On Universality of Critical Behavior in the Focusing Nonlinear Schrodinger Equation, Elliptic Umbilic Catastrophe and the Tritronquee Solution to the Painleve-I Equation

摘要

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