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Landau-Lifshitz方程的反散射方法
【作者】 凌黎明；
【导师】 郭柏灵；
【作者基本信息】 中国工程物理研究院 ， 基础数学， 2013， 博士
【摘要】 本论文研究可积Landau-Lifshitz方程的反散射方法.经典可积情形下的Landau-Lifshitz方程早在上个世纪70年代末就已有大量的研究工作.国内的很多学者,在80年代,90年代初期亦研究过此经典模型的相关问题.经典可积的模型,虽然有其理想的地方,但是它为与之相关非可积模型研究提供一些思路.从中发现一些新现象和新问题亦为非可积的方程的研究指明方向.本文的主要工作是运用发展的反散射方法—Riemann-Hilbert方法研究此经典模型.由于近年来反散射方法的发展,利用此方法研究此经典模型,仍然可以得到一些新的结果.首先,我们可以得到在反散射框架下的解的存在唯一性.其次对于离散散射数据和精确解的求解,我们利用的主要工具是推广的Darboux变换.首次将推广的Darboux变换结合反散射方法巧妙地处理反散射方法中多重极点的散射数据的求解问题.据此,我们可以给出一般的孤立子求解公式,进而给孤立子解以完全的分类.除此之外,我们可以运用Deift-Zhou方法分析解的长时间渐近行为.反散射方法作为求解可积偏微分方程柯西问题的重要方法,亦提供了研究微分方程的重要手段.近年来,非线性科学领域内出现了新的研究对象—怪波.它的产生机制为调制不稳定性.此时,由于涉及到微分方程的不适定性,微分方程的一些理论已经无法分析.反散射方法为这些新现象的研究亦提供了重要手段.第一章为背景介绍.首先介绍方程的背景知识,研究进展.其次介绍反散射方法的历史背景,重要进展和技巧,以及一些最新结果.最后给出了本文的创新点和主要结果.第二章研究经典Landau-Lifshitz方程的反散射方法.首先运用规范变换和反散射方法适定性理论的新结果,得到Landau-Lifshitz方程在一个带权的Soblev空间的适定性.其次,运用推广的Darboux变换方法将孤立了解给予完全的分类.最后利用Deift-Zhou方法分析一般孤立子解的长时问渐近行为.第三章研究球对称情形可积的Landau-Lifshitz方程,此模型的可积性研究早在1994年由Lakshamnan等人提出.关于此模型精确解的研究已有些结果,但是利用反散射方法研究该问题尚属首次.主要因为其相应的谱问题为非等谱的半线问题.我们需要对谱问题进行适当的延拓.这里对于奇数维和偶数维系统运用不同的延拓方式.除此之外,我们还对方程解的动力学行为进行分析.同时我们顺便给出了推广的NLS方程的孤子解的动力学行为研究.第四章研究经典可积Landau-Lifshitz方程在自旋波背景下的反散射问题.运用的主要工具仍然是规范变换和反散射.首先,我们利用规范变换将其变为聚焦的NLS方程在平面波背景下的反散射问题.利用规范变换理论,我们研究其守恒律以及Galilean变换.最后我们利用推广的Darboux变换得到其一般的孤立子解公式.并且具体给出了呼吸子解和怪波解的表达式,同时通过画图分析这些具体解的动力学行为.
【Abstract】 In this dissertation, we consider the inverse scattering method of Landau-Lifshitz equation. There are lots of works concerned the integrable Landau-Lifshitz equation in the70s of the last century. Domestic scholars research the relevant issues of this model also in the80s and90s of the last century. Although the classical integrable model is idealization, it provides research idea of the other non-integrable models. Therefore, finding some new phenomenon and new problems in the integrable models can be used to define the research orientation for the non-integrable ones. The main tasks in this dissertation are researching the classical model by inverse scattering method or Riemann-Hilbert approach. On account of the recent developments of inverse scattering method, we can obtain some new results by this approach. Firstly, we can obtain the well-posedness in the frame of inverse scattering. Secondly, we take advantage of the generalized Darboux transformation to tackle with the discrete scattering data and exact solution. We combine generalized Darboux transformation with inverse scattering method for the first time. By this cohesion, we ingeniously deal with the high order pole scattering coefficient. Moreover, the general soliton formula and the complete classification of soliton solution can be obtained. Finally, we can analysis the long time asymptotics of this model by Deift-Zhou method. These results can not obtained by classical PDE analysis. Therefore, the inverse scattering method, as a method to solve the Cauchy problems of integrable partial differential equation, provides the important ways to research it. Recently, there is a hot topic-rogue wave in the nonlinear science. An important physical mechanism to explain the rogue wave is the modulational instability. Owning to the ill-posednees of the model, we can not use the theory of analysis of PDEs in this instance. The inverse scattering method provides new ways to tackle with it.In chapter1, we briefly introduce the physical background and historical results of the model. Besides, we present the historical background, some new results and important progress with respect to the inverse scattering method. The main innovative points are also presented in this part.In chapter2, we concern on the inverse scattering method of the classical integrable Landau-Lifshitz equation. Firstly, we obtain the well-posedness in a weighted Soblev space by combining the gauge transformation and some known results. Secondly, we present the completely classification of soliton solution by combining the generalized Darboux transformation and inverse scattering method. Finally, we analyse the long time asymptotics of Landau-Lifshitz equation by Deift-Zhou nonlinear steepest descent method.In chapter3, we concern on the spherical symmetry Landau-Lifshitz (ssL-L) equa-tion. The integrability of this model is given in by Lakshamnan group. There are some results about exact solution. But there is no result about the Cauchy problem about this model, because it is the non-isospectral half line problem. We need extend the spectral problem on the line properly. Besides, we analyse the dynamics of solution of ssL-L equation. As a by-product, we show the dynamics of the generalized NLS equation.In chapter4, we consider the inverse scattering problem of Landau-Lifshitz equa-tion on the spin wave background. We give the gauge transformation and inverse scattering analysis. Firstly, we take advantage of gauge transformation to convert this problem into focusing NLS equation on the plane wave background. Besides, we give the Galilean transformation to Landau-Lifshitz equation and conversation laws. Finally, we utilize the generalized Darboux transformation to give the generalized soli-ton formula. Moreover, the breather solutions and rogue wave solutions are presented explicitly. The dynamics of solution is obtained by plotting picture.