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Publication - Professor Tamara Grava

    On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations

    Citation

    Grava, T, Dubrovin, B, Klein, C & Moro, A, 2015, ‘On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations’. Journal of Nonlinear Science, vol 25., pp. 631-707

    Abstract

    We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (PI ) equation or its fourth-order analogue P2I . As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

    Full details in the University publications repository