Department of Mathematics
Abstract. I will present a collection of joint works with J. Bona, G. Iooss, J-C. Saut and Jie Shen. I will start with the introduction of model equations for the three dimensional water wave equations, which include the Euler equations, the Boussinesq systems and the KP equations. The derivation from Euler equation to Boussinesq systems, from Boussinesq systems to KP-type equations will be briefly described. The comparison between different models are made with respect to the dispersion relations, physically relevant properties and computational complexities. The existing results on initial value problems, Boundary-initial -value problems for various Boussinesq systems are reviewed.
I'll also show the existence of solitary wave solutions, multi-pulsed wave solutions, cnoidal wave solutions, standing wave solutions and two-diemsional periodic wave patterns for the Boussinesq systems. The numerical simulations with various initial wave profiles will also be presented.