Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies

Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies
Author: Wolfgang Bulla
Publisher: American Mathematical Soc.
Total Pages: 97
Release: 1998
Genre: Mathematics
ISBN: 0821808087

In this work, the authors provide a self-contained discussion of all real-valued quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies of completely integrable evolution equations. The approach utilizes algebro-geometric methods, factorization techniques for finite difference expressions, as well as Miura-type transformations. Detailed spectral theoretic properties of Lax pairs and theta function representations of the solutions are derived. Features: Simple and unified treatment of the topic. Self-contained development. Novel results for the Kac-van Moerbeke hierarchy and its algebro-geometric solutions.

Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-Van Moerbeke Hierarchies

Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-Van Moerbeke Hierarchies
Author: Wolfgang Bulla
Publisher: American Mathematical Society(RI)
Total Pages: 97
Release: 2014-09-11
Genre: MATHEMATICS
ISBN: 9781470402303

In this work, the authors provide a self-contained discussion of all real-valued quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies of completely integrable evolution equations. The approach utilizes algebro-geometric methods, factorization techniques for finite difference expressions, as well as Miura-type transformations. Detailed spectral theoretic properties of Lax pairs and theta function representations of the solutions are derived.

(m)KdV Solitons on the Background of Quasi-Periodic Finite-Gap Solutions

(m)KdV Solitons on the Background of Quasi-Periodic Finite-Gap Solutions
Author: Fritz Gesztesy
Publisher: American Mathematical Soc.
Total Pages: 102
Release: 1995
Genre: Mathematics
ISBN: 0821804065

In the introductory section, we review the formulation of the Korteweg-de Vries (KdV) equation and of the modified KdV (mKdV) equation as a compatibility condition for a Lax pair of linear operators. We then illustrate Miura's transformation, which maps solutions of the mKdV into solutions of the KdV. We then give a general overview of the concept of soliton solutions relative to general backgrounds, and of the single and double commutation methods. Finally, we present the main results of the article. To avoid the clutter of too many technical details, the paper is organized in four sections and five appendices.

Soliton Equations and Their Algebro-Geometric Solutions: Volume 2, (1+1)-Dimensional Discrete Models

Soliton Equations and Their Algebro-Geometric Solutions: Volume 2, (1+1)-Dimensional Discrete Models
Author: Fritz Gesztesy
Publisher: Cambridge University Press
Total Pages: 438
Release: 2008-09-04
Genre: Mathematics
ISBN: 1139473778

As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices. Detailed notes for each chapter, together with an exhaustive bibliography, enhance understanding of the main results.

Modern Analysis and Applications

Modern Analysis and Applications
Author: Vadim Adamyan
Publisher: Springer Science & Business Media
Total Pages: 518
Release: 2009-08-29
Genre: Mathematics
ISBN: 376439921X

This is the second of two volumes containing peer-reviewed research and survey papers based on talks at the International Conference on Modern Analysis and Applications. The papers describe the contemporary development of subjects influenced by Mark Krein.

Differential Equations and Mathematical Physics

Differential Equations and Mathematical Physics
Author: Rudi Weikard
Publisher: American Mathematical Soc.
Total Pages: 491
Release: 2000
Genre: Mathematics
ISBN: 0821821571

This volume contains the proceedings of the 1999 International Conference on Differential Equations and Mathematical Physics. The contributions selected for this volume represent some of the most important presentations by scholars from around the world on developments in this area of research. The papers cover topics in the general area of linear and nonlinear differential equations and their relation to mathematical physics, such as multiparticle Schrödinger operators, stability of matter, relativity theory, fluid dynamics, spectral and scattering theory including inverse problems. Titles in this series are co-published with International Press, Cambridge, MA.

Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy's 60th Birthday

Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy's 60th Birthday
Author: Helge Holden
Publisher: American Mathematical Soc.
Total Pages: 409
Release: 2013-07-08
Genre: Mathematics
ISBN: 0821875744

This volume contains twenty contributions in the area of mathematical physics where Fritz Gesztesy made profound contributions. There are three survey papers in spectral theory, differential equations, and mathematical physics, which highlight, in particu

Operator Theory, Analysis and Mathematical Physics

Operator Theory, Analysis and Mathematical Physics
Author: Jan Janas
Publisher: Springer Science & Business Media
Total Pages: 261
Release: 2007-04-29
Genre: Mathematics
ISBN: 3764381353

This volume contains lectures delivered at the International Conference Operator Theory and its Applications in Mathematical Physics (OTAMP 2004), held at the Mathematical Research and Conference Center in Bedlewo near Poznan, Poland. The idea behind these lectures was to present interesting ramifications of operator methods in current research of mathematical physics.

Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows

Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows
Author: Wenxian Shen
Publisher: American Mathematical Soc.
Total Pages: 111
Release: 1998
Genre: Mathematics
ISBN: 0821808672

This volume is devoted to the study of almost automorphic dynamics in differential equations. By making use of techniques from abstract topological dynamics, it is shown that almost automorphy, a notion which was introduced by S. Bochner in 1955, is essential and fundamental in the qualitative study of almost periodic differential equations.

Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations

Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations
Author: Donald J. Estep
Publisher: American Mathematical Soc.
Total Pages: 125
Release: 2000
Genre: Mathematics
ISBN: 0821820729

This paper is concerned with the computational estimation of the error of numerical solutions of potentially degenerate reaction-diffusion equations. The underlying motivation is a desire to compute accurate estimates as opposed to deriving inaccurate analytic upper bounds. In this paper, we outline, analyze, and test an approach to obtain computational error estimates based on the introduction of the residual error of the numerical solution and in which the effects of the accumulation of errors are estimated computationally. We begin by deriving an a posteriori relationship between the error of a numerical solution and its residual error using a variational argument. This leads to the introduction of stability factors, which measure the sensitivity of solutions to various kinds of perturbations. Next, we perform some general analysis on the residual errors and stability factors to determine when they are defined and to bound their size. Then we describe the practical use of the theory to estimate the errors of numerical solutions computationally. Several key issues arise in the implementation that remain unresolved and we present partial results and numerical experiments about these points. We use this approach to estimate the error of numerical solutions of nine standard reaction-diffusion models and make a systematic comparison of the time scale over which accurate numerical solutions can be computed for these problems. We also perform a numerical test of the accuracy and reliability of the computational error estimate using the bistable equation. Finally, we apply the general theory to the class of problems that admit invariant regions for the solutions, which includes seven of the main examples. Under this additional stability assumption, we obtain a convergence result in the form of an upper bound on the error from the a posteriori error estimate. We conclude by discussing the preservation of invariant regions under discretization.