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. Using Riemann–Hilbert analysis, we rigorously studied the asymptotics near a certain gradient catastrophe. In accordance with a conjecture made by Dubrovin et. al., the asymptotics in the this region is universally (insensitive to initial condition) described by the tritronquee solution to the Painleve-I equation. Furthermore, we are able to universally characterize the shapes of the spike-like local structures (rogue wave on periodic background) on top of the poles of the tritronquee solution. Joint work with Peter Miller.

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