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Resonant dispersive shock waves in nonlinear optics and shallow water waves

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HY2W05 - Physical applications

Dispersive hydrodynamics, mathematical models formulated by hyperbolic conservation laws with dispersive corrections, govern fluid or fluid-like phenomena that naturally emerge in media whose dissipative effect (e.g., viscosity) is negligible or non-existent relative to wave dispersion. The framework of dispersive hydrodynamics possesses a plethora of nonlinear dispersive wave phenomena, including solitary waves and dispersive shocks waves (DSWs). This talk focuses on the latter wave phenomenon arising in two contexts: nonlocal nonlinear medium of defocusing nematic liquid crystals and shallow water waves with surface tension. In nematic liquid crystals, distinct non-classical DSW regimes are generated from discontinuous initial conditions for the optical field and are resonant due to the effect of non-convex dispersion, resulting in a resonant radiation propagating ahead of the DSW itself. The classical DSW structure disappears and a Whitham shock emerges as the initial jumps become sufficiently large. Analytical methods based on Whitham modulation theory and asymptotic techniques are used to determine solutions for these DSW regimes. The transition effect from the nonlocal (low power) to local (high power) limits will also be considered. In the end of the talk, a particular non-classical resonant shallow water DSW regime, so-called Cross-over DSW (CDSW), governed by Kawahara and extended KdV equations will be discussed. The analytical solutions obtained are verified via numerical solutions and excellent agreement is found. Co-authors: names and affiliations: Noel F. Smyth (University of Edinburgh), Dimitri Frantzeskakis (National and Kapodistrian University of Athens), Theodoros Horikis (University of Ioannina), Tim Marchant (University of Melbourne) 

This talk is part of the Isaac Newton Institute Seminar Series series.

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