双线性方法在求解非线性方程中的应用

发布时间：2018-06-10 07:27

本文选题：Hirota双线性方法 + 非线性演化方程　；参考：《山东科技大学》2017年硕士论文

【摘要】：近年来,离散可积系统已经被广泛地应用到很多领域,例如光学、流体力学、磁流体学等.相比连续可积系统,离散可积系统可以更好的刻画自然界物质的运动规律,成为近年来研究热点问题.除此之外,孤子方程有理解的研究可以深刻的描述许多物理现象,具有重要的潜在应用价值.在本文中,我们构造了一个新的离散谱算子,得到了离散晶格方程的可积辛映射和Hamiltonian结构.随后运用Hirota双线性方法分别得到了两类非线性发展方程的有理解、Lump解和怪波解.全文结构如下:1介绍了孤立子理论的产生以及发展现状,简单概述了 Hirota双线性求解方法的思想和应用.2依据离散可积系统的屠格式理论,构造了一个新的离散谱算子,通过离散可积系统的零曲率方程,得到了离散可积系统的Hamiltonian结构并列举了一组可积耦合方程.随后借助于Bargmann对称约束条件,得到了两组显示的对称约束,从而给出了离散可积系统的可积辛映射.最后运用对称理论求解的思想,对离散可积耦合方程进行求解研究,并分析了步长对解的动力学形态的影响.3根据双线性求解方法的思维和符号计算的技巧,首先运用多项式方法得到了(3+1)-维类浅水波方程的5组有理解,通过选取合适的参数,展示出有理解的动力学特征,随后借助二次函数思想,得到了约化(3+1)维类浅水波方程的lump解,通过参数约束确保此lump解在(x,y)平面的任何方向上均是局部有理的.4首先根据二次函数理论和双线性方法得到了约化(3+1)维非线性发展方程的lump解,更进一步地,将二次函数方法延拓为二次函数与指数函数的结合,从而得到了lump解和线孤子解的相互作用,由于lump解和线孤子解运动方向的不同,展示了两种动力学行为,孤子融合和孤子裂变.随后将二次函数延拓到二次函数与双曲余弦函数的结合,得到了二元函数的怪波解,相比于以前二维函数的线怪波解,此处得到的怪波解在(x,y)平面上均是局部有理的.需要强调的是,第四章中,非线性发展方程的怪波解在(x,y)平面上任何方向均有怪波的特性,将一维怪波推广到二维怪波,改善了传统意义上的二维线怪波解.
[Abstract]:In recent years, discrete integrable systems have been widely used in many fields, such as optics, hydrodynamics, magnetohydrology and so on. Compared with continuous integrable systems, discrete integrable systems can better describe the motion laws of natural materials, and have become a hot topic in recent years. In addition, soliton equations can describe many physical phenomena deeply and have important potential application value. In this paper we construct a new discrete spectral operator and obtain the integrable symplectic mapping and Hamiltonian structure of discrete lattice equations. Then, the Hirota bilinear method is used to obtain two kinds of nonlinear evolution equations, which are the understood stump solution and the odd wave solution, respectively. The structure of the paper is as follows: 1. The generation and development of soliton theory are introduced. The idea of Hirota bilinear solution method and its application .2 A new discrete spectral operator are constructed according to the Tu format theory of discrete integrable system. Through the zero curvature equation of discrete integrable system, the Hamiltonian structure of discrete integrable system is obtained and a set of integrable coupling equations are listed. Then, by means of Bargmann's symmetric constraint condition, two sets of shown symmetric constraints are obtained, and then the integrable symplectic mapping of discrete integrable systems is given. Finally, the solution of discrete integrable coupled equations is studied by using the idea of symmetry theory, and the influence of step size on the dynamic shape of solution is analyzed. 3. According to the thinking of bilinear solution method and the technique of symbolic calculation, Firstly, five groups of shallow water wave equations are obtained by using polynomial method. By selecting appropriate parameters, the dynamic characteristics with understanding are shown, and then the idea of quadratic function is used. In this paper, the lump solution of the shallow water wave equation with reduced dimension is obtained. It is ensured that the lump solution is locally rational in any direction of the plane by parameter constraints. At first, according to the quadratic function theory and bilinear method, the lump solution of the reduced 31) dimensional nonlinear evolution equation is obtained. The quadratic function method is extended to the combination of quadratic function and exponential function, and the interaction between the lump solution and the linear soliton solution is obtained. Due to the difference of the motion direction between the lump solution and the linear soliton solution, two dynamic behaviors are shown. Soliton fusion and soliton fission. Then, by combining the quadratic function with the hyperbolic cosine function, the singular wave solution of the binary function is obtained. Compared with the linear anomalous wave solution of the previous two-dimensional function, the singular wave solution obtained here is locally rational in the plane of the hyperbolic cosine function. It should be emphasized that in chapter 4, the solution of nonlinear evolution equation has the characteristic of strange wave in any direction of the plane. The one-dimensional strange wave is extended to the two-dimensional strange wave, which improves the traditional 2-D linear strange wave solution.
【学位授予单位】：山东科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.7

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