节点文献
孤子理论若干问题研究
【作者】 刘宏准；
【导师】 潘祖梁；
【作者基本信息】 浙江大学 ， 应用数学， 2007， 博士
【摘要】 本文围绕孤子理论的Hirota直接方法，扰动，对称，相似解等问题做了以下三方面的主要工作：1、首先研究了双线性算子的一些性质，得到了在对数变换意义下KdV类的双线性方程所直接相对应的非线性pde(即过渡方程)的一些性质。利用这些性质，本文首次得出了线性项非空的多项式形式的原方程与相应的过渡方程之间的线性项非空的多项式形式的变换的具体可能形式，并构造例子给以证实；得到了任意给定的线性项非空的多项式形式非线性pde具有线性项非空的多项式形式变换的一些必要条件。如果所给定的线性项非空的多项式形式的非线性PDE能化成KdV类的双线性方程，我们可以发现这个原方程线性部分算子与其相对应的KdV类的双线性方程中的双线性算子的关系，从而直接写出该方程所对应的双线性方程。这些结果使我们对多项式形式的线性项非空的KdV类的非线性PDE有直观的了解，从而能直接的改进一些义献的结果。2、对一阶非线性扰动方程的情形，用泰勒级数关于扰动参数ε在零点附近直接展开方法推导了该扰动方程渐近解析解所应具有的条件等式。通过这个条件等式，我们就有可能获得一些变换，这些变换把给定的未扰动方程的解析解直接的映成扰动方程的渐近解析解。为了获得更多的这类变换，我们引入了Lie-Baclund(LB)对称，从而利用可能存在的未扰动方程的递归算子和LB对称来构造大量的此类变换，最终从未扰动方程的精确解出发，通过这些变换，直接获得—大批的渐近解析解。扰动的KdV方程和扰动的Burgers方程将被作为例子给出。至于高阶的情形，我们仅给出理论上的一些推导结果和一个简单的例子。值得注意的是，一阶情形与高阶情形的理论结果的推导与变换的计算无须用到任何的群论知识。另外我们用Hirota直接方法研究了一种扰动的KdV方稗的N-孤立子解，并研究了扰动情形下N-孤立子解在的图像情况。3、对Broer-Kaup(BK)方程组所得到的单一方程作了经典的李对称相似约化，并给出部分相似解。对其中一类特殊的情形，通过观察，我们发现了BK方程绢的相似解与著名的Burgers方程之间的关系，从而，我们可以从已知的大量的Burgers方程和热方程的精确解中得到大量的BK方程的相似解。
【Abstract】 In this paper, we discuss mainly three aspects about the Hirota’s direct method, perturbation, symmetries and similarity solutions of nonlinear partial differential equations, and they read as follows:1. Firstly the significant characters of the bilinear operator are studied, then we investigate the characters of the nonlinear PDE directly obtained by the logarithmic transformation for the KdV-type bilinear equation. According to these characters, the polynomial transformations with nonempty linear term between the original polynomial equation with nonempty linear term and its KdV-type bilinear equation are obtained in details and also are confirmed with some examples. Some necessary conditions for the given polynomial nonlinear PDE with nonempty linear term, which owns corresponding polynomial transformation with nonempty linear term, are found. If the given nonlinear PDE owns its Kdv-type bilinear form, according to these necessary conditions, we can find the relationships between linear operators of the original equation and the bilinear operators of its bilinear form. And the bilinear form of the given original equation can be obtained directly in virtue of the relationships. These results make us straightforth understanding of the KdV-type nonlinear PDE, which help us improve directly some results appearing in literature.2、As for the 1~st-order perturbed nonlinear PDE, we obtain the conditional equality that the coefficient of the 1~st-order term of the Taylor series expansion aroundε=0 for any asymptotical analytical solution of the perturbed PDE with perturbing parameterεmust be admitted. By making use of the conditional equality, we may obtain some transformations, which directly map the analytical solutions of the given unperturbed PDE to the asymptotical analytical solutions of the corresponding perturbed one. The Lie-Baclund symmetries is introduced in order to obtaining more transformations. Hence, we can directly create more transformations in virtue of known Lie-Baclund symmetries and recursion operators of corresponding unperturbed equation. The perturbed Burgers equation and the perturbed Korteweg-de Vries(KdV) equation are used as examples. As for the higher-order cases, we only give the main results with a simple example. We should notice that not any group theoretical knowledge is introduced in the deduce process for obtaining these theoretical results and transformations. In addition, the N-solitons solutions of the perturbed KdV equations are investigated by Hirota’s direct method approach and the 1-soliton diagram are illustrated.3. By a known transformation, the Broer-Kaup(BK) equations are combined to a single one. After the classical Lie symmetry analysis and similarity reductions are performed for the single one, new similar solutions are obtained. From the special kind of its similarity reductions, we find the relationship between the BK equations and the famous Burgers equation and the heat equation by observation. By making use of these relationship, more similarity solution can be created directly by abundant known solutions of the Burgers equation and the heat equation.