函数展开法在求解非线性偏微分方程中的应用

发布时间：2018-04-14 20:06

本文选题：函数展开法 + 非线性偏微分方程　；参考：《宁波大学》2015年硕士论文

【摘要】：物理学上许多的问题都可以被归结为数学模型。因此,寻找由数学模型带来的非线性偏微分方程的精确解不仅在孤子理论中有着十分重要的地位,而且有助于我们加深对物理学本质的认识。函数展开法在求解非线性偏微分方程中具有不可替代的作用,其中推广的tanh函数展开法和相容的Riccati展开法是两种十分简捷有效的方法。本文主要运用这两种方法来研究Gardner方程、(2+1)维Konopelchenko-Dubrovsky方程以及一类耦合Boussinesq型方程解的问题。在Gardner方程的研究方面,本文首先简单地介绍了Painlev′e截断展开法和tanh函数展开法的基本思想和步骤,然后利用这两种方法得到了Gardner方程的解和相容性条件方程,然后通过将解写成椭圆函数形式,对相容性条件方程进行求解,最终得到了孤立子与椭圆周期解的相互作用。在Gardner方程研究方面,本文首先简要地介绍了相容的Riccati展开法,并给出了相容的Riccati展开可解性的概念,然后对Gardner方程应用相容的Riccati展开法求得其解与相容性条件,求解相容性条件和利用椭圆函数的定义,得到Gardner方程新的精确解以及一些具有特殊结构的解。在(2+1)维Konopelchenko-Dubrovsky方程研究中,本文运用相容的Riccati展开法,先求得(2+1)维Konopelchenko-Dubrovsky方程的解和相容性条件,然后通过对相容性条件进行求解,观察到(2+1)维Konopelchenko-Dubrovsky方程新的解之间存在着具有周期性扭结的相互作用。在一类耦合Boussinesq型系统研究中,本文同样运用相容的Riccati展开法,求得一类耦合Boussinesq型系统的解和相容性条件,然后通过求解其相容性条件,得到一类耦合Boussinesq型系统新的孤立波与其他非线性激发之间的相互作用。值得关注的是,我们可以得到一类耦合Boussinesq型系统的正弦孤立波解结构,这主要是利用了雅可比椭圆函数和第三类不完全可积椭圆积分。
[Abstract]:Many problems in physics can be reduced to mathematical models.Therefore, finding exact solutions of nonlinear partial differential equations brought by mathematical models is not only very important in soliton theory, but also helpful to our understanding of the essence of physics.The function expansion method plays an irreplaceable role in solving nonlinear partial differential equations. The generalized tanh function expansion method and the compatible Riccati expansion method are two very simple and effective methods.In this paper, we mainly use these two methods to study the solution of the Gardner equation / 21) -dimensional Konopelchenko-Dubrovsky equation and a class of coupled Boussinesq type equations.In the research of Gardner equation, this paper introduces the basic ideas and steps of Painlev'e truncation expansion method and tanh function expansion method, and then obtains the solution and compatibility condition equation of Gardner equation by using these two methods.Then, by writing the solution as an elliptic function, the compatibility condition equation is solved and the interaction between the soliton and the elliptic periodic solution is obtained.In the study of Gardner equation, the compatible Riccati expansion method is introduced briefly, and the concept of compatible Riccati expansion solvability is given. Then the compatible Riccati expansion method is applied to the Gardner equation to obtain its solution and compatibility conditions.By using the definition of elliptic function, a new exact solution of Gardner equation and some solutions with special structure are obtained.In this paper, the compatible Riccati expansion method is used to obtain the solution and compatibility condition of the Konopelchenko-Dubrovsky equation in the first place, and then the compatibility condition is solved by the compatibility condition.It is observed that there is a periodic kink interaction between the new solutions of the Konopelchenko-Dubrovsky equation.In the study of a class of coupled Boussinesq type systems, the compatible Riccati expansion method is also used to obtain the solution and compatibility conditions of a class of coupled Boussinesq type systems.The interaction between new solitary waves and other nonlinear excitations for a class of coupled Boussinesq type systems is obtained.It is worth noting that we can obtain the structure of sinusoidal solitary wave solutions for a class of coupled Boussinesq type systems, which mainly uses Jacobian elliptic functions and the third class of incomplete integrable elliptic integrals.
【学位授予单位】：宁波大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.29

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