This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like $$ \partial^2\phi+V(\phi)=\delta^D(x). $$ I do not know if a real meaning can be attached to something like this. But I can think to a gradient expansion. E.g., consider the nonlinear Klein-Gordon equation $$ \partial^2\phi+\lambda\phi^3=\delta^D(x). $$ A gradient expansion has a leading order $$ \partial_t^2\phi_0+\lambda\phi_0=\delta^D(x) $$ and I know the exact solution to $\partial_t^2\phi'+\lambda\phi'=\delta(t)$. Can I attach a meaning to something like this from the exact solution of this latter equation maybe using Coulombeau functions?

This is more than an exercise as can have some applications in physics. But what I am really interested to is if all this can have a mathematical meaning.

Thanks.

This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like $$ \partial^2\phi+V(\phi)=\delta^D(x). $$ I do not know if a real meaning can be attached to something like this. But I can think to a gradient expansion. E.g., consider the nonlinear Klein-Gordon equation $$ \partial^2\phi+\lambda\phi^3=\delta^D(x). $$ We can rewrite this equation as $$ \partial_t^2\phi+\lambda\phi^3=\delta(t)\delta^{D-1}(x) + \epsilon\Delta\phi. $$ and to treat the Laplacian as a perturbation (for this aim I introduced an arbitrary $\epsilon$ that I will set to 1 at the end of computation). Then, I have a gradient expansion in term of powers $\epsilon$. This has for the leading order $$ \partial_t^2\phi_0+\lambda\phi_0=\delta^D(x) $$ and I know the exact solution to $\partial_t^2\phi'+\lambda\phi'=\delta(t)$. Can I attach a meaning to something like this from the exact solution of this latter equation maybe using Coulombeau functions?

There is also a less singular approach. I just rescale time as $\tau=\sqrt{\lambda}t$ and I get $$ \lambda\partial_\tau^2\phi+\lambda\phi^3=\sqrt{\lambda}\delta(\tau)\delta^{D-1}(x) + \Delta\phi $$ that is $$ \partial_\tau^2\phi+\phi^3=\frac{1}{\sqrt{\lambda}}\delta(\tau)\delta^{D-1}(x) + \frac{1}{\lambda}\Delta\phi. $$ This is again a gradient expansion in powers of $\frac{1}{\sqrt{\lambda}}$ but the leading order has the form $$ \partial_\tau^2\phi_0+\phi_0^3=0 $$ that has a known exact solution. Next-to-leading order is no more singular giving a linear equation.

This is more than an exercise as can have some applications in physics. But what I am really interested to is if all this can have a mathematical meaning.

Thanks.