On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the \it tritronquée solution to the Painlevé-I equation

2007/04/04 by Dubrovin, B., Grava, T., Klein, C. · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.0704.0501

Abstract

We argue that the critical behaviour near the point of ``gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation iεψ_t +\fracε²2ψ_xx+ |ψ|² ψ=0 with analytic initial data of the form ψ(x,0;ε) =A(x) e^\fraciε S(x) is approximately described by a particular solution to the Painlevé-I equation.

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• The Painlevé I hierarchy: Correspondence between the isomonodromic approach and the minimal models of the KP hierarchy

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