3:00–4:00 pm Eckhart 206
Title: 'Simplification' in Partial Differential Equations
Abstract:
We will recall the origins of Fourier analysis and its connection to partial differential equations, through the work of Fourier on heat conduction, in the early 19’th century. This led to the representation of solutions of linear evolution equations by the Fourier method, as a superposition of plane waves, a remarkable “simplification” that transformed the study of linear partial differential equations and led to fundamental technical advances in the 19th century. In the middle of the 20’th century, through the breakthrough computer simulations of Fermi-‐Pasta-‐Ulam (mid50s) and Kruskal-‐Zabusky (mid 60s) it was observed computationally that some nonlinear equations modeling wave propagation, asymptotically, also exhibit a “simplification”, this time as superposition of “traveling waves” and “radiation”. This has become known as the “soliton resolution conjecture”. Recently, there have been important advances in obtaining mathematical proofs of these types of computer simulations in the context of nonlinear wave equations, which I will discuss.