Fundamental solution of nonlinear PDE

Question

A fundamental solution of a linear PDE (in sense of Schwartz), $Lu=0$ is defined as a distribution $E$ such that $LE=\delta$. Now I wish to find fundamental solution of nonlinear PDE, such as the burger equation. Could anyone suggest a possible procedure?

Answer 1

I would not agree with what you have written. Fundamental solution is indeed a distribution that solves $$ LE=\delta. $$ This is not the fundamental solution to $$ Lu=0. $$ The primary use of the fundamental solution is that given an inhomogeneous equation $$ Lu=f, $$ and the fundamental solution $E$, we have $$ u=E\ast f. $$ This is true because $L$ is a linear operator, for which the superposition principle works. This will not work for a nonlinear equation, and hence the notion of the fundamental solution for a nonlinear equation does not make much sense.