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Soliton solutions of nonlinear partial differential equations using variational approximations and inverse scattering techniques.

机译：使用变分近似和逆散射技术的非线性偏微分方程的孤子解。

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摘要

Throughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of variational approximations and inverse scattering transform. Additionally, a new technique for estimating the error of a variational approximation is established. Note that the material in chapter 2, "Quantitative Measurements of Variational Approximations" has recently been published [20].;Variational problems have long been used to mathematically model physical systems. Their advantage has been the simplicity of the model as well as the ability to deduce information concerning the functional dependence of the system on various parameters embedded in the variational trial functions. However, the only method in use for estimating the error in a variational approximation has been to compare the variational result to the exact solution. In this work, it is demonstrated that one can computationally obtain estimates of the errors in a one-dimensional variational approximation, without any a priori knowledge of the exact solution. Additionally, this analysis can be done by using only linear techniques. The extension of this method to multidimensional problems is clearly possible, although one could expect that additional diffculties would arise.;One condition for the existence of a localized soliton is that the propagation constant does not fall into the continuous spectrum of radiation modes. For a higher order dispersive systems, the linear dispersion relation exhibits a multiple branch structure. It could be the case that in a certain parameter region for which one of the components of the solution has oscillations (i.e., is in the continuous spectrum), there exists a discrete value of the propagation constant, kES, for which the oscillations have zero amplitude. The associated solution is referred to as an embedded soliton (ES). This work examines the ES solutions in a chi(2) : chi (3), type II system. The method employed in searching for the ES solutions is a variational method recently developed by Kaup and Malomed [Phys. D 184, 153-61 (2003)] to locate ES solutions in a SHG system. The variational results are validated by numerical integration of the governing system.;A model used for the 1-D longitudinal wave propagation in microstructured solids is a KdV-type equation with third and fifth order dispersions as well as first and third order nonlinearities. Recent work by Ilison and Salupere (2004) has identified certain types of soliton solutions in the aforementioned model. The present work expands the known family of soliton solutions in the model to include embedded solitons . The existence of embedded solitons with respect to the dispersion parameters is determined by a variational approximation. The variational results are validated with selected numerical solutions.
机译：在过去的几十年中，开发了许多技术来建立非线性偏微分方程（NPDE）的解决方案。这些技术的特点是解决大型NPDE的范围有限。本工作将使用上个世纪开发的两种最普遍的技术研究NPDE的分析。在这项工作中，本文的分析和技术适用于变分逼近和逆散射变换领域中尚未解决的物理问题。另外，建立了用于估计变分近似的误差的新技术。请注意，第2章“变化近似的定量测量”中的材料最近已发布[20]。变化问题长期以来一直用于对物理系统进行数学建模。它们的优点是模型的简单性以及推断有关系统功能依赖于可变试验函数中嵌入的各种参数的信息的能力。但是，用于估计变化近似误差的唯一方法是将变化结果与精确解进行比较。在这项工作中，证明了人们可以以一维变分近似的方式从计算上获得误差的估计，而无需任何先验知识的精确解。此外，可以仅使用线性技术来完成此分析。尽管可以预料还会出现更多的困难，但显然可以将这种方法扩展到多维问题上。局部孤子存在的一个条件是，传播常数不属于辐射模式的连续谱。对于高阶色散系统，线性色散关系表现出多分支结构。可能的情况是，在溶液的成分之一具有振荡（即处于连续光谱中）的某个参数区域中，存在传播常数kES的离散值，其振荡为零振幅。相关的解决方案称为嵌入式孤子（ES）。这项工作检查了chi（2）：chi（3）类型II系统中的ES解决方案。用于搜索ES解决方案的方法是Kaup和Malomed [Phys。 D 184，153-61（2003）]在SHG系统中定位ES解决方案。通过控制系统的数值积分验证了变分结果。；用于微结构固体中一维纵波传播的模型是具有三阶和五阶色散以及一阶和三阶非线性的KdV型方程。 Ilison和Salupere（2004）的最新工作已经在上述模型中确定了某些类型的孤子解。当前的工作扩展了模型中已知的孤子解决方案系列，以包括嵌入式孤子。关于色散参数的嵌入式孤子的存在是通过变分近似来确定的。变异结果用选定的数值解进行了验证。

著录项

  • 作者

    Vogel, Thomas K.;

  • 作者单位

    University of Central Florida.;

  • 授予单位 University of Central Florida.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2007
  • 页码 94 p.
  • 总页数 94
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;
  • 关键词

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