Date: Wednesday, April 03, 2013
Location: 4088 East Hall (3:10 PM to 4:00 PM)
Abstract: Whitham modulation or modulation theory describes the evolution of slowly modulated wave-trains of nonlinear, dispersive partial differential equations (PDEs). In practice, this involves considering the PDE in a certain asymptotic limit so that quantities such as the wavenumber, phase speed, amplitude, etc. of the solution are locally well defined. The theory then consists in obtaining evolution equations, arising from the PDE, for these parameters: the modulation equations. These modulation equations may be found through multiple-scale expansions, through the averaging of conservation laws, or the averaging of the Lagrangian of the system. The method is grounded on intuition and remarkably useful in physical applications such as dispersive shock waves.
I will begin the talk with physical examples of dispersion and the mathematics used to study dispersive waves. I will then discuss the foundations of modulation theory and the construction of the modulation equations. I will finish by showing an application of Whitham modulation theory to solve a dispersive shock problem in a manner akin to the method of matched asymptotic expansions used in ODEs. The talk is intended to be mostly self-contained.
Speaker: Alfredo Wetzel
Institution: University of Michigan