反散射变换与指数函数法的三个问题研究

发布时间：2018-04-01 23:38

本文选题：反散射变换　切入点：指数函数法　出处：《渤海大学》2017年硕士论文

【摘要】：反散射变换法和指数函数法是孤子理论近些年发展起来的求解非线性偏微分方程的重要方法.反散射变换法首先通过正散射求出t(28)0时刻的散射数据,然后利用时间发展式求出散射数据随时间t的变化规律,最后通过位势重构得到非线性偏微分方程的解.指数函数法首先设所求非线性偏微分方程的指数函数有理拟解,然后通过平衡最高阶导数项和最高次非线性项以及收集指数函数同次幂系数确定拟解中待定参数的值.本文一方面研究如何将反散射变换法推广应用于求解谱参数分别按照正弦函数和有理式发展的两个新非等谱AKNS方程组的问题.另一方面研究如何解决指数函数法在运算过程中出现的“中间表达式膨胀”问题和如何确定指数函数法求解非线性晶格方程的最简拟解问题.本文的主要工作有:首先,通过推广AKNS线性谱问题及其时间发展式推导出谱参数按照正弦函数发展以及按照有理式发展的两个新非等谱AKNS方程组,然后推广反散射变换法分别对其求解,结果得到这两个非等谱方程组的新精确解和新n孤子解,并对所得部分解的局域空间结构和动力演化行为进行模拟.其次,通过给指数函数法拟解的新形式提出指数函数法的一个直接算法,作为算法的两个例子,我们将其应用于KdV方程和Jimbo-Miwa方程.算例表明我们的算法能在较大程度上解决指数函数法的“中间表达式膨胀”问题.最后,通过定义有理指数函数拟解的正负方幂给出指数函数法在求解一类变系数非线性晶格方程时最简拟解的一个定理及其证明,应用我们所给定理可以省略利用平衡方程中最高阶导数项和最高次非线性项的方式确定拟解的过程,从而将求解这类非线性晶格方程的指数函数法进行改进.作为算例,我们利用最简拟解求解了变系数mKdV晶格方程,从中展示出最简拟解的有效性.
[Abstract]:Inverse scattering transform method and exponential function method are important methods for solving nonlinear partial differential equations developed in recent years. Then the law of scattering data with time t is obtained by time evolution. Finally, the solution of nonlinear partial differential equation is obtained by potential reconstruction. The exponential function method first establishes the rational quasi-solution of exponential function of nonlinear partial differential equation. Then, by balancing the highest derivative term and the highest order nonlinear term and collecting the same power coefficient of the exponential function, the value of the parameters to be determined in the quasi solution is determined. On the one hand, this paper studies how to extend the backscattering transformation method to solve the spectral parameters. The problems of two new nonisospectral AKNS equations developed according to sinusoidal function and rational formula respectively are discussed. On the other hand, how to solve the problem of "intermediate expression expansion" in the operation of exponential function method and how to determine it are studied. Exponential function method is used to solve the most simple quasi solution problem of nonlinear lattice equation. The main work of this paper is as follows: first of all, By extending the AKNS linear spectrum problem and its time evolution, two new nonisospectral AKNS equations with spectral parameters developed according to sinusoidal function and rational formula are derived, and then the generalized inverse scattering transformation method is used to solve them respectively. Results the new exact solutions and new n-soliton solutions of these two nonisospectral equations are obtained, and the local spatial structure and dynamic evolution behavior of the obtained partial solutions are simulated. This paper presents a direct algorithm of exponential function method by giving a new form of solution to exponential function method, as two examples of the algorithm. We apply it to KdV equation and Jimbo-Miwa equation. The example shows that our algorithm can solve the problem of "intermediate expression expansion" in exponential function method to a large extent. Finally, By defining the positive and negative power of the quasi-solution of rational exponential function, a theorem and proof of the simplest quasi-solution of exponential function method for a class of nonlinear lattice equations with variable coefficients are given. By using the theorem we give, we can omit the process of determining the quasi solution by means of the highest derivative term and the highest subnonlinearity term in the equilibrium equation, and then improve the exponential function method for solving the nonlinear lattice equation of this kind of equation. In this paper, we solve the mKdV lattice equation with variable coefficients by using the simplest quasi solution, which shows the validity of the simplest quasi solution.
【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】：O175.29

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