函数展开法在两个非线件方程中的应用

发布时间：2018-05-07 09:37

本文选题：精确解 + Boiti-Leon-Pempinelli　；参考：《西北大学》2017年硕士论文

【摘要】：随着近代科学技术的发展,非线性物理方程在非线性科学许多领域都有着不可替代的地位,在等离子体、流体力学、医学、生物学等方面都具有十分广泛的应用.因此,为了更好地将理论与实际相结合,非线性物理方程的求解问题已经成为数学研究者的最活跃的课题之一.到目前为止,数学研究者已经提出了许多研究精确解的方法,例如:反散射变换,Darboux变换,tanh-coth展开法,齐次平衡法,双线性法,指数函数展开法,sine-cosine展开法等等.人们在应用这些方法求非线性物理方程解的过程中,获得了丰富的结果,与此同时,也获得了些许有实际意义的新解.本文主要应用tanh-coth展开法,结合Riccati方程,在行波解为根号假设,正级数假设以及正负级数对称假设三种假设情况下,研究了两类非线性物理方程,并求得了丰富的行波解.本文首先叙述了相容Riccati展开法与tanh-coth 展开法求解非线性物理方程的具体步骤,其次第二章应用相容 Riccati 展开法,证明了 Boiti-Leon-Pempinelli方程的CRE可积性,并在tanh-coth展开法的基础上,求得了此方程组的一些精确解并获得了其孤立子与其他激发之间的交互作用解.最后利用tanh-coth展开法求解了 Jimbo-Miwa方程,获得了 Jimbo-Miwa方程的双曲函数形式解,三角函数形式解,有理数形式解。
[Abstract]:With the development of modern science and technology, nonlinear physical equations play an irreplaceable role in many fields of nonlinear science. They are widely used in plasma, fluid dynamics, medicine, biology and so on. Therefore, in order to better combine theory with practice, the problem of solving nonlinear physical equations has become one of the most active topics of mathematics researchers. Up to now, many mathematical researchers have put forward many methods to study exact solutions, such as the inverse scattering transformation Darboux transform and the tanh-coth expansion method, the homogeneous equilibrium method, the bilinear method, the exponential function expansion method and the sine-cosine expansion method, and so on. In the process of applying these methods to the solution of nonlinear physical equations, a lot of results have been obtained. At the same time, some new solutions of practical significance have also been obtained. In this paper, using tanh-coth expansion method and Riccati equation, two kinds of nonlinear physical equations are studied under the assumption that the traveling wave solution is the root, the positive series hypothesis and the positive and negative series symmetry hypothesis, and the rich traveling wave solutions are obtained. In this paper, the concrete steps of the compatible Riccati expansion method and the tanh-coth expansion method for solving nonlinear physical equations are described. In the second chapter, the CRE integrability of the Boiti-Leon-Pempinelli equation is proved by using the compatible Riccati expansion method, and on the basis of the tanh-coth expansion method, the CRE integrability of the Boiti-Leon-Pempinelli equation is proved. Some exact solutions of the equations are obtained and the interaction solutions between the soliton and other excitations are obtained. Finally, the Jimbo-Miwa equation is solved by tanh-coth expansion method. The hyperbolic function solution, trigonometric function solution and rational number form solution of Jimbo-Miwa equation are obtained.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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