Dispersive shock waves

ICERM - March 2022
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Dispersive shock waves

Shock waves can be observed both experimentally and theoretically, and manifest as a train of localized, highly oscillating waves. They are observed in water waves, plasma, optics, or Bose-Einstein condensates. Dispersive shock waves can appear when one of the relevant quantities of the model changes abruptly compared to the others. Mathematically, it is possible to grasp the formation of a dispersive shock wave by letting a particular parameter of the modeling equation tend to zero: this is called the zero-dispersion limit.
The Benjamin-Ono equation is a model of water waves describing a certain regime of internal waves at the interface of two fluids of different densities when the lower layer is very deep. The Benjamin-Ono equation accurately predicts concrete models, for instance it can be used to describe the submerged “plumes” of oil during oil spills [CM11]. In the zero-dispersion limit, there exists a shock time after which an oscillating zone appears and grows with time [MX11].

Figure 1: Formation of a dispersive shock for the Benjamin-Ono equation. Credit: Elliot Blackstone and Peter D. Miller
During the Fall 2021 ICERM program “Hamiltonian Methods in Dispersive and Wave Evolution Equations”, Louise Gassot, Institute Postdoc at ICERM for the academic year, studied the zero dispersion limit for the Benjamin-Ono equation [Gas21]. A precise description of the dispersive shock waves for the Benjamin-Ono equation after the shock time is the goal of a collaboration with Elliot Blackstone (University of Michigan) and Peter D. Miller (University of Michigan), who also participated in the ICERM program. The methods rely on integrable systems [GK21], numerical analysis, partial differential equations, complex analysis, semi-classical analysis and matrix theory.

[CM11] R. Camassa and R. McLaughlin, “How do underwater oil plumes form?”, online video
from http://www.youtube.com/watch?v=6Cp6fHINQ94.
[Gas21] L. Gassot, Zero-dispersion limit for the Benjamin-Ono equation on the torus with single
well initial data, arXiv preprint arXiv:2111.06800, 2021.
[GK21] P. Gérard and T. Kappeler, On the Integrability of the Benjamin‐Ono Equation on the Torus, Communications on Pure and Applied Mathematics 74(8): 1685-1747, 2021.
[MX11] P. D. Miller and Z. Xu, On the zero-dispersion limit of the Benjamin-Ono Cauchy problem
for positive initial data, Communications on Pure and Applied Mathematics, 64(2):205–270, 2011.