What about some engaging PDE topics for undergraduates? Does anyone have any suitable PDE topics (research or otherwise) for an undergraduate math student? Consider the student has only completed an introductory class in PDE. The student also has experience in basic real analysis 
Is nonlinear PDE just out of this student's league?
I am willing to listen to both sides: Applied or Pure, but I am leaning towards the pure. Also, I'd like to stay away from Numerical Methods
 A: Here are a few ideas.  There isn't really an "answer" to this question, but it's too long for a comment.
Some fairly tractable nonlinear PDE: 


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*1D Burger's equation $u_t+uu_x=\epsilon u_{xx}$.  With this example in mind they could study the method of characteristics, weak solutions, shock waves, etc. If they need an application, this equation models traffic flow and shallow water waves.

*Nonlinear diffusion equation $u_t=Du_{xx}+au-bu^3$.  Some nice "qualitative" analysis can be done with this equation - phase planes, dimensional analysis and nondimensionalization, self-similar solutions, travelling wave solutions.

*1D KdV equation $u_t+(u_x)^3+6uu_x=0$.  Dispersion relations, solitons, integrability, inverse scattering transform (ambitious for the average undergrad)


Here are some general methods that are invaluable to nearly all PDE:


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*The Fourier Transform, dispersion relations

*Green's functions, fundamental solutions

*Functional analysis as it applies to PDE: Hilbert spaces and orthonormal expansions, basic operator & spectral theory, Fredholm integral operators

*The method of characteristics, shocks, nonlinear conservation laws (i.e. PDE of the form $u_t+(f(u))_x=0$)

*Calculus of variations (ambitious)

*Ill-posed problems like the backwards heat equation $u_t+\triangle u=0$ (more ambitious)


Some more "classical" PDE topics that still require interesting analysis: 


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*Perron's method for the Laplace equation (hard)

*The transport equation $u_t+cu_x=0$ and transport-diffusion equation $u_t+cu_x=\epsilon u_{xx}$.  Both are solved using the method of characteristics.

*The 3D wave equation (most introductory PDE courses only cover 1D and 2D).

*Derivation of the heat equation using random walks & Brownian motion

A: In a famous Russian novel "Master and Margarita" Mikhail Bulgakov wrote

'No,' replied Margarita, ' what really puzzles me is where you have
  found the space for all this.' With a wave of her hand Margarita emphasised
  the vastness of the hall they were in.
  Koroviev smiled sweetly, wrinkling his nose.
  'Easy!' he replied. ' For anyone who knows how to handle the fifth
  dimension it's no problem to expand any place to whatever size you please.

It was always engaging for me to speculate whether the author knew about the properties of wave propagation in the spaces of even dimensions, or he picked up the number five over a more obvious "four" due to some other reasons :) 
