School of Mathematics and Statistical Sciences Colloquium talk by Peter Miller

About this event

Title of the talk: Universality in the small-dispersion limit of the Benjamin-Ono equation

Abstract: This talk concerns the Benjamin-Ono (BO) equation of internal wave theory, and properties of the solution of the Cauchy initial-value problem in the situation that the initial data is fixed but the coefficient of the nonlocal dispersive term in the equation is allowed to tend to zero (i.e., the zero-dispersion limit). It is well-known that existence of a limit requires the weak topology because high-frequency oscillations appear even though they are not present in the initial data. Physically, this phenomenon corresponds to the generation of a dispersive shock wave. In the setting of the Korteweg-de Vries (KdV) equation, it has been shown that dispersive shock waves exhibit a universal form independent of initial data near the two edges of the dispersive shock wave, and also near the gradient catastrophe point for the inviscid Burgers equation from which the shock wave forms. In this talk, we will present corresponding universality results for the BO equation. These have quite a different character than in the KdV case; while for KdV one has universal wave profiles expressed in terms of solutions of Painlevé-type equations, for BO one instead has expressions in terms of classical Airy functions and Pearcey integrals. These results are proved for general rational initial data using a new approach based on an explicit formula for the solution of the Cauchy problem for BO. This is joint work with Elliot Blackstone and Matthew Mitchell, based on other work with Blackstone, Louise Gassot, and Patrick Gérard.

Short bio of the speaker: Peter D. Miller is a Professor of Mathematics at the University of Michigan. He earned his Ph.D. in applied mathematics from the University of Arizona in 1994. His research focuses on nonlinear waves, integrable systems, asymptotic analysis, special functions, and random matrix theory, with particular interest in universal phenomena in nonlinear wave propagation. He was named a Fellow of the Society for Industrial and Applied Mathematics in 2026.