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We argue that the critical behavior near the point of " gradient catastrophe " of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation iiΨ t + 2 2 Ψ xx + |Ψ | 2 Ψ = 0, 1, with analytic initial data of the form Ψ (x, 0;) = A(x)e i S(x) is approximately described by a particular solution to the Painlevé-I equation.

This article is concerned with a conjecture in [8] on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasilinear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is… (More)

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