A formal representation of systems governed by ODEs is:
dot{x} = f(x,u)
y = g(x)
Is there any formal way to define a nonlinear system with PDEs?
This is a good question.
The standard reference for nonlinear infinite dimensional systems is
V. Barbu, Analysis and control of nonlinear infinite dimensional systems, Mathematics in
science and engineering, vol. 190, Academic Press, Inc., 1993.
A sample representation of nonlinear infinite dimensional systems on 4-manfolds is given in Part III in
Timothy Nguyen, The Seiberg-Witten Equations on Manifolds with Boundary, Ph.D. thesis, MIT, 2011:
http://www-math.mit.edu/~timothyn/papers/thesis.pdf
I think that many nonlinear PDE have the form (I use d/dt for partial derivative)
du/dt = F(d/dx, u),
where F is a nonlinear operator that can include partial derivatives w.r.t. x and multiplication by function of u. Very often instead of F there is equation
du/dt = A u,
where A is nonlinear differential operator. While there are many possible types of PDE, interesting physical applications concentrate over the first derivative over time. Here we have KdV equation, nonlinear Schrodinger, Sin-Gordon and others.
A good starting point is the excellent textbook "Partial Differential Equations" by L.C. Evans. Part III of the book is devoted to nonlinear pde's.
@book{evans2010partial,
title={Partial Differential Equations},
author={Evans, L.C.},
isbn={9780821849743},
lccn={2009044716},
series={Graduate studies in mathematics},
url={https://books.google.fr/books?id=Xnu0o\_EJrCQC},
year={2010},
publisher={American Mathematical Society}
}
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