五阶和七阶Korteweg-de Vries方程的行波解：Sech和Cn解
Traveling Wave Solutions to Fifth- and Seventh-order Korteweg-de Vries Equations: Sech and Cn Solutions
摘要：
本文综述了在应用科学和工程中应用的一个含高阶色散项的Korteweg-de Vries（KdV）方程的物理意义。我们还提出了精确的行波解决方案，这个广义KdV方程使用椭圆函数的方法，可以很容易地应用到任何标量演化或波动方程的多项式项只涉及奇导数。我们发现，广义KdV方程仍然支持驼峰形孤立波以及椭圆余弦波的解决方案，提供的系数满足特定的代数约束。封闭形式的解析解可作为数值求解器或与实验数据比较的基准。它们通常对应于相空间中的同宿轨道，并作为稳定区域和不稳定区域的分界线。本文中提出的一些解决方案的正确，补充，并说明以前在文献中报道的结果，而另一些是新颖的。
英文摘要：
In this paper we review the physical relevance of a Korteweg-de Vries (KdV) equation with higher-order dispersion terms which is used in the applied sciences and engineering. We also present exact traveling wave solutions to this generalized KdV equation using an elliptic function method which can be readily applied to any scalar evolution or wave equation with polynomial terms involving only odd derivatives. We show that the generalized KdV equation still supports hump-shaped solitary waves as well as cnoidal wave solutions provided that the coefficients satisfy specific algebraic constraints. Analytical solutions in closed form serve as benchmarks for numerical solvers or comparison with experimental data. They often correspond to homoclinic orbits in the phase space and serve as separatrices of stable and unstable regions. Some of the solutions presented in this paper correct, complement, and illustrate results previously reported in the literature, while others are novel.
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