Solution of nonlinear partial differential equations from base equations

Abstract

It is well known that certain nonlinear ordinary differential equations can be solved in terms of the solution of a related linear ordinary differential equation. Although Kamke [l] lists at least twenty such nonlinear differential equations, a note by Pinney [2] seems to be the starting point of recent interest [3-lo] in this method of solution. Ames [ll] gives a summary of results and references through 1965. Solutions of specific nonlinear differential equations [lo] have also served as the basis of solution for nonlinear equations. The differential equation whose solutions are used to solve another differential equations will be called the base equation. Compared .to ordinary differential equations, the use of base equations for solving nonlinear partial differential equations is not as extensive. Ames [12] has discussed certain equations equivalent to linear forms. Along the same lines, Dasarathy [13] has shown through a transformation technique the equivalence of a class of second-order nonlinear equations to third-order linear equations. Montroll [14] h as used the idea of a base equation in solving nonlinear equations describing population growth and diffusion. Recently, Reid and Pritchard [15] have extended a nonlinear differential equation due to Herbst [ 16, Eq. IV] to one of n-dimensions, for which initial value problems may be posed if certain conditions are met by the solution of the base equation. Without being so designated, the base equation technique has been used in constructing nonlinear quantum field theories in which the KleinGordon equation becomes the base equation (see, e.g. [17-191). We mention that Chambers [20] employed solutions of the scalar wave equation to form solutions of the Klein-Gordon equation. In this paper we apply the base equation method to obtain a class of nonlinear partial differential equations of the second order. Rather elaborate analyses [lO-111 have been used to establish forms of nonlinear ordinary differential equations. However, to derive a number of important equations it