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MiS Preprint
54/2007
On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation
Boris Dubrovin, Tamara Grava and Christian Klein
Abstract
We argue that the critical behaviour near the point of "gradient catastroph" of the solution to the Cauchy problem for the focusing nonlinear Schr\"odinger equation $ i\epsilon\,\Psi_t +\frac{\epsilon^2}2\Psi_{xx}+ |\Psi|^2 \Psi =0$, $\epsilon\ll 1$, with analytic initial data of the form $\Psi(x,0;\epsilon) =A(x) \, e^{\frac{i}{\epsilon} S(x)}$ is approximately described by a particular solution to the Painlevé-I equation.
On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the Tritronquée solution to the Painlevé-I equation
In: Journal of nonlinear science, 19 (2009) 1, pp. 57-94
