Algebro Geometric Approach To Nonlinear Integrable Equations
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Author | : Eugene D. Belokolos |
Publisher | : |
Total Pages | : 360 |
Release | : 1994 |
Genre | : Differential equations |
ISBN | : |
A brief but self-contained exposition of the basics of Riemann surfaces and theta functions prepares the reader for the main subject of this text, namely, the application of these theories to solving nonlinear integrable equations for various physical systems. Physicists and engineers involved in studying solitons, phase transitions or dynamical (gyroscopic) systems and mathematicians with some background in algebraic geometry and abelian and automorphic functions, are the targeted audience. This book is suitable for use as a supplementary text to a course in mathematical physics.
Author | : Eugene D. Belokolos |
Publisher | : Springer |
Total Pages | : 337 |
Release | : 1994-06-28 |
Genre | : Science |
ISBN | : 9783540502654 |
A brief but self-contained exposition of the basics of Riemann surfaces and theta functions prepares the reader for the main subject of this text, namely the application of these theories to solving nonlinear integrable equations for various physical systems. Physicists and engineers involved in studying solitons, phase transitions, or dynamical (gyroscopic) systems, and mathematicians with some background in algebraic geometry and abelian and automorphic functions, are the targeted audience. This book is suitable for use as a supplementary text to a course in mathematical physics.
Author | : Fritz Gesztesy |
Publisher | : Cambridge University Press |
Total Pages | : 522 |
Release | : 2003-06-05 |
Genre | : Mathematics |
ISBN | : 9781139439411 |
The focus of this book is on algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions, also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses techniques from the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces). The presentation is rigorous, detailed, and self-contained, with ample background material provided in various appendices. Detailed notes for each chapter together with an exhaustive bibliography enhance the presentation offered in the main text.
Author | : Norbert Euler |
Publisher | : CRC Press |
Total Pages | : 367 |
Release | : 2021-09-07 |
Genre | : Mathematics |
ISBN | : 1000423301 |
The third volume in this sequence of books consists of a collection of contributions that aims to describe the recent progress in nonlinear differential equations and nonlinear dynamical systems (both continuous and discrete). Nonlinear Systems and Their Remarkable Mathematical Structures: Volume 3, Contributions from China just like the first two volumes, consists of contributions by world-leading experts in the subject of nonlinear systems, but in this instance only featuring contributions by leading Chinese scientists who also work in China (in some cases in collaboration with western scientists). Features Clearly illustrate the mathematical theories of nonlinear systems and its progress to both the non-expert and active researchers in this area Suitable for graduate students in Mathematics, Applied Mathematics and some of the Engineering sciences Written in a careful pedagogical manner by those experts who have been involved in the research themselves, and each contribution is reasonably self-contained
Author | : Gerald Teschl |
Publisher | : American Mathematical Soc. |
Total Pages | : 373 |
Release | : 2000 |
Genre | : Mathematics |
ISBN | : 0821819402 |
This volume serves as an introduction and reference source on spectral and inverse theory of Jacobi operators and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.
Author | : Ron Donagi |
Publisher | : Cambridge University Press |
Total Pages | : 421 |
Release | : 2020-04-02 |
Genre | : Mathematics |
ISBN | : 110880358X |
Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between algebraic geometry and integrable systems, differential equations, mathematical physics, and many other areas. The authors, many of whom have been at the forefront of research into these topics for the last decades, have all been influenced by Previato's research, as her collaborators, students, or colleagues. The diverse articles in the book demonstrate the wide scope of Previato's work and the inclusion of several survey and introductory articles makes the text accessible to graduate students and non-experts, as well as researchers. This first volume covers a wide range of areas related to integrable systems, often emphasizing the deep connections with algebraic geometry. Common themes include theta functions and Abelian varieties, Lax equations, integrable hierarchies, Hamiltonian flows and difference operators. These powerful tools are applied to spinning top, Hitchin, Painleve and many other notable special equations.
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.
Author | : Mark Pinsky |
Publisher | : Cambridge University Press |
Total Pages | : 405 |
Release | : 2008-03-17 |
Genre | : Mathematics |
ISBN | : 0521895278 |
Reflects the range of mathematical interests of Henry McKean, to whom it is dedicated.
Author | : Alexander I. Bobenko |
Publisher | : Springer Science & Business Media |
Total Pages | : 268 |
Release | : 2011-02-12 |
Genre | : Mathematics |
ISBN | : 3642174124 |
This volume offers a well-structured overview of existent computational approaches to Riemann surfaces and those currently in development. The authors of the contributions represent the groups providing publically available numerical codes in this field. Thus this volume illustrates which software tools are available and how they can be used in practice. In addition examples for solutions to partial differential equations and in surface theory are presented. The intended audience of this book is twofold. It can be used as a textbook for a graduate course in numerics of Riemann surfaces, in which case the standard undergraduate background, i.e., calculus and linear algebra, is required. In particular, no knowledge of the theory of Riemann surfaces is expected; the necessary background in this theory is contained in the Introduction chapter. At the same time, this book is also intended for specialists in geometry and mathematical physics applying the theory of Riemann surfaces in their research. It is the first book on numerics of Riemann surfaces that reflects the progress made in this field during the last decade, and it contains original results. There are a growing number of applications that involve the evaluation of concrete characteristics of models analytically described in terms of Riemann surfaces. Many problem settings and computations in this volume are motivated by such concrete applications in geometry and mathematical physics.
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.