Integrability properties and multi-kink solutions of a generalised Fokker-Planck equation

TL;DR

This paper investigates a generalized Fokker-Planck equation, revealing its integrability, constructing multi-kink solutions, and exploring shock wave dynamics, fusion, fission, and Bäcklund transformations.

Contribution

It introduces a geometric approach to derive multi-kink solutions and analyzes shock interactions in the generalized Fokker-Planck framework.

Findings

Solutions include time-dependent shock waves and multi-kinks.

Shock interactions exhibit fusion and fission dynamics.

Bäcklund transformations relate different solutions.

Abstract

We analyse a generalised Fokker-Planck equation by making essential use of its linearisability through a Cole-Hopf transformation. We determine solutions of travelling wave and multi-kink type by resorting to a geometric construction in the regime of small viscosity. The resulting asymptotic solutions are time-dependent Heaviside step functions representing classical (viscous) shock waves. As a result, line segments in the space of independent variables arise as resonance conditions of exponentials and represent shock trajectories. We then discuss fusion and fission dynamics exhibited by the multi-kinks by drawing parallels in terms of shock collisions and scattering processes between particles, which preserve total mass and momentum. Finally, we propose B\"acklund transformations and examine their action on the solutions to the equation under study.