Solitons, Nonlinear Evolution Equations and Inverse Scattering

Author: Mark J. Ablowitz,M. A. Ablowitz,P. A. Clarkson,Peter A.. Clarkson

Publisher: Cambridge University Press

ISBN: 9780521387309

Category: Mathematics

Page: 516

View: 7521

This book brings together several aspects of soliton theory currently available only in research papers. Emphasis is given to the multi-dimensional problems which arise and includes inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multi-dimensions and the dbar method.

Direct and Inverse Methods in Nonlinear Evolution Equations

Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5–12, 1999

Author: Robert M. Conte,Franco Magri,Micheline Musette,Junkichi Satsuma,Pavel Winternitz

Publisher: Springer Science & Business Media

ISBN: 9783540200871

Category: Science

Page: 279

View: 5307

Many physical phenomena are described by nonlinear evolution equation. Those that are integrable provide various mathematical methods, presented by experts in this tutorial book, to find special analytic solutions to both integrable and partially integrable equations. The direct method to build solutions includes the analysis of singularities à la Painlevé, Lie symmetries leaving the equation invariant, extension of the Hirota method, construction of the nonlinear superposition formula. The main inverse method described here relies on the bi-hamiltonian structure of integrable equations. The book also presents some extension to equations with discrete independent and dependent variables. The different chapters face from different points of view the theory of exact solutions and of the complete integrability of nonlinear evolution equations. Several examples and applications to concrete problems allow the reader to experience directly the power of the different machineries involved.

Solitons and the Inverse Scattering Transform

Author: Mark J. Ablowitz,Harvey Segur

Publisher: SIAM

ISBN: 089871477X

Category: Mathematics

Page: 425

View: 1275

A study, by two of the major contributors to the theory, of the inverse scattering transform and its application to problems of nonlinear dispersive waves that arise in fluid dynamics, plasma physics, nonlinear optics, particle physics, crystal lattice theory, nonlinear circuit theory and other areas. A soliton is a localised pulse-like nonlinear wave that possesses remarkable stability properties. Typically, problems that admit soliton solutions are in the form of evolution equations that describe how some variable or set of variables evolve in time from a given state. The equations may take a variety of forms, for example, PDEs, differential difference equations, partial difference equations, and integrodifferential equations, as well as coupled ODEs of finite order. What is surprising is that, although these problems are nonlinear, the general solution that evolves from almost arbitrary initial data may be obtained without approximation.

Nonlinear Dynamics

Integrability, Chaos and Patterns

Author: Muthusamy Lakshmanan,Shanmuganathan Rajaseekar

Publisher: Springer Science & Business Media

ISBN: 3642556884

Category: Mathematics

Page: 620

View: 8659

This self-contained treatment covers all aspects of nonlinear dynamics, from fundamentals to recent developments, in a unified and comprehensive way. Numerous examples and exercises will help the student to assimilate and apply the techniques presented.

Integrable Hamiltonian Hierarchies

Spectral and Geometric Methods

Author: Vladimir Gerdjikov,Gaetano Vilasi,Alexandar Borisov Yanovski

Publisher: Springer Science & Business Media

ISBN: 3540770534

Category: Science

Page: 643

View: 3160

This book presents a detailed derivation of the spectral properties of the Recursion Operators allowing one to derive all the fundamental properties of the soliton equations and to study their hierarchies.

Introduction to Nonlinear Dispersive Equations

Author: Felipe Linares,Gustavo Ponce

Publisher: Springer

ISBN: 1493921819

Category: Mathematics

Page: 301

View: 5912

This textbook introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, with special focus on two key models, the Korteweg–de Vries equation and the nonlinear Schrödinger equation. A concise and self-contained treatment of background material (the Fourier transform, interpolation theory, Sobolev spaces, and the linear Schrödinger equation) prepares the reader to understand the main topics covered: the initial-value problem for the nonlinear Schrödinger equation and the generalized Korteweg–de Vries equation, properties of their solutions, and a survey of general classes of nonlinear dispersive equations of physical and mathematical significance. Each chapter ends with an expert account of recent developments and open problems, as well as exercises. The final chapter gives a detailed exposition of local well-posedness for the nonlinear Schrödinger equation, taking the reader to the forefront of recent research. The second edition of Introduction to Nonlinear Dispersive Equations builds upon the success of the first edition by the addition of updated material on the main topics, an expanded bibliography, and new exercises. Assuming only basic knowledge of complex analysis and integration theory, this book will enable graduate students and researchers to enter this actively developing field.

Nonlinear Wave Equations

Author: Satyanad Kichenassamy

Publisher: CRC Press

ISBN: 9780824793289

Category: Science

Page: 296

View: 7578

This work examines the mathematical aspects of nonlinear wave propagation, emphasizing nonlinear hyperbolic problems. It introduces the tools that are most effective for exploring the problems of local and global existence, singularity formation, and large-time behaviour of solutions, and for the study of perturbation methods.

Nonlinear Waves in Integrable and Nonintegrable Systems

Author: Jianke Yang

Publisher: SIAM

ISBN: 0898719682

Category: Nonlinear waves

Page: 430

View: 9153

Presents cutting-edge developments in the theory and experiments of nonlinear waves. Its comprehensive coverage of analytical and numerical methods for nonintegrable systems is the first of its kind.

