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 evolutionary equations by the Fourier method, as a superposition of plane waves, a remarkable "simplification" that transformed the study of linear partial differential equations. With the advent of computers in the middle of the 20’th century, through the remarkable computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed numerically that 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”. The only proofs available have been for “integrable” equations, which can be reduced to a collection of linear equations. The proof of such results, in the non-integrable case, has been one of the grand challenges in the study of nonlinear differential equations. Recently, there have been important breakthroughs in obtaining mathematical proofs of these types of numerical observations, in the context of nonlinear wave equations, which I will discuss.