Fundamental solution of nonlinear PDE 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?
 A: 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. 
A: Just three notes if the question is still active.


*

*In the paper "A contribution to the theory of generalized functions" by Yurii Egorov it is mentioned (see p. 9) that using the Poisson kernel
$$
\delta_k(x) = \frac{1}{\pi} \frac{k}{1 + k^2 x^2},
$$
it is possible to show that for any regular compactly supported function $\varphi$, there is an arbitrary constant $C$ such that
$$
\int_{-\infty}^\infty \delta^2_k(x) \varphi(x) dx \to C \varphi(0) ~~ {\rm as} ~~ k \to \infty.
$$
This means that in the sense of distributions,
$$
\delta_k^2 \to C \delta ~~ {\rm as} ~~ k \to \infty.
$$
This weak limit justifies the following notion (in the sense of distributions):
$$
\delta^2(x) = C \delta(x).
$$
Note that this result is to the prominent mathematician V. S. Vladimirov. It, in principle, shows that it is "possible" to consider nonlinear generalized functions as weak limits of $\delta$-like sequences.

*I have recently came accross the interesting book "A Nonlinear Theory of Generalized Functions" by Biagioni, where the author develops a nonlinear theory of generalized functions. He also solves nonlinear partial differential equations using generalized functions.

*You can also take a look at "Multiplication of Distributions: A tool in mathematics, numerical engineering and theoretical physics" by Colombeau. You can find generalized solution of nonlinear equations.
