Superposition solutions to the extended KdV equation for water surface waves
Superposition solutions to the extended KdV equation for water surface waves
复制标题
水面波扩展KdV方程的叠加解
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
E. Infeld
中科院分区：
文献类型：
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作者：
P. Rozmej;A. Karczewska;E. Infeld
KdV方程可以由欧拉方程的浅水极限推导出来。在过去的几十年里，这个方程已经扩展到包括高阶效应。尽管该方程只有一个守恒定律，但存在精确的周期解和孤子解。 Khare 和 Saxena（Phys Lett A 377:2761–2765, 2013；J Math Phys 55:032701, 2014；J Math Phys 56:032104, 2015）证明了通过组合几个基本方程（例如 Korteweg–de Vries、非线性薛定谔）的已知方程来生成新精确解的可能性。在这里我们发现，对于这些方程的高阶、不可积扩展，可以重复这种构造。与文献中的许多陈述相反，可积性和非线性一变量波解的数量之间似乎没有相关性。
The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include higher-order effects. Although this equation has only one conservation law, exact periodic and solitonic solutions exist. Khare and Saxena (Phys Lett A 377:2761–2765, 2013; J Math Phys 55:032701, 2014; J Math Phys 56:032104, 2015) demonstrated the possibility of generating new exact solutions by combining known ones for several fundamental equations (e.g., Korteweg–de Vries, nonlinear Schrödinger). Here we find that this construction can be repeated for higher-order, non-integrable extensions of these equations. Contrary to many statements in the literature, there seems to be no correlation between integrability and the number of nonlinear one variable wave solutions.