退化达布变换的行列式表示及其应用

发布时间：2018-06-16 08:05

本文选题：退化达布变换 + Positon解　；参考：《中国科学技术大学》2016年博士论文

【摘要】：本文研究退化达布变换的行列式表示及其应用。第二章,针对复mKdV方程,详细推导n阶达布变换的行列式表示,再通过其中的特征值退化来定义退化达布变换并给出其行列式表示。第三章,利用退化达布变换构造复mKdV方程的positon解的显式解析表达。利用模平方分解方法,把复mKdV方程的positon解分解为多个孤立子之和,并讨论轨迹以及碰撞后变换的“相移”。这与实mKdV方程的奇异positon解有很大的差异。第四章,针对高阶非线性薛定谔方程,利用退化达布变换构造n阶怪波解的行列式表示,并通过二阶到五阶怪波解的图形来考虑怪波解的分解规律,得到高阶怪波解的基本形式：基本模式、三角形模式、环形模式。这些解包含参数α和6,这两个参数表示高阶非线性薛定谔方程的高阶项的作用(高阶色散和非线性效应)。另外为了分析怪波解局域性,本文利用等高线方法定义一阶怪波解的长度和宽度。通过前述各阶怪波的演化图像,发现高阶项对怪波解在t方向上有明显的压缩效应。第五章,提出非局域LPD方程及其Lax对,并构造其n阶达布变换和1阶退化达布变换。利用退化达布变换,构造非局域LPD方程的有理孤立子解,研究该解的分类和对应的PT势函数的时间演化。
[Abstract]:In this paper, the determinant representation of degenerate Darboux transform and its application are studied. In the second chapter, for complex mKdV equation, the determinant representation of n-order Darboux transform is derived in detail, and then the degenerate eigenvalue is used to define the degenerate Darboux transform and its determinant representation is given. In chapter 3, the explicit analytic expression of the positon solution of the complex mKdV equation is constructed by using the degenerate Darboux transform. The positon solution of the complex mKdV equation is decomposed into the sum of several solitons by means of the modular square decomposition method, and the "phase shift" of the trajectory and the post-collision transformation is discussed. This is quite different from the singular positon solution of the real mKdV equation. In chapter 4, for the higher order nonlinear Schrodinger equation, the determinant representation of the n-order odd wave solution is constructed by using the degenerate Darboux transform, and the decomposition law of the strange wave solution is considered by the graph of the second-order to fifth-order strange wave solution. The basic forms of the higher order strange wave solutions are obtained: the basic model, the triangular model, and the annular model. These solutions include parameters 伪 and 6, which represent the effects of higher order terms (higher order dispersion and nonlinear effect) of the higher order nonlinear Schrodinger equation. In addition, in order to analyze the localization of strange wave solutions, the length and width of the first order strange wave solutions are defined by using the contour method. It is found that the higher order term has obvious squeezing effect on the solution of the strange wave in the t direction through the evolutional images of the above order strange waves. In chapter 5, the nonlocal LPD equation and its lax pair are proposed, and the n-order Darboux transformation and the first-order degenerate Darboux transformation are constructed. The rational soliton solutions of nonlocal LPD equations are constructed by degenerate Darboux transform. The classification of the solutions and the time evolution of the corresponding PT potential functions are studied.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.29

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