Generalization of SUSY intertwining relations: New exact solutions of Fokker-Planck equation

¹ Saint Petersburg State University - 7/9 Universitetskaya nab., St.Petersburg, 199034 Russia
² Akaki Tsereteli State University - 4600 Kutaisi, Republic of Georgia

^(a) m.ioffe@spbu.ru
^(b) cutaisi@yahoo.com

Received: 28 January 2020
Accepted: 31 March 2020

Abstract

It is commonly known that the Fokker-Planck equation is exactly solvable only for some particular systems, usually with time-independent drift coefficients. To extend the class of solvable problems, we use the intertwining relations of SUSY Quantum Mechanics but in new —asymmetric— form. It turns out that this form is just useful for the solution of the Fokker-Planck equation. As usual, intertwining provides a partnership between two different systems both described by the Fokker-Planck equation. Due to the use of an asymmetric kind of intertwining relations with a suitable ansatz, we managed to obtain a new class of analytically solvable models. What is important, this approach allows us to deal with the drift coefficients depending on both variables, x and t. An illustrating example of the proposed construction is given explicitly.