Darboux transformations and linear parabolic partial differential equations

Solutions for a class of linear parabolic partial differential equation are provided. These solutions are obtained by first solving a system of (n + 1) nonlinear partial differential equations. This system arises as the coefficients of a Darboux transformation and is equivalent to a matrix Burgers' equation. This matrix equation is solved using a generalized Hopf-Cole transformation. The. solutions for the original equation are given in terms of solutions of the heat equation. These results are applied to the (1 + 1)-dimensional Schrodinger equation where all bound state solutions are obtained for a 2n-parameter family of potentials. As a special case, the solutions for integral members of the regular and modified Poschl-Teller potentials are recovered.