若干非线性演化方程精确求解法的研究
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摘要
本文主要研究了在物理学领域中提出的一些非线性演化方程或方程组的精确解的求解问题。考虑的问题主要为：如何从待求解的方程或方程组出发，寻找有效的变换，将原方程(组)变为简单的、易解的方程，通过求解简单的方程，而获得原方程或方程组的精确解。主要工作如下：
(1)．第二章介绍了非线性微分方程精确求解的一般原则，并通过实例说明了这一原则的使用方式和适用范围。
(2)．第三章利用齐次平衡法研究了一些高维非线性演化方程和方程组的求解问题，通过对该方法的某些关键步骤的处理，将原方程转化为一线性的微分方程(组)，从而获得了这些方程大量的多种精确解。
(3)．通过引入一个新的变量，将(2+1)-维破裂孤子方程简化为一个单个方程，或利用Bcklund变换将一类(2+1)-维的耦合方程组化为仅有其一个未知量的(1+1)-维非线性演化方程，这时方程不仅维数降低，而且阶数也降低。这使得近年来使用的非常有效的推广的tanh-函数法、投影Riccati方程法等求解法使用起来更简单，特别是在求这类非线性演化方程组的类孤波解时，其求解的简炼程度十分显著，甚至比求原方程的行波解的原有方法更简单(分别见第四五章)。
(4)．第六章对楼森岳教授提出的求解非线性演化方程或方程组的分离变量解的分离变量法作了进一步的研究，利用Bcklund变换和Cole-Hopf变换，一些非线性演化方程或方程组可被化为一个具有一个或两个分别以{x，t}和{y，t}为自变量的任意函数的线性偏微分方程。利用与楼森岳教授的分离变量法相同的思想，发现了一类新的分离变量解，同时也得到了一些与楼森岳教授的分离变量法所确定的同类分离变量解，但这里的方法更简单。
本文的安排如下，第一章简要的介绍了非线性演化方程(组)精确求解法的发展情况。第二章以张鸿庆教授于1978年提出的偏微分方程求解的构造性的机械化算法，即“AC=BD”的理论模式为指导，介绍微分方程求解的一般原则及其在非线性演化方程精确求解中的应用。第三章至第六章则分别就上面所提到的
(l)一(4)这四个方面进行详细的讨论.
This dissertation mainly studies finding exact solutions of some nonlinear evolution equation(s) arrising from physics. The main problems under consideration is how to seek efficient transformations that can reduce the equation(s) to be solved to one(s) which is (are) simpler and easily solved. By solving the latter, the exact solutions of the original equation(s) can be then obtained. The main results derived are as follows:
(1). In chapter 2, we introduce the general principle to solve nonlinear partial differential equations. Some illustrative examples are presented to show how to use the principle and the application range.
(2). In chapter 3, homogeneous balance method is used to deal with some problems of finding solutions of higher dimensional nonlinear evolution equation(s). By improving some key procedure of the method, the original equation(s) is (are) reduced to a linear equation(s), hence a great number of exact solutions for this (these) are obtained.
(3). By introducing a new variable, the (2+l)-dimensional breaking soliton equation is simplified to be a sigle one, or using Backlund transformation to change a class of (2+l)-dimensional coupled equations to a (1+1)-dimensional nonlinear equation with only an unknow variable. The latter one is not only with a low dimension, but also a low order. This makes it much simpler to use the efficient extended tanh-function method and the projective Riccati equation method used in recent years in the literature. In particular, when seeking soliton-like solutions of this kind of nonlinear evolution equations, the concise degree is much notable, even more simpler than the method existing to find travelling wave solutions. For more details, please consult chapter 4-5, respectively.
(4). Chapter 6 further investigates the variable separation method to solve nonlinear evolution equation(s) presented by professor Lou senyue. By using
Backlund transformation and Cole-Hopf transformation, some nonlinear equation(s) can be reduced to a linear partial equation including one or two arbitrary functions of two varable (x,t) or (y,t), respectively. Simlar to professor Lou's variable separation method, a new type of variable separation solutions are found. At the same time, some same type solutions obtained by professor Lou are also derived, but the method used here is more simpler.
The dissertation is organized as follows. In chapter 1 the development of solving methods for finding exact solutions of nonlinear evolution equations is introduced briefly. In chapter 2, under the guidance of the theory of AC=BD presented by professor Zhang Hongqing in 1978, we introduce general principle of seeking solutions for differential equation and its application to searching for exact solutions of nonlinear evolution equations. The four aspects mentioned above are discussed in details in chapter 3-6, respectively.
