On critical behaviour in systems of Hamiltonian partial differential equations

Boris Dubrovin, Tamara Grava, Christian Klein, Antonio Moro

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16 Citations (Scopus)
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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.
Original languageEnglish
Pages (from-to)631-707
JournalJournal of Nonlinear Science
Volume25
Issue number3
Early online date11 Feb 2015
DOIs
Publication statusPublished - Jun 2015

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