Colin Thomson (UNC-CH), GMA Seminar

Title: Applications of the biHamiltonian structure of the Korteweg-de Vries equation

Abstract: Hamiltonian mechanics (along with its complement, Lagrangian mechanics) is a common formulation of classical dynamics. After reviewing the geometric structure of finite-dimensional Hamiltonian systems, we generalize to Hamiltonian partial differential equations, and show that the Korteweg-de Vries (KdV) equation for waves in shallow water has two distinct, but compatible, Hamiltonian structures. These independent structures may be used to establish two amazing facts about the KdV equation: there is an infinite sequence of conserved quantities and the equation can be solved exactly despite its nonlinearity. We conclude by applying the exact solution to multi-soliton dynamics.