arXiv 0909.10202009-09-05math-phmath.MP

Camassa-Holm和Whitham方程小色散极限的数值解及多尺度展开

Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions

S. Abenda, T. Grava, C. Klein

备注：: 25 pages, 15 figures

摘要：

Camassa-Holm（CH）方程解的小色散极限的特征在于出现快速调制振荡区。给出了这些振荡的渐近描述，在短时间内，由CH方程的单相解，其中相应的椭圆曲线的分支点取决于通过Whitham方程的物理坐标。我们提出了一个猜想的渐近解的相位。光滑驼峰状的初始数据的数值研究这一限制提供了有力的证据，这一猜想的有效性。我们提出了一个定量的CH和渐近解之间的数值比较。在Whitham区的内部和边界上的小色散参数$ε$的依赖性进行了研究。在区域的内部，CH和渐近解之间的差是$ε$阶，在后缘是$\sqrtε$阶，在前缘是$ε^{1/3}$阶。对于后者，我们提出了一个多尺度的扩展，它描述的振幅的振荡的Hastings-McLeod解决方案的Painlevé II方程。我们数值计算表明，这种多尺度解提供了一个增强的渐近描述附近的前缘。

英文摘要：

The small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We present a conjecture for the phase of the asymptotic solution. A numerical study of this limit for smooth hump-like initial data provides strong evidence for the validity of this conjecture. We present a quantitative numerical comparison between the CH and the asymptotic solution. The dependence on the small dispersion parameter $ε$ is studied in the interior and at the boundaries of the Whitham zone. In the interior of the zone, the difference between CH and asymptotic solution is of the order $ε$, at the trailing edge of the order $\sqrtε$ and at the leading edge of the order $ε^{1/3}$. For the latter we present a multiscale expansion which describes the amplitude of the oscillations in terms of the Hastings-McLeod solution of the Painlevé II equation. We show numerically that this multiscale solution provides an enhanced asymptotic description near the leading edge.

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