The large-time development of the solution to an initial-value problem for the Korteweg–de Vries equation: I. Initial data has a discontinuous expansive step
The large-time development of the solution to an initial-value problem for the Korteweg–de Vries equation: I. Initial data has a discontinuous expansive step
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Korteweg–de Vries 方程初值问题解的大规模发展： I. 初始数据具有不连续的扩展步骤
DOI：
10.1088/0951-7715/21/10/010
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发表时间：
2008
期刊：
影响因子：
1.7
通讯作者：
D. Needham
中科院分区：
文献类型：
--
作者：
J. Leach;D. Needham
本文考虑Korteweg-de Vries方程的一个初值问题。所考虑的归一化Korteweg-de Vries方程由给出，其中x和τ分别表示无量纲距离和时间。特别地，我们考虑了当初始数据具有不连续的扩展步长时的情形，其中，对于x⩾0，u(x，0)=u0，对于x<；0，u(x，0)=0。利用匹配的渐近坐标展开法得到了该问题解的大τ渐近结构，它在x⩾0中形成一个展开波，而解在x<；0中是振荡的，振荡包络为τ→∞.
In this paper, we consider an initial-value problem for the Korteweg–de Vries equation. The normalized Korteweg–de Vries equation considered is given by where x and τ represent dimensionless distance and time, respectively. In particular, we consider the case when the initial data has a discontinuous expansive step, where u(x, 0) = u0 for x ⩾ 0 and u(x, 0) = 0 for x < 0. The method of matched asymptotic coordinate expansions is used to obtain the large-τ asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in x ⩾ 0, while the solution is oscillatory in x < 0, with the oscillatory envelope being of as τ → ∞.