变系数KdV族的BT与类热—波动方程的精确解

发布时间：2018-03-31 08:31

本文选题：变系数KdV方程族　切入点：双线性形式B?cklund变换　出处：《渤海大学》2017年硕士论文

【摘要】：随着孤子理论研究的深入,人们发现非线性系统中完全可积的方程都存在孤子解。近几十年来出现了许多求孤子方程精确解的方法,如:反散射方法、Hirota双线性方法、B?cklund变换(BT)等。其中反散射方法、Hirota双线性方法对孤子理论的发展起着不可磨灭的作用,人们发现能用反散射法求精确解的方程也能用BT求解。Hirota双线性方法属于构造性求解,这种求解法不依附于方程的Lax对或者谱问题,具备简便、直接的特点。相对而言,BT可以由求出的一个解可以利用关系式直接写出另一个解,进而获得许多新解,但难点在于找出其中的变换关系式。学者们通过研究发现了双线性形式的BT,使求解更简单明了。近几年来人们愈加关注分数阶问题,精确求解分数阶方程比较难,仍处于初始阶段。本文的主要工作概括如下:首先,从Schr?dinger线性谱问题及其时间发展式(即Lax对)出发,利用本征函数的相容性条件,通过引入有限多个只与时间有关的任意函数,推出一个新的Lax可积变系数KdV族。同时得到变系数KdV族中一个变系数KdV方程的双线性形式BT,基于找到的BT,得到这个变系数KdV方程的单孤子解、双孤子解、三孤子解,进而归纳出n孤子解。其次,通过利用分离变量法和Mittag-Leffler函数的性质求解一类变系数Caputo时间-分数阶类热-波动方程的初边值问题,最终得到此变系数时间-分数阶类热-波动方程新精确解的一个统一表达式。为进一步确定所求得的精确解,考虑了三个带有初边值条件的具体类热-波动方程,并分别得到其精确解,所获得的精确解包括一些已知解为特例。研究结果表明,变量分离法可以用来求解科学与工程中的一些其他时间-分数阶类热-波动方程。
[Abstract]:With the further study of soliton theory, it has been found that soliton solutions exist in completely integrable equations in nonlinear systems. In recent decades, there have been many methods for finding exact solutions of soliton equations, such as backscattering method and Hirota bilinear method. The inverse scattering method plays an indelible role in the development of soliton theory. It is found that the equation which can be solved accurately by backscattering method can also be solved by BT bilinear method. This method is not dependent on the Lax pair or spectral problem of the equation, and has the characteristics of simplicity and directness. But the difficulty lies in finding out the transformation relation. Through the research, the scholars have found the bilinear form of BTT, which makes the solution more simple and clear. In recent years, people pay more attention to the fractional order problem, and it is difficult to solve the fractional order equation accurately. The main work of this paper is summarized as follows: first, from Schr? The dinger linear spectrum problem and its time evolution (i.e. Lax pair) are discussed. By introducing finite number of arbitrary functions related only to time, by using the compatibility condition of eigenfunctions, A new Lax integrable variable coefficient KdV family is derived. At the same time, a bilinear form of the variable coefficient KdV equation in the KdV family is obtained. Based on the obtained KdV equation, the single soliton solution, double soliton solution and three soliton solution of the KdV equation are obtained. Secondly, by using the method of separating variables and the properties of Mittag-Leffler function, we solve the initial boundary value problem of a class of Caputo time-fractional thermo-wave equations with variable coefficients. Finally, a unified expression for the new exact solution of the time-fractional thermo-wave equation with variable coefficients is obtained. In order to further determine the exact solution obtained, three specific thermo-wave equations with initial boundary conditions are considered. The exact solutions obtained include some known solutions as special cases. The results show that the method of variable separation can be used to solve some other time-fractional thermo-wave equations in science and engineering.
【学位授予单位】：渤海大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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