Universality Near the Gradient Catastrophe Point in the Semiclassical Sine-Gordon Equation
Bing-Ying Lu and Peter D. Miller
Department of Mathematics, University of Michigan, Ann Arbor
Abstract:
We study the semiclassical limit of the sine-Gordon (sG) equation with below threshold pure impulse initial data of Klaus-Shaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of Dubrovin, Grava and Klein, we found that in a \(\mathcal{O}(\epsilon^{4/5})\) neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are universally described by the Painlevé I tritronquée solution. A linear map can be explicitly made from the tritronquée solution to this neighborhood. Under this map: away from the tritronquée poles, the first correction of sG is universally given by the real part of the Hamiltonian of the tritronquée solution; localized defects appear at locations mapped from the poles of tritronquée solution; the defects are proved universally to be a two parameter family of special localized solutions on a periodic background for the sG equation. We are able to characterize the solution in detail. Our approach is the rigorous steepest descent method for matrix Riemann-Hilbert problems, substantially generalizing Bertola and Tovbis's results on the nonlinear Schrödinger equation to establish universality beyond the context of solutions of a single equation.
Left: The cosine of the sG solution near the catastrophe point. Center: the real part of the PI tritronquée Hamiltonian. Right: Mathematical model for a "defect".