Both the Nonlinear Schroedinger Equation (NLS) and the Korteweg-de Vries Equation (KdV) are integrable nonlinear PDEs. This means that though they are nonlinear (so typical PDE integration techniques don't work), exact solutions can be found via a method called the Inverse Scattering Transform, and the solutions are made of solitons (solitary wave packets that look like sech(x)).
In this project, we will seek to simulate these multi-soliton solutions with a combination of Mathematica and either Python or Matlab, creating figures and animations of how two or three solitons interact, looking closely what happens when they cross, and comparing to some "fake" solutions that are more trivial to generate.
Finally, we will do the same with solutions to an NLS equation that has nonzero boundary conditions when x goes to infinity.
Completion of Differential Equations is necessary. Completion or enrollment in PDE and/or Linear Algebra are helpful but not required. Some experience with Matlab or Python is also necessary.