Supersymmetric pairing of kinks for polynomial nonlinearities

TL;DR

This paper introduces a supersymmetric-inspired factorization method to derive and pair kink solutions in polynomial nonlinear differential equations, revealing new solutions with different widths but identical velocities, applicable to reaction-diffusion and oscillator models.

Contribution

It presents a novel factorization approach based on supersymmetry principles to generate and pair kink solutions in polynomial nonlinear equations, expanding solution space and potential applications.

Findings

Derived kink solutions for polynomial nonlinear equations.

Established a pairing mechanism linking different kink solutions.

Applied method to models like Fisher, FitzHugh-Nagumo, and microtubule polymerization.

Abstract

We show how one can obtain kink solutions of ordinary differential equations with polynomial nonlinearities by an efficient factorization procedure directly related to the factorization of their nonlinear polynomial part. We focus on reaction-diffusion equations in the travelling frame and damped-anharmonic-oscillator equations. We also report an interesting pairing of the kink solutions, a result obtained by reversing the factorization brackets in the supersymmetric quantum mechanical style. In this way, one gets ordinary differential equations with a different polynomial nonlinearity possessing kink solutions of different width but propagating at the same velocity as the kinks of the original equation. This pairing of kinks could have many applications. We illustrate the mathematical procedure with several important cases, among which the generalized Fisher equation, the FitzHugh-Nagumo equation, and the polymerization fronts of microtubules