Australian Mathematical Sciences Institute - Research Publications

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    Numerical and analytical study of undular bores governed by the full water wave equations and bidirectional Whitham-Boussinesq equations
    Vargas-Magana, RM ; Marchant, TR ; Smyth, NF (AMER INST PHYSICS, 2021-06)
    Undular bores, also termed dispersive shock waves, generated by an initial discontinuity in height as governed by two forms of the Boussinesq system of weakly nonlinear shallow water wave theory, the standard formulation and a Hamiltonian formulation, two related Whitham–Boussinesq equations, and the full water wave equations for gravity surface waves are studied and compared. It is found that the Whitham–Boussinesq systems give solutions in excellent agreement with numerical solutions of the full water wave equations for the positions of the leading and trailing edges of the bore up until the onset on modulational instability. The Whitham–Boussinesq systems, which are far simpler than the full water wave equations, can then be used to accurately model surface water wave undular bores. Finally, comparisons with numerical solutions of the full water wave equations show that the Whitham–Boussinesq systems give a slightly lower threshold for the onset of modulational instability in terms of the height of the initial step generating the undular bore.
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    Nematic Dispersive Shock Waves from Nonlocal to Local
    Baqer, S ; Frantzeskakis, DJ ; Horikis, TP ; Houdeville, C ; Marchant, TR ; Smyth, NF (MDPI, 2021-06)
    The structure of optical dispersive shock waves in nematic liquid crystals is investigated as the power of the optical beam is varied, with six regimes identified, which complements previous work pertinent to low power beams only. It is found that the dispersive shock wave structure depends critically on the input beam power. In addition, it is known that nematic dispersive shock waves are resonant and the structure of this resonance is also critically dependent on the beam power. Whitham modulation theory is used to find solutions for the six regimes with the existence intervals for each identified. These dispersive shock wave solutions are compared with full numerical solutions of the nematic equations, and excellent agreement is found.