Colloquium

Emergent coherent structures in integrable nonlinear dispersive PDEs

 Dr. Robert Jenkins

Department of Mathematics

Colorado State University

ABSTRACT: The interplay between dispersion and nonlinearity creates rich and complicated dynamics in many systems of physical and mathematical interest. For general nonlinear systems, finding precise descriptions of their evolution is a non-tractable problem and numerical methods are necessary. In this talk, I'll focus on integrable models, which while specialized, are often canonical models of weakly nonlinear interactions. In this integrable setting we can use the tools developed from the inverse scattering transform to rigorously derive analytic formulae for the coherent structures which emerge in different asymptotic regimes. After introducing the basic ideas and some background, I'll discuss some of my previous results on both the large time behavior and zero dispersion limit of various integrable systems. I'll end by briefly discussing some very recent and ongoing work in which we add randomness to the picture to study the statistical behavior of integrable systems given random initial data. The talk will be largely non-technical; in particular no familiarity with integrability will be assumed.