Algebraic Approaches to Partial Differential Equations

Author: Xiaoping Xu

Publisher: Springer Science & Business Media

ISBN: 3642368743

Category: Mathematics

Page: 394

View: 2147

This book presents the various algebraic techniques for solving partial differential equations to yield exact solutions, techniques developed by the author in recent years and with emphasis on physical equations such as: the Maxwell equations, the Dirac equations, the KdV equation, the KP equation, the nonlinear Schrodinger equation, the Davey and Stewartson equations, the Boussinesq equations in geophysics, the Navier-Stokes equations and the boundary layer problems. In order to solve them, I have employed the grading technique, matrix differential operators, stable-range of nonlinear terms, moving frames, asymmetric assumptions, symmetry transformations, linearization techniques and special functions. The book is self-contained and requires only a minimal understanding of calculus and linear algebra, making it accessible to a broad audience in the fields of mathematics, the sciences and engineering. Readers may find the exact solutions and mathematical skills needed in their own research.

Introduction to non-Kerr Law Optical Solitons

Author: Anjan Biswas,Swapan Konar

Publisher: CRC Press

ISBN: 1420011405

Category: Science

Page: 216

View: 339

Despite remarkable developments in the field, a detailed treatment of non-Kerr law media has not been published. Introduction to non-Kerr Law Optical Solitons is the first book devoted exclusively to optical soliton propagation in media that possesses non-Kerr law nonlinearities. After an introduction to the basic features of fiber-optic communications, the book outlines the nonlinear Schrödinger equation (NLSE), conserved quantities, and adiabatic dynamics of soliton parameters. It then derives the NLSE for Kerr law nonlinearity from basic principles, the inverse scattering transform, and the 1-soliton solution. The book also explains the variational principle and Lie transform. In each case of non-Kerr law solitons, the authors develop soliton dynamics, evaluated integrals of motion, and adiabatic dynamics of soliton parameters based on multiple-scale perturbation theory. The book explores intra-channel collision of optical solitons in both Hamiltonian and non-Hamiltonian type perturbations. In addition, it examines the stochastic perturbation of optical solitons, the corresponding Langevin equations, and optical couplers, followed by an introduction to optical bullets. Establishing a basis in an important yet insufficiently documented subject, Introduction to non-Kerr Law Optical Solitons will help fuel advances in optical communication systems.


Author: Boling Guo,Xiao-Feng Pang,Yu-Feng Wang,Nan Liu

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 3110549638

Category: Mathematics

Page: 376

View: 1000

This book provides an up-to-date overview of mathematical theories and research results on solitons, presenting related mathematical methods and applications as well as numerical experiments. Different types of soliton equations are covered along with their dynamical behaviors and applications from physics, making the book an essential reference for researchers and graduate students in applied mathematics and physics. Contents Introduction Inverse scattering transform Asymptotic behavior to initial value problems for some integrable evolution nonlinear equations Interaction of solitons and its asymptotic properties Hirota method Bäcklund transformations and the infinitely many conservation laws Multi-dimensional solitons and their stability Numerical computation methods for some nonlinear evolution equations The geometric theory of solitons Global existence and blow up for the nonlinear evolution equations The soliton movements of elementary particles in nonlinear quantum field The theory of soliton movement of superconductive features The soliton movements in condensed state systemsontents

Nonlinear Evolution Equations and Dynamical Systems

Author: Sandra Carillo,Orlando Ragnisco

Publisher: Springer Science & Business Media

ISBN: 3642840396

Category: Science

Page: 233

View: 1899

Nonlinear Evolution Equations and Dynamical Systems (NEEDS) provides a presentation of the state of the art. Except for a few review papers, the 40 contributions are intentially brief to give only the gist of the methods, proofs, etc. including references to the relevant litera- ture. This gives a handy overview of current research activities. Hence, the book should be equally useful to the senior resercher as well as the colleague just entering the field. Keypoints treated are: i) integrable systems in multidimensions and associated phenomenology ("dromions"); ii) criteria and tests of integrability (e.g., Painlev test); iii) new developments related to the scattering transform; iv) algebraic approaches to integrable systems and Hamiltonian theory (e.g., connections with Young-Baxter equations and Kac-Moody algebras); v) new developments in mappings and cellular automata, vi) applications to general relativity, condensed matter physics, and oceanography.

Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrodinger Equations

Author: Victor A. Galaktionov,Enzo L. Mitidieri,Stanislav I. Pohozaev

Publisher: CRC Press

ISBN: 1482251736

Category: Mathematics

Page: 569

View: 1687

Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations shows how four types of higher-order nonlinear evolution partial differential equations (PDEs) have many commonalities through their special quasilinear degenerate representations. The authors present a unified approach to deal with these quasilinear PDEs. The book first studies the particular self-similar singularity solutions (patterns) of the equations. This approach allows four different classes of nonlinear PDEs to be treated simultaneously to establish their striking common features. The book describes many properties of the equations and examines traditional questions of existence/nonexistence, uniqueness/nonuniqueness, global asymptotics, regularizations, shock-wave theory, and various blow-up singularities. Preparing readers for more advanced mathematical PDE analysis, the book demonstrates that quasilinear degenerate higher-order PDEs, even exotic and awkward ones, are not as daunting as they first appear. It also illustrates the deep features shared by several types of nonlinear PDEs and encourages readers to develop further this unifying PDE approach