On critical behaviour in systems of Hamiltonian partial differential equations

Dubrovin, Boris, Grava, Tamara, Klein, Christian and Moro, Antonio (2015) On critical behaviour in systems of Hamiltonian partial differential equations. Journal of Nonlinear Science, 25 (3). pp. 631-707. ISSN 0938-8974

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Official URL: http://dx.doi.org/10.1007/s00332-015-9236-y

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.

Item Type: Article
Additional Information: Published online 11-2-2015.
Uncontrolled Keywords: Hamiltonian PDEs, Hyperbolic and Elliptic systems, Gradient catastrophe and elliptic umbilic catastrophe, Quasi-integrable systems, Painlevé equations, 35Q55, 37K05, 34M55
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
Depositing User: Antonio Moro
Date Deposited: 02 Jun 2015 11:47
Last Modified: 13 May 2017 18:09
URI: http://nrl.northumbria.ac.uk/id/eprint/22430

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