Time-dependent Darboux transformation and supersymmetric hierarchy of Fokker-Planck equations

TL;DR

This paper introduces a method to solve non-stationary Fokker-Planck equations using Darboux transformations and supersymmetric quantum mechanics, establishing a hierarchy of solutions for specific drift conditions.

Contribution

It extends supersymmetric quantum mechanics to non-stationary Fokker-Planck equations via Darboux transformations, enabling systematic solution generation.

Findings

Developed a procedure for solving time-dependent Fokker-Planck equations.

Established a hierarchy of solutions for shape-invariant drift coefficients.

Linked solutions of Fokker-Planck and Schrödinger equations through supersymmetry.

Abstract

A procedure is presented for solving the Fokker-Planck equation with constant diffusion but non-stationary drift. It is based on the correspondence between the Fokker-Planck equation and the non-stationary Schr\"odinger equation. The formalism of supersymmetric quantum mechanics is extended by applying the Darboux transformation directly to the non-stationary Schr\"odinger equation. From a solution of a Fokker-Planck equation a solution of the Darboux partner is obtained. For drift coefficients satisfying the condition of shape invariance, a supersymmetric hierarchy of Fokker-Planck equations with solutions related by the Darboux transformation is obtained.