Differential transformations of parabolic second-order operators in the plane
Abstract
Description
Here, Darboux's classical results about transformations with differential substitutions for hyperbolic equations are extended to the case of parabolic equations of the form $L u = \big(D^2_{x} + a(x,y) D_x + b(x,y) D_y + c(x,y)\big)u=0$.
We prove a general Theorem that provides a way to determine transformations for parabolic equations shown above. It turnes out that transforming operators $M$ of some higher order can be always represented as a composition of some first-order operators that consecutively define a series of transformations.
Existence of inverse transformations implies some differential constrains on the coefficients of the initial operator. We show that these relations can imply famous integrable equations, in particular, the Boussinesq equation.
12 pages, LaTeX 2e, v2: small changes, the dedication added. Accepted for publication in: Proceesings Steklov Inst. Math. (Moscow), 2009, v. 266
12 pages, LaTeX 2e, v2: small changes, the dedication added. Accepted for publication in: Proceesings Steklov Inst. Math. (Moscow), 2009, v. 266