Speaker: Professor Dmitry Pelinovsky (McMaster University)
Date: 31st Jul 2024 - 9:00 to 10:00
Venue: External
Title: Plenary talk: Dispersive hydrodynamics in the modified KdV equation
Event: (PDWW01) Frontiers of Dispersive Hydrodynamics
Abstract: I will overview the rich dispersive hydrodynamics phenomena within just one integrable model, the modified Korteweg–de Vries (KdV) equation. In spite of being a basic integrable equation related to the well-studied AKNS spectral problem, it offers many new questions to be explored. Traveling periodic waves of the mKdV equation can be analyzed with the theory of elliptic functions. Compared to the similar solutions of the NLS equations, these elliptic functions are degenerations of Riemann Theta function of genus two. I will show based on [1] how these elliptic functions can be useful to give a variational characterization of traveling periodic functions and to prove their stability or instability with respect to the temporal dynamics.
I will then switch to understanding breathers on modulationally stable elliptic background and rogue waves on modulationally unstable background. Some preliminary results were obtained some time ago in [2]. In the recent study [3], we have extended the construction of bright and dark breathers from the KdV theory, see [4] and references therein, to the defocusing mKdV theory.
I will conclude my talk with the state of art on the construction of kink breathers for the defocusing mKdV equation and general rogue waves for the focusing mKdV equation.
[1] U. Le and D.E. Pelinovsky, SIAM J. Appl. Dynam. Syst. 21 2518–2534 (2022)[2] J. Chen and D.E. Pelinovsky, J. Nonlinear Sci. 29 2797–2843 (2019)[3] A. Mucalica, and D. E. Pelinovsky, arXiv:2312.08969[4] M. Hoefer, A. Mucalica, and D. E. Pelinovsky, J. Phys. A: Mathem. Theor. 56 185701 (2023)
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24 сен 2024
