# Non-linear Basis Functions for PDE

An idea popped into my head awhile ago while doing a project on non-linear effects of systems of coupled oscillators, but I'm not an expert on this subject so I don't know if it's any good. From cursory searching I couldn't seem to find anything, which most likely means it doesn't work, or I don't know the correct jargon.

From my understanding, the foundation of the theory of linear systems can be attributed to the fact you can construct a basis in the space of solutions and you can use that basis to find other solutions subject to boundary conditions. This all rests on the fact that for some linear operator $$L$$ and two functions $$f$$ and $$g$$ that satisfy:

$$L(f)=0$$ $$L(g)=0$$

Then we have:

$$L(f+g)=0$$

The problem with non-linear systems is that you can't construct a basis this way. Suppose we have a linear operator $$L$$, but also a non-linear operator $$N$$, and two functions $$f$$ and $$g$$ that satisfy:

$$L(f)=N(f)$$ $$L(g)=N(g)$$

Then: $$L(f+g)\ne N(f+g)$$

We can't construct a basis by addition. But perhaps there is still a way in general to construct a basis? Can we find some operator $$\phi(f, g)$$ that satisfies:

$$L(\phi(f, g))=N(\phi(f, g))$$

Where $$\phi$$ could be how to "combine" solutions into other solutions. For linear systems we'd simply have $$\phi(f,g)=f+g$$, but I thought that in general for this idea to be consistent you'd want to make sure $$\phi$$ obeys the following conditions:

$$\phi(f, g)=\phi(g, f)$$ $$\phi(f, 0)=f$$ $$\phi(f, \phi(g, h))=\phi(\phi(f, g), h)$$

The most general way I see to do this is to define $$\phi$$ as:

$$\phi(f,g)=F^{-1}(F(f)+F(g))$$

Where $$F$$ is some transformation. However, this implies:

$$F(\phi(f, g))=F(f)+F(g)$$

So that the system becomes linear under a suitable transformation $$F$$. I didn't attempt to figure out if a transformation $$F$$ of this type always exists, and it seems like in most practical cases trying to find $$F$$ requires you to solves another non-linear equation, so I don't know if it can be called progress.

Is there literature I can read that explores the gist of this idea, or is it simply considered trivial in the cases where a system can be made linear? Can it be shown that in some systems there is no such $$F$$?