The small dispersion limit of the Korteweg‐de Vries equation. III
The small dispersion limit of the Korteweg‐de Vries equation. III
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DOI：
10.1002/cpa.3160360606
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发表时间：
1983-11
影响因子：
3
通讯作者：
P. Lax;C. Levermore
中科院分区：
文献类型：
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作者：
P. Lax;C. Levermore
在第一部分和第二部分中，我们得到了当uxxx系数趋于零时KdV方程的解的分布极限u的显式公式。该公式包含n个参数β1、…在第四节中，我们证明了对于t<；Tb，n=1，并且确定了β的值β的值。在第五节中，我们已经证明了参数βi满足一个非线性偏微分方程组。在第三部分第六节中，我们证明了当t大时，n=3，并且我们确定了当t大时β1，β2，β3以及u和u2的渐近行为。显式公式表明u和u2分别为O(t−1)和O(t-2)(见公式(6.2)和(6.24))。在第7节中，我们研究其值趋于零的初始数据作为x+∞，以及趋于-1的值作为x−∞。如果人们接受一些关于具有这样的初始数据的解的行为的似是而非的猜测，我们推导出解的显式公式，并确定解的大规模渐近行为：函数S(ζ)可以用完全椭圆积分来表示；对于U2，也可以得到类似的公式。在第8节中，我们指出了如何将这一系列论文的处理扩展到多峰(但仍然是负的)初始数据。
In Parts I and II we have derived explicit formulas for the distribution limit u of the solution of the KdV equation as the coefficient of uxxx tends to zero. This formula contains n parameters β1, …, βn whose values, as well as whose number, depends on x and t. In Section 4 we have shown that for t<tb, n=1, and the value of β, was determined. In Section 5 we have shown that the parameters βi satisfy a nonlinear system of partial differential equations. In Part III, Section 6 we show that for t large, n=3, and we determine the asymptotic behavior of β1, β2, β3, and of u and u2, for t large. The explicit formulas show that u and u2 are O(t−1) and O(t-2) respectively (see formulas (6.2) and (6.24)). In Section 7 we study initial data whose value tends to zero as x+∞, and to -1 as x−∞. If one accepts some plausible guesses about the behavior of solutions with such initial data, we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution: . The function s(ζ) is expressible in terms of complete elliptic integrals; a similar formula is derived for U2. In Section 8 we indicate how to extend the treatment of this series of papers to multihumped (but still negative) initial data.