Purpose

The purpose of this paper is to demonstrate that the numerical method is not everything for nonlinear equations. Some properties cannot be revealed numerically; an example is used to elucidate the fact.

Design/methodology/approach

A variational principle is established for the generalized KdV – Burgers equation by the semi-inverse method, and the equation is solved analytically by the exp-function method, and some exact solutions are obtained, including blowup solutions and discontinuous solutions. The solution morphologies are studied by illustrations using different scales.

Findings

Solitary solution is the basic property of nonlinear wave equations. This paper finds some new properties of the KdV–Burgers equation, which have not been reported in open literature and cannot be effectively elucidated by numerical methods. When the solitary solution or the blowup solution is observed on a much small scale, their discontinuous property is first found.

Originality/value

The variational principle can explain the blowup and discontinuous properties of a nonlinear wave equation, and the exp-function method is a good candidate to reveal the solution properties.

中文翻译：

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不同比例下的差分方程与差分方程

目的

本文的目的是证明数值方法并不是非线性方程式的全部。有些属性无法用数字显示；举一个例子来说明这一事实。

设计/方法/方法

通过半逆方法建立了广义KdV-Burgers方程的变分原理，并通过exp函数方法对方程进行了解析求解，并获得了一些精确的解，包括爆破解和不连续解。通过使用不同比例的插图研究溶液的形态。

发现

孤立解是非线性波动方程的基本性质。本文发现了KdV–Burgers方程的一些新性质，尚未在公开文献中进行报道，也无法通过数值方法有效地加以阐明。当以小规模观察到单独溶液或爆炸溶液时，首先发现它们的不连续性。

创意/价值

变分原理可以解释非线性波动方程的爆破和不连续性质，而exp函数法是揭示解性质的理想选择。