A superposition property for a class of nonlinear partial differential equations

Abstract

This note complements a previous paper by the author in two ways. Firstly errors in this previous paper for the sufficient conditions for sums of two solutions of specific equations to satisfy the equation are underlined. Secondly, in this work, the class of nonlinear partial differential equations that can be decomposed into sums of terms that are products of not necessarily the same number of linear partial differential operators is considered and necessary and sufficient conditions on the operators and on the solutions are established for the linear combinations of two solutions to satisfy the equation. In the mentioned previous paper a necessary and sufficient condition was obtained for the same superposition property for equations that are composed of sums of products of the same number of linear partial differential operators. Also an example is included for the problem taken up.