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Stability and existence of traveling wave solutions of the two-dimensional nonlinear Schrodinger equation and its higher-order generalizations.

机译：二维非线性Schrodinger方程行波解的稳定性和存在性及其高阶推广。

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摘要

The two-dimensional cubic nonlinear Schrödinger equation (NLSE) is a partial differential equation (PDE) that can be used as a model of phenomena in physical systems ranging from waves on deep water to pulses in optical fibers. Traveling waves often play an important role in studies of these systems. In this thesis, we study analytically and numerically the stability and existence of traveling wave solutions of the NLSE and its higher-order generalizations.; There are three main sections of this thesis. In the first section, we review the properties of the entire class of one-dimensional, bounded, traveling wave solutions of the NLSE and we prove the existence and construct the form of the one-dimensional, bounded, traveling wave solutions of two higher-order generalizations of the NLSE.; In the second section, we show asymptotically that all one-dimensional, bounded, traveling wave solutions of the two-dimensional NLSE are linearly unstable with respect to long-wave transverse perturbations. We compare these asymptotic results with results from numerical simulations of the two-dimensional NLSE with perturbed one-dimensional traveling wave solutions used as initial conditions. We complete the section by making a prediction for physical experiments of waves on deep water.; In the final section, we introduce two new symplectic PDE integration schemes. The first of these schemes can be used to solve PDEs with nonlinear parts solvable by the method of characteristics. The second scheme can be used to solve PDEs that require three-way splitting. We use these schemes to study the one-dimensional bounded solutions of the PDEs introduced in the first section.
机译：二维三次非线性Schrödinger方程（NLSE）是一个偏微分方程（PDE），可以用作物理系统中现象的模型，其范围从深水中的波到光纤的脉冲。行波通常在这些系统的研究中起重要作用。本文对NLSE及其高阶推广的行波解的稳定性和存在性进行了分析和数值研究。本论文主要分为三个部分。在第一部分中，我们回顾了NLSE的整个一维有界行波解的性质，并证明了存在性并构造了两个更高阶的一维有界行波解的形式。 NLSE的命令概括。在第二部分中，我们渐近显示了二维NLSE的所有一维有界行波解相对于长波横向扰动都是线性不稳定的。我们将这些渐近结果与二维NLSE数值模拟的结果进行比较，并以一维行波解作为初始条件。我们通过对深水波的物理实验进行预测来完成本节。在最后一节中，我们介绍了两种新的辛PDE集成方案。这些方案中的第一个可用于求解具有可通过特征方法求解的非线性零件的PDE。第二种方案可用于解决需要三向拆分的PDE。我们使用这些方案来研究第一部分中介绍的PDE的一维有界解。

著录项

  • 作者

    Carter, John David.;

  • 作者单位

    University of Colorado at Boulder.;

  • 授予单位 University of Colorado at Boulder.;
  • 学科 Mathematics.; Physics General.
  • 学位 Ph.D.
  • 年度 2001
  • 页码 147 p.
  • 总页数 147
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;物理学;
  • 关键词

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