Universality Near the Gradient Catastrophe Point in the Semiclassical Sine‐Gordon Equation
Universality Near the Gradient Catastrophe Point in the Semiclassical Sine‐Gordon Equation
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DOI：
10.1002/cpa.22018
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发表时间：
2019-12
影响因子：
3
通讯作者：
Bing-ying Lu;P. Miller
中科院分区：
文献类型：
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作者：
Bing-ying Lu;P. Miller
研究了具有Klaus-Shaw型阈下纯脉冲初值的sine-Gordon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的一个猜想，我们发现在梯度突变点附近的一个O ~ 4/5邻域内，sG解的渐近性普遍地由Painlevé I tritronquée解描述.从tritronquée解到这个邻域可以显式地得到一个线性映射。在此地图下：在远离tritronquée极点的地方，sG的第一个修正普遍地由tritronquée解的哈密顿量的真实的部分给出;局部缺陷出现在从tritronquée解的极点映射的位置;缺陷普遍地被证明是sG方程在周期背景上的特殊局部解的双参数族。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，大大推广了[5]，以建立超越单个方程解的普遍性。© 2021 Wiley Periodicals LLC.
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 Oϵ4/5 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.