In this article, numerical solutions of the seven different forms of the single soliton and double soliton solutions of the Korteweg‐de Vries equation are investigated. Since numerical solution of the six test problems for small‐times do not exist in the literature, the present numerical results firstly are reported with exact solutions. Besides small‐time solutions, long‐time solutions of all test problems are obtained and compared with some of the earlier works. Present algorithm which is based on combination of finite difference method and differential quadrature method have obtained superior results than those in the given literature. Numerical and exact solutions for small‐time of all test problems are plotted together for all test problems. Since the numerical results are too close to exact solutions the graphs are indistinguishable. Numerical simulations for long‐time solutions are plotted and the error graphs are plotted for the end of the simulations of all test problems.

中文翻译：

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二次非线性Korteweg-de Vries方程的单孤子和双孤子解的小时和长时性

在本文中，研究了Korteweg-de Vries方程的单孤子和双孤子解的七种不同形式的数值解。由于文献中不存在六个短期测试问题的数值解，因此首先以精确解报道了目前的数值结果。除了短期解决方案之外，还获得了所有测试问题的长期解决方案，并将其与一些早期工作进行了比较。目前的基于有限差分法和差分正交法相结合的算法取得了比给定文献更好的结果。针对所有测试问题，针对小型测试问题的数值解和精确解被绘制在一起。由于数值结果太接近于精确解，因此图是无法区分的。