One-parameter isospectral solutions for the Fokker–Planck equation

Abstract

Using the fact that the one-dimensional Fokker–Planck (1DFP) equation can be set into an imaginary time-dependent Schrödinger equation form, in this work we propose an approach to find the generalized solutions of the 1DFP equation. To do that, we assume that the involved potential in the one-dimensional Schrödinger equation can be identified from a proper ansatz, or a Witten superpotential W( x), in the corresponding Riccati equation. Consequently, using the generalized W g ±( x) as particular solutions of the nonlinear differential equations involved, we find the generalized partner potentials which allow simultaneously to obtain the associated generalized solutions of the corresponding stationary and time-dependent 1DFP equations. Such generalized solutions are one-parameter isospectral solutions as result of the supersymmetric quantum mechanics or Darboux transform approaches. As an useful application of the proposed equations we found the generalized solutions and the isospectral potential partners resulting on the 1DFP equation for the cases of a parabolic and an inverted drift potentials. Besides, the proposed method can be used for searching other solvable drift potentials as well as to find their corresponding generalized probability densities.