Abstract: In this talk I will discuss the universality near a gradient catastrophe point and a phase transition curve in the semiclassical limit of the sine-Gordon (sG).

We study the semiclassical sG 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 an O(ϵ^4/5) neighbourhood 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 characterise the solution in detail.

Apart from the gradient catastrophe, one can also observe that the solution transitions from locally periodic into more complicated behaviour. In fact, there exist a transition curve which marks the transition from genus one to higher genus solution. The gradient catastrophe is a non differentiable point on this curve. Near the transition curve but away from the gradient catastrophe, the solution exhibits another type of universality.

Our approach is the rigorous steepest descent method for matrix Riemann--Hilbert problems, generalising Bertola and Tovbis's results on the nonlinear Schrödinger equation to establish universality beyond the context of solutions of a single equation.