一个高阶非线性薛定谔方程的精确解研究

发布时间：2018-05-09 02:25

本文选题：薛定谔方程 + 双线性　；参考：《华北电力大学(北京)》2017年硕士论文

【摘要】：非线性发展方程可以描述等离子体、流体力学、非线性光学等自然现象。本文分别以一个高阶非线性薛定谔方程和相应的变系数高阶非线性薛定谔方程为数学模型,从解的角度对孤立波在非线性系统中传播的特性进行理论研究。研究对象带有高阶奇数项和偶数项,其在解释相关自然现象的基本规律时比一般非线性薛定谔方程更准确。首先,本文通过双线性的方法得到单孤子解和二孤子解,并且研究了高阶项对孤子解的影响和两孤子之间的相互作用。其次,根据广义的达布变换,我们获得了方程的呼吸子解和怪波解。通过选取不同的谱参数讨论了二阶呼吸子的动力学性质,同时也研究了呼吸子和怪波之间的碰撞。最后,文章以高阶非线性薛定谔方程的解为基础,构造出了非均匀高阶非线性薛定谔方程的精确解。
[Abstract]:Nonlinear evolution equations can describe natural phenomena such as plasma hydrodynamics nonlinear optics and so on. In this paper, a higher order nonlinear Schrodinger equation and a corresponding variable coefficient high order nonlinear Schrodinger equation are used as mathematical models to study theoretically the propagation characteristics of solitary waves in nonlinear systems from the point of view of solution. The object of study has higher order odd and even terms, which is more accurate than the general nonlinear Schrodinger equation in explaining the basic laws of related natural phenomena. Firstly, the single soliton solution and the two-soliton solution are obtained by bilinear method, and the influence of higher order terms on the soliton solution and the interaction between the two solitons are studied. Secondly, according to the generalized Darboux transform, we obtain the respiratory subsolutions and odd wave solutions of the equation. The dynamical properties of the second order respiration are discussed by selecting different spectral parameters, and the collision between the respiratory and strange waves is also studied. Finally, based on the solution of the higher order nonlinear Schrodinger equation, the exact solution of the nonuniform high order nonlinear Schrodinger equation is constructed.
【学位授予单位】：华北电力大学(北京)
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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