Non-linear Basis for PDE's

Question

Asked this on stack exchange and got no response, so I'll try here.

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 $F$?

Answer 1

In your question, I can't make sense of what you want $L$ to be. Suppose you take your favourite nonlinear PDE to be $N$. What would $L$ be?

But I think you want the theory of nonlinear superposition principles for exterior differential systems, as in the paper https://arxiv.org/abs/0708.0679. However, you probably can't read that paper without reading Bryant et. al. Exterior Differential Systems.

Keep in mind that we don't really have anything like a basis of solutions of very many linear partial differential operators, even when we understand the operators very well. What is a basis of solutions of the Laplace operator in the plane?

Answer 2

A natural way to think about this is via Koopman (or composition) operator, corresponding to the nonlinear operator. This linear koopman operator acts on the $\it{functions}$ of state, and hence is necessarily infinite-dimensional, even when the underlying space is finite-dimensional (i.e., if you have an ODE, rather than a PDE). In the extended space, nonlinear basis are simply the eigenfunctions of this operator.

Here's a link to get you started:

https://faculty.missouri.edu/~liyan/Coll.pdf

Take the case of nonlinear system with hyperbolic attractive points (sinks). Locally the nonlinear system is linearizable, and hence has linear basis.

Koopman theoretic tools extend this notion, and formalize "linearization in the large", i.e., in the whole basin of attraction of a given fixed point.

Answer 3

In the spirit close to Ben's answer there also is nonlinear superposition coming from the Backlund transformations for systems that are integrable in the sense of soliton theory, see e.g. this introductory paper for starters.