(1)．第二章介绍了非线性微分方程精确求解的一般原则，并通过实例说明了这一原则的使用方式和适用范围。
(2)．第三章利用齐次平衡法研究了一些高维非线性演化方程和方程组的求解问题，通过对该方法的某些关键步骤的处理，将原方程转化为一线性的微分方程(组)，从而获得了这些方程大量的多种精确解。
(3)．通过引入一个新的变量，将(2+1)-维破裂孤子方程简化为一个单个方程，或利用Bcklund变换将一类(2+1)-维的耦合方程组化为仅有其一个未知量的(1+1)-维非线性演化方程，这时方程不仅维数降低，而且阶数也降低。这使得近年来使用的非常有效的推广的tanh-函数法、投影Riccati方程法等求解法使用起来更简单，特别是在求这类非线性演化方程组的类孤波解时，其求解的简炼程度十分显著，甚至比求原方程的行波解的原有方法更简单(分别见第四五章)。
(4)．第六章对楼森岳教授提出的求解非线性演化方程或方程组的分离变量解的分离变量法作了进一步的研究，利用Bcklund变换和Cole-Hopf变换，一些非线性演化方程或方程组可被化为一个具有一个或两个分别以{x，t}和{y，t}为自变量的任意函数的线性偏微分方程。利用与楼森岳教授的分离变量法相同的思想，发现了一类新的分离变量解，同时也得到了一些与楼森岳教授的分离变量法所确定的同类分离变量解，但这里的方法更简单。
本文的安排如下，第一章简要的介绍了非线性演化方程(组)精确求解法的发展情况。第二章以张鸿庆教授于1978年提出的偏微分方程求解的构造性的机械化算法，即“AC=BD”的理论模式为指导，介绍微分方程求解的一般原则及其在非线性演化方程精确求解中的应用。第三章至第六章则分别就上面所提到的
(l)一(4)这四个方面进行详细的讨论.
This dissertation mainly studies finding exact solutions of some nonlinear evolution equation(s) arrising from physics. The main problems under consideration is how to seek efficient transformations that can reduce the equation(s) to be solved to one(s) which is (are) simpler and easily solved. By solving the latter, the exact solutions of the original equation(s) can be then obtained. The main results derived are as follows:
(1). In chapter 2, we introduce the general principle to solve nonlinear partial differential equations. Some illustrative examples are presented to show how to use the principle and the application range.
(2). In chapter 3, homogeneous balance method is used to deal with some problems of finding solutions of higher dimensional nonlinear evolution equation(s). By improving some key procedure of the method, the original equation(s) is (are) reduced to a linear equation(s), hence a great number of exact solutions for this (these) are obtained.
(3). By introducing a new variable, the (2+l)-dimensional breaking soliton equation is simplified to be a sigle one, or using Backlund transformation to change a class of (2+l)-dimensional coupled equations to a (1+1)-dimensional nonlinear equation with only an unknow variable. The latter one is not only with a low dimension, but also a low order. This makes it much simpler to use the efficient extended tanh-function method and the projective Riccati equation method used in recent years in the literature. In particular, when seeking soliton-like solutions of this kind of nonlinear evolution equations, the concise degree is much notable, even more simpler than the method existing to find travelling wave solutions. For more details, please consult chapter 4-5, respectively.
(4). Chapter 6 further investigates the variable separation method to solve nonlinear evolution equation(s) presented by professor Lou senyue. By using
Backlund transformation and Cole-Hopf transformation, some nonlinear equation(s) can be reduced to a linear partial equation including one or two arbitrary functions of two varable (x,t) or (y,t), respectively. Simlar to professor Lou's variable separation method, a new type of variable separation solutions are found. At the same time, some same type solutions obtained by professor Lou are also derived, but the method used here is more simpler.
The dissertation is organized as follows. In chapter 1 the development of solving methods for finding exact solutions of nonlinear evolution equations is introduced briefly. In chapter 2, under the guidance of the theory of AC=BD presented by professor Zhang Hongqing in 1978, we introduce general principle of seeking solutions for differential equation and its application to searching for exact solutions of nonlinear evolution equations. The four aspects mentioned above are discussed in details in chapter 3-6, respectively.
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