Soliton Equations and Higher Order Nonlinear Partial Differential Equations

Abstract

Some nonlinear PDEs partial differential equations are exactly solvable. As an example, nonlinear PDEs such as the Soliton equation were shown to be exactly solvable by quantum operator methods. More recently the Boltzmann equation was solved exactly as well. A question is are other equations such as general arbitrary order nonlinear PDEs solvable. Also recent generalizations of nonextensive statistics has introduced a nonlinear in power of the distribution PDE equation, which are solved for linear drift coefficients by the power-law distribution derived from the Tsallis nonextensive statistics and for which we have recently presented an exact solution for arbitrary nonlinear drift coefficients [3]. Are there possible general solution methods for nonlinear partial differential equations. One possible approach that we present in this letter depends on the ability to transform generally nonlinear PDEs of arbitrary order to 2nd order standard form Fokker-Planck PDEs which have been shown to have exact short time transition probability solutions irregardless of the nonlinear form of the drift and diffusion coefficients. We discuss these questions briefly in the following derivation of a solution to the KdV type of third order nonlinear PDE.