teaching a little nonlinear PDE to an undergraduate

Question

I would like to teach a little nonlinear PDE to an undergraduate who is taking a course in second-order linear boundary value problems. I have never taught nonlinear PDE before, although it is my research specialty. I have a decent book on solitons, but my research specialty is critical point theory in elliptic PDE. Several of my colleagues are experts on solitons. The critical point theory would probably require some Sobolev spaces, which I would like to avoid. I have Evans's great book on PDE, but I think it is too advanced for him. I would like to know if there are any free resources on the Web which I can use, to get readings for myself and the student, and exercises for the student. One more thing: I am interested in PDE that have explicit analytical solutions, or at least can be written as infinite sums of specific functions (I don't want to get into numerical methods for this project).

Answer 1

I would suggest to study equations that admit special solutions that can be found by reducing the original problem to an ODE. Let me illustrate this with some examples:

1. Non-linear boundary value problems with radial symmetry.
2. Travelling wave solutions of semi-linear heat or wave equations
3. Self-similar solutions of semi-linear heat or wave equations, like Burger's equation with viscosity.