The existence of three-dimensional generalized solitary waves

Applied and Computational Mathematics Seminar
Monday, October 6, 2008 - 1:00pm for 1 hour (actually 50 minutes)
Skiles 255
Shengfu Deng – School of Mathematics, Georgia Tech
Haomin Zhou
We consider the three-dimensional gravity-capillary waves on water of finite-depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamical system in stead of using Hamiltonian formulation method. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants.