Presented By: Applied Interdisciplinary Mathematics (AIM) Seminar  Department of Mathematics
AIM Seminar: Universality in the SmallDispersion Limit of the BenjaminOno Equation
Peter D. Miller, University of Michigan
This talk concerns the BenjaminOno (BO) equation of internal wave theory, and properties of the solution of the Cauchy initialvalue 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 zerodispersion limit). It is wellknown that existence of a limit requires the weak topology because highfrequency 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 Kortewegde 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 PainlevĂ©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, Louise Gassot, Patrick GĂ©rard, and Matthew Mitchell.
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