Water waves

Water wave equations describe the evolution of waves on the surface of oceans, lakes, etc. They form a class of nonlinear partial differential equations with dispersive effects, meaning that waves of different wavelengths propagate at different speeds.

Our group studies integrable water wave equations and focuses on qualitative properties of solutions, including blow-up phenomena and long-time behavior. Our group also studies nonlinear dispersive equations beyond water wave models. 

Such problems require tools from complex analysis and harmonic analysis, including nonlinear Fourier–type transforms. 

The Korteweg–De Vries (KdV) equation

The Boussinesq equation

The good Boussinesq equation