Mathematics 697WA - ST-NonlinWaves & Appl/Continua

The aim of this course will be to give an overview of the mathematical background, physical applications and numerical computations associated with a number of prototypical wave systems both at the continuum and at the lattice level. We will start from finite dimensional Hamiltonian systems, discuss their symmetries and Lagrangian/Hamiltonian structure, and then extend considerations to infinite dimensional systems of partial differential and differential-difference equations. Prototypical case examples will be the continuum and the discrete nonlinear Schrodinger equation, the Korteweg-de Vries equation, the sine-Gordon equation, and the Fermi-Pasta-Ulam lattice, among others. We will examine the symmetries, conservation laws, solitary wave solutions, linearization spectral properties and dynamics of such equations and attempt to connect them with physical applications from nonlinear optics, fluid mechanics, materials science and atomic physics, as well as develop computational tools (such as bifurcation analysis and time-stepping algorithms) about how to address them. Time permitting, we will also make short excursions to systems of multiple components, higher dimensions or of dissipative character (e.g. reaction-diffusion type) to discuss some similarities and differences with these.

Open to Graduate students only. Note: Prerequisites: Math 532H or equivalent (required) and Math 534H or equivalent (required)