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Publication - Professor Tamara Grava

    Shock formation in the dispersionless Kadomtsev–Petviashvili equation


    Grava, T, Eggers, JG & Klein, C, 2016, ‘Shock formation in the dispersionless Kadomtsev–Petviashvili equation’. Nonlinearity, vol 29., pp. 1384?1416


    The dispersionless Kadomtsev–Petviashvili (dKP) equation
    is one of the simplest nonlinear wave equations describing
    two-dimensional shocks. To solve the dKP equation numerically we use a
    coordinate transformation inspired by the method of characteristics for
    the one-dimensional Hopf equation .
    We show numerically that the solutions to the transformed equation
    stays regular for longer times than the solution of the dKP equation.
    This permits us to extend the dKP solution as the graph of a multivalued
    function beyond the critical time when the gradients blow up. This
    overturned solution is multivalued in a lip shape region in the (x, y)
    plane, where the solution of the dKP equation exists in a weak sense
    only, and a shock front develops. A local expansion reveals the
    universal scaling structure of the shock, which after a suitable change
    of coordinates corresponds to a generic cusp catastrophe. We provide a
    heuristic derivation of the shock front position near the critical point
    for the solution of the dKP equation, and study the solution of the dKP
    equation when a small amount of dissipation is added. Using
    multiple-scale analysis, we show that in the limit of small dissipation
    and near the critical point of the dKP solution, the solution of the
    dissipative dKP equation converges to a Pearcey integral. We test and
    illustrate our results by detailed comparisons with numerical
    simulations of both the regularized equation, the dKP equation, and the
    asymptotic description given in terms of the Pearcey integral.

    Full details in the University publications repository