【作者】 贾强；

【导师】 卢殿臣；

【作者基本信息】 江苏大学 ， 应用数学， 2003， 硕士

【摘要】 本文主要研究了一类带有小扰动参数的非线性Schr（?）dinser方程的求解问题，讨论了自伴算子的本征函数的正交性和完备性，介绍了寻求微分方程的近似解常用的摄动方法，并从带有某种扰动项的NLS方程出发，利用多重尺度的摄动方法得到了方程的零级近似方程和一级近似方程，通过对近似方程中算子的特征态的讨论，引入适当的“导出态”，建立了算子在L₂空间的特征态的完备性。利用这种完备性，得到了近似方程中相应的量在该完备集中的展开式，并得到了展开式中系数的演化方程，再通过对这些演化方程的讨论，求得了该方程在微小扰动下的一级近似解。本文给出了这种方法的基础——算子在L₂空间的特征态的完备性和多重尺度的摄动技巧。由于在计算的过程中采用了不同的时间尺度，所得的近似解具有较好的精确性，值得指出的是该方法也同样适用于扰动项为一般形式的情况。更多还原

【Abstract】 This paper mainly studies the solutions of the nonlinear Schrodinger equation with a small parameter; gives the properties of the eigenstates for the self-adjoint operator, namely, the orthogonality and completeness; introduces the perturbation theory in which people get the approximate solution of differential equations. And beginning with a perturbed NLS equation, using a multi-scales perturbation expansion, we get the zero order and the first order equations, discuss the eigenstates of the operator in the equations, induct relevant "derivative states", form the completeness of the bounded eigenstates of the associated operator in LI space, and expand the corresponding parameters in the closure, get a series evolution equations of the coefficients in the expanded formulas, find the first order approximate solution by researching the evolution equations. This paper also gives the basis of this method-the completeness we have formed and the singular perturbation technique. Since we use different time-scales in the calculation, the solution we get is more precise. It is worth pointing out that this method also can be used in other NLS equations with any perturbed terms.更多还原