【作者】 李伟；

【导师】 黄晔辉；

【作者基本信息】 华北电力大学(北京) ， 运筹学与控制论， 2018， 硕士

【摘要】 非线性发展方程可以描述流体力学、等离子体、非线性光学中的自然现象。孤立子理论是非线性科学的重要分支之一,带源的孤立子方程是对原孤子方程的一种可积耦合推广。本文分别以带自相容源的KdV方程和带自相容源的非线性薛定谔方程为数学模型,研究带源方程的解并对孤立波在非线性系统中传播的特性进行理论研究。研究对象带有自相容源,其反应了不同波之间的相互作用,在解释相关自然现象的基本规律时比KdV方程和非线性薛定谔方程更丰富。首先,本文通过Lie变换群方法得到方程的无穷小变换和对应的向量空间及最优系统,进一步求得方程的孤子解。其次,根据幂级数法,我们获得了方程的精确解。再次,通过选取不同的谱参数讨论了孤子解的动力学性质。最后,应用乘数法求得带自相容源的非线性方程的守恒律。更多还原

【Abstract】 Nonlinear evolution equations can describe natural phenomena of fluid mechanics,plasma theory and nonlinear optical.Soliton theory is one of the important branches of Nonlinear Science,soliton equation with self-consistent source is a integrable coupling extension of the soliton equation.Based on the KdV equation with a self-consistent source and the nonlinear Schrodinger equation with a self-consistent source,the solutions of the source equation are studied and the propagation characteristics of solitary waves in a nonlinear system are investigated theoretically,The equations with self-consistent sources reflect the interaction between different solitary waves.In explaining the basic rule of natural phenomenon,the equations with self-consistent source terms are more abundant than the normal nonlinear equations.Firstly,the infinitesimal transformation and the corresponding vector space and the optimal systems are derived by Lie symmetry method.Further,the soliton solutions of the equations are obtained.Secondly,the exact solutions of the equations are constructed via the power series method.Specially,the interactions between soliton solutions are studied by adjusting the spectral parameters.Finally,we use the multiplier method to obtain the conservation laws of nonlinear equations with the self-consistent source.更多还原