbolas in the velocity versus damping coefficient plane.

The factorization approach offers a new analytical tool for nonlinear wave equations.

The results connect solution existence to geometric constraints in parameter space.

Abstract

The recent factorization scheme that we introduced for nonlinear polynomial ODEs in math-ph/0401040 is applied to the interesting case of damped wave equations with cubic nonlinearities. Traveling kink solutions are possible in the plane defined by the kink velocity versus the damping coefficient only along hyperbolas that are plotted herein