The Semi-Classical Sine-Gordon Equation, Universality at the Gradient Catastrophe and the Painleve-I Equation

sine-Gordon equation;Inverse scattering transform;Integrable system;Riemann–Hilbert Problem;Mathematics;Science;Mathematics

Lu, BingyingWu, Sijue ;

University of Michigan

关键词: sine-Gordon equation;    Inverse scattering transform;    Integrable system;    Riemann–Hilbert Problem;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/147676/bylu_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This thesis concerns the semi-classical sine-Gordon equation with pure impulse initial data below the threshold of rotation: begin{equation*}begin{gathered}epsilon ^2 u_{tt} - epsilon ^2 u_{xx} + sin {u} = 0,u(x,0) equiv 0, quadepsilon u_t(x,0) = G(x) leq 0, quad text{and} quad|G(0)|< 2.end{gathered}end{equation*} We consider a wide class of solutions that decays at infinity.A dispersively-regularized shock forms in finite time. We study the universality of the solutions near a certain catastrophe point. In accordance with a conjecture by Dubrovin–Grava–Klein on Hamiltonian perturbations near the gradient catastrophe point of an elliptic system, we found that the asymptotics of the sine-Gordon solution is described by the tritronqu;;ee solution to the Painlev;;e-I equation. Furthermore, we are able to describe the local peak-like structures corresponing to where the tritronqu;;ee solution, well-known to have singularities, fails to describe the asymptotics. Our result is universal in the sense that the local asymptotics is not sensitive to the initial conditions as long as it falls into a large class of functions; it is only the space-time location of the transition that depends on the initial data. Our main tool is the Riemann--Hilbert technique for integrable systems, in particular the Deift–Zhou steepest descent method. The approach is inspired by the work of Bertola–Tovbis on the focusing NLS equation.