We study the semiclassical limit of the sine-Gordon (sG) equation 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 a neighborhood 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 the 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 characterize the solution in detail. Our approach is the rigorous steepest descent method for matrix Riemann-Hilbert problems, substantially generalizing [5] to establish universality beyond the context of solutions of a single equation. © 2021 Wiley Periodicals LLC.

中文翻译：

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半经典正弦-戈登方程梯度突变点附近的普遍性

我们研究了具有低于阈值的 Klaus-Shaw 型纯脉冲初始数据的 sine-Gordon (sG) 方程的半经典极限。该系统的 Whitham 平均近似在有限时间内表现出梯度突变。根据 Dubrovin、Grava 和 Klein 的猜想，我们发现在在梯度突变点附近的邻域，sG 解的渐近性普遍用 Painlevé I tritronquée 解来描述。可以从该邻域的 tritronquée 解决方案显式地制作线性地图。在这张图下：远离 tritronquee 极点，sG 的第一次修正普遍由 tritronquee 解的哈密顿量的实部给出；局部缺陷出现在从 tritronquée 解决方案的两极映射的位置；这些缺陷被普遍证明是 sG 方程周期性背景上的特殊局部解的二参数族。我们能够详细描述解决方案。我们的方法是矩阵 Riemann-Hilbert 问题的严格最速下降法，基本上概括 [5] 以建立超越单个方程解的普遍性。© 2021 威利期刊有限责任公司。