Soliton Equations And Their Algebro Geometric Solutions Volume 1 1 1 Dimensional Continuous Models
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Soliton Equations And Their Algebro Geometric Solutions Volume 1 1 1 Dimensional Continuous Models
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Author : Fritz Gesztesy
language : en
Publisher: Cambridge University Press
Release Date : 2003-06-05
Soliton Equations And Their Algebro Geometric Solutions Volume 1 1 1 Dimensional Continuous Models written by Fritz Gesztesy and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-06-05 with Mathematics categories.
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.
Soliton Equations And Their Algebro Geometric Solutions Volume 1 1 1 Dimensional Continuous Models
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Author : Fritz Gesztesy
language : en
Publisher: Cambridge University Press
Release Date : 2003-06-05
Soliton Equations And Their Algebro Geometric Solutions Volume 1 1 1 Dimensional Continuous Models written by Fritz Gesztesy and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-06-05 with Mathematics categories.
This book is about 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 and 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).
Soliton Equations And Their Algebro Geometric Solutions Volume 2 1 1 Dimensional Discrete Models
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Author : Fritz Gesztesy
language : en
Publisher: Cambridge University Press
Release Date : 2008-09-04
Soliton Equations And Their Algebro Geometric Solutions Volume 2 1 1 Dimensional Discrete Models written by Fritz Gesztesy and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-09-04 with Mathematics categories.
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.
Soliton Equations And Their Algebro Geometric Solutions Volume 2 1 1 Dimensional Discrete Models
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Author : Fritz Gesztesy
language : en
Publisher: Cambridge University Press
Release Date : 2003
Soliton Equations And Their Algebro Geometric Solutions Volume 2 1 1 Dimensional Discrete Models written by Fritz Gesztesy and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Mathematics categories.
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.
The Weierstrass Sigma Function In Higher Genus And Applications To Integrable Equations
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Author : Shigeki Matsutani
language : en
Publisher: Springer Nature
Release Date : 2025-03-25
The Weierstrass Sigma Function In Higher Genus And Applications To Integrable Equations written by Shigeki Matsutani and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-03-25 with Mathematics categories.
This book’s area is special functions of one or several complex variables. Special functions have been applied to dynamics and physics. Special functions such as elliptic or automorphic functions have an algebro-geometric nature. These attributes permeate the book. The “Kleinian sigma function”, or “higher-genus Weierstrass sigma function” generalizes the elliptic sigma function. It appears for the first time in the work of Weierstrass. Klein gave an explicit definition for hyperelliptic or genus-three curves, as a modular invariant analogue of the Riemann theta function on the Jacobian (the two functions are equivalent). H.F. Baker later used generalized Legendre relations for meromorphic differentials, and brought out the two principles of the theory: on the one hand, sigma uniformizes the Jacobian so that its (logarithmic) derivatives in one direction generate the field of meromorphic functions on the Jacobian, therefore algebraic relations among them generate the ideal of the Jacobian as a projective variety; on the other hand, a set of nonlinear PDEs (which turns out to include the “integrable hierarchies” of KdV type), characterize sigma. We follow Baker’s approach. There is no book where the theory of the sigma function is taken from its origins up to the latest most general results achieved, which cover large classes of curves. The authors propose to produce such a book, and cover applications to integrable PDEs, and the inclusion of related al functions, which have not yet received comparable attention but have applications to defining specific subvarieties of the degenerating family of curves. One reason for the attention given to sigma is its relationship to Sato's tau function and the heat equations for deformation from monomial curves. The book is based on classical literature and contemporary research, in particular our contribution which covers a class of curves whose sigma had not been found explicitly before.
Topics In Differential Geometry
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Author : Peter W. Michor
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
Topics In Differential Geometry written by Peter W. Michor and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
"This book treats the fundamentals of differential geometry: manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections, Riemann manifolds, isometric actions, and symplectic and Poisson geometry. It gives the careful reader working knowledge in a wide range of topics of modern coordinate-free differential geometry in not too many pages. A prerequisite for using this book is a good knowledge of undergraduate analysis and linear algebra."--BOOK JACKET.
Methods Of Spectral Analysis In Mathematical Physics
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Author : Jan Janas
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-12-16
Methods Of Spectral Analysis In Mathematical Physics written by Jan Janas and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-12-16 with Science categories.
The volume contains the proceedings of the OTAMP 2006 (Operator Theory, Analysis and Mathematical Physics) conference held at Lund University in June 2006. The conference was devoted to the methods of analysis and operator theory in modern mathematical physics. The following special sessions were organized: Spectral analysis of Schrödinger operators; Jacobi and CMV matrices and orthogonal polynomials; Quasi-periodic and random Schrödinger operators; Quantum graphs.
Rogue Waves
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Author : Boling Guo
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2017-06-26
Rogue Waves written by Boling Guo and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-06-26 with Mathematics categories.
This book gives an overview of the theoretical research on rogue waves and discusses solutions to rogue wave formation via the Darboux and bilinear transformations, algebro-geometric reduction, and inverse scattering and similarity transformations. Studies on nonlinear optics are included, making the book a comprehensive reference for researchers in applied mathematics, optical physics, geophysics, and ocean engineering. Contents The Research Process for Rogue Waves Construction of Rogue Wave Solution by the Generalized Darboux Transformation Construction of Rogue Wave Solution by Hirota Bilinear Method, Algebro-geometric Approach and Inverse Scattering Method The Rogue Wave Solution and Parameters Managing in Nonautonomous Physical Model
Analytical Properties Of Nonlinear Partial Differential Equations
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Author : Alexei Cheviakov
language : en
Publisher: Springer Nature
Release Date : 2024-03-22
Analytical Properties Of Nonlinear Partial Differential Equations written by Alexei Cheviakov and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-03-22 with Mathematics categories.
Nonlinear partial differential equations (PDE) are at the core of mathematical modeling. In the past decades and recent years, multiple analytical methods to study various aspects of the mathematical structure of nonlinear PDEs have been developed. Those aspects include C- and S-integrability, Lagrangian and Hamiltonian formulations, equivalence transformations, local and nonlocal symmetries, conservation laws, and more. Modern computational approaches and symbolic software can be employed to systematically derive and use such properties, and where possible, construct exact and approximate solutions of nonlinear equations. This book contains a consistent overview of multiple properties of nonlinear PDEs, their relations, computation algorithms, and a uniformly presented set of examples of application of these methods to specific PDEs. Examples include both well known nonlinear PDEs and less famous systems that arise in the context of shallow water waves and far beyond. The book will beof interest to researchers and graduate students in applied mathematics, physics, and engineering, and can be used as a basis for research, study, reference, and applications.
Nonlinear Partial Differential Equations And Hyperbolic Wave Phenomena
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Author : Norske videnskaps-akademi. Research Program on Nonlinear Partial Differential Equations
language : en
Publisher: American Mathematical Soc.
Release Date : 2010-10-01
Nonlinear Partial Differential Equations And Hyperbolic Wave Phenomena written by Norske videnskaps-akademi. Research Program on Nonlinear Partial Differential Equations and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-10-01 with Mathematics categories.
This volume presents the state of the art in several directions of research conducted by renowned mathematicians who participated in the research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway, during the academic year 2008-09. The main theme of the volume is nonlinear partial differential equations that model a wide variety of wave phenomena. Topics discussed include systems of conservation laws, compressible Navier-Stokes equations, Navier-Stokes-Korteweg type systems in models for phase transitions, nonlinear evolution equations, degenerate/mixed type equations in fluid mechanics and differential geometry, nonlinear dispersive wave equations (Korteweg-de Vries, Camassa-Holm type, etc.), and Poisson interface problems and level set formulations.