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arXiv 1608.03057nlin.SI
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短脉冲方程的长时间渐近性

Long-time asymptotics for the short pulse equation

Jian Xu

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摘要：

本文研究了短脉冲（SP）方程初值问题（IVP）解的长时间行为。由于SP方程是一个完备的可积系统，它包含Wadati-Konno-Ichikawa（WKI）型拉克斯对，我们利用逆散射方法将该IVP问题转化为一个$2\times 2$矩阵Riemann-Hilbert问题.由于谱变量$k$在WKI型拉克斯对中是相同的阶数，因此我们在新尺度$(y,t)$中参数化地构造了这个IVP的解，而原始尺度$(x,t)$是根据新尺度中的函数给出的，就这个Riemann-Hilbert问题的解而言。然而，对于振荡Riemann-Hilbert问题，利用Deift和Zhou的非线性最速下降法，我们可以得到当时间$t$趋于无穷大时，短脉冲方程解在原尺度$(x,t)$上的显式首阶渐近解.

英文摘要：

In this paper, we analyze the long-time behavior of the solution of the initial value problem (IVP) for the short pulse (SP) equation. As the SP equation is a complete integrable system, which posses a Wadati-Konno-Ichikawa (WKI)-type Lax pair, we formulate a $2\times 2$ matrix Riemann-Hilbert problem to this IVP by using the inverse scattering method. Since the spectral variable $k$ is the same order in the WKI-type Lax pair, we construct the solution of this IVP parametrically in the new scale $(y,t)$, whereas the original scale $(x,t)$ is given in terms of functions in the new scale, in terms of the solution of this Riemann-Hilbert problem. However, by employing the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problem, we can get the explicit leading order asymptotic of the solution of the short pulse equation in the original scale $(x,t)$ as time $t$ goes to infinity.

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