Speaker : 1.Dr Derchyi Wu (Academia Sinica) 2. Dr. Bing-Ying Lu (Academia Sinica)
Title : 1.The Direct Problem of perturbed Kadomtsev-Petviashvili II 1-line solitons 2. The universality of the semi-classical sine-Gordon equation at the gradient catastrophe
Time : 2018-12-14 (Fri) 10:00 - 12:00
Place : Seminar Room 638, Institute of Mathematics (NTU Campus)
Abstract: 1.Boiti-Pempinelli-Pogrebkov's inverse scattering theories on the KPII equation provide an integrable approach to solve the Cauchy problem of the perturbed KPII multi-line soliton solutions and the stability problem of KPII multi-line solitons.
In this talk, we will present rigorous analysis for the direct scattering theory of perturbed KPII one line solitons, the simplest case in Boiti-Pempinelli-Pogrebkov's theories. Namely, for generic small perturbation of the one line soliton, the existence of the eigenfunction is proved by establishing uniform estimates of the Green function and the Cauchy integral equation for the eigenfunction is justified by non-uniform estimates of the spectral transform.

2.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 Painleve-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 Schrodinger equation) on top of the poles of the tritronquee 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 Schrodinger equation by Bertola-Tovbis.