On nonlinear equations related to the distributions of composition of processes

The relationship between non-trivial linear partial differential equations and the probability densities of compositions of relevant processes has been pointed out in the recent literature.In this note, we construct exact solutions for nonlinear partial differential equations starting from these linear equations by using simple transformations. In this way, we have an interesting bridge between the fundamental solutions of linear equations with a clear probabilistic meaning and the construction of exact interesting solutions for the corresponding nonlinear equations.This method can be obviously generalized to many other cases.