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A challenge like this frequently requires that you try various approaches, just to get sufficient grasp on the problem, such that you discover a new approach that can solve the challenge.
But you might not have reached that point except by trying the unsuccessful approaches first.
Are you familiar with the story how the equations came about for generating a fractal image on a computer, pixel by pixel?
There is ABSOLUTELY no common methods for nonlinear PDE's (even linear). Consider this fact, each nonlinear PDE almost represents a branch. Quantum mechanics is all about Schrodinger equation; general relativity is all about the Einstein equation; magnetic- electronics is about Maxwell equation;look at shallow water waves which are represented by KdV equation, ... .Every has a completely different method.
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