PDE Seminar

DATE2018-12-20 15:10-16:00


SPEAKER陸冰瑩 博士(中央研究院

TITLEThe Universality of the Semi-Classical Sine-Gordon Equation at the Gradient Catastrophe

ABSTRACT We study the semi-classical sine-Gordon equation with pure impulse initial data below the threshold of rotation: $\epsilon^2 u_{tt}-\epsilon^2 u_{xx}+\sin(u)=0$, $u(x,0) \equiv 0$, $\epsilon u_t(x,0)=G(x)\leq 0$, and $|G(0)|<2$. A dispersively-regularized shock forms in finite time. We found, in accordance with a conjecture made by Dubrovin et. al., that the asymptotics near a certain gradient catastrophe is universally (insensitive to initial condition) described by the tritronqu\’ee solution to the Painlev\’e-I equation. Furthermore, we are able to universally characterize the shapes of the spike-like local structures (similar to rogue wave on periodic background for the focusing nonlinear Schr\"odinger equation) on top of the poles of the tritronqu\’ee solution. Our technique is the Deift-Zhou steepest descent analysis of the Riemann-Hilbert problem associated with the sine-Gordon equation. Our approach is inspired by a study of universality for the focusing nonlinear Schr\”odinger equation by Bertola-Tovbis. (joint work with Peter Miller)