Undergraduate Project Suggestions

Question

A student of mine has expressed interest in doing an independent project next quarter with me. This would not be for credit and it is purely for her own educational stimulation. She wants to study thermodynamics/fluid dynamics in the future, but she is a (very smart and precocious) freshman who will have only completed Calculus II. At our school, this means she will know up to techniques of integration and the very basics of sequences and series.

My question is: Is there anything that will be both interesting and accessible for her, which she will not eventually have to learn in the standard Linear Algebra/ODE/Vector Calc sequence?

Any suggestions are greatly appreciated!

Answer 1

I have a couple suggestions that have PDEs but only require calculus.

1. She could try to write some Matlab/Python/C/C++ code numerically integrating Navier-Stokes equation. It can never hurt to learn a little programming.

2. Derive the form of the soliton of the KDV equation.

   For the KDV equation $$ u_t+u_{xxx}+6uu_x = 0 $$ determine its moving solitary wave solution of the form $u=u(x-ct)$.

3. Derive the Lax pairs for the KDV.

   By showing the for the following two equations \begin{align} L\phi &= \lambda\phi\\ \phi_t &= M\phi \end{align} to be compatible where $L$ and $M$ are operators and $\lambda$ a constant, the compatibility condition is $$ L_t + [L,M] = $$ where $[L,M]$ is the commutator. Then verify the KDV equation is the compatibility condition for the follwoing Lax pair \begin{align} L & =\partial_{xx} + u\\ M &= -4\partial_{xxx}-3(2u\partial_x-u_x) \end{align}

4. Verify the first three conserved quantities for the NLS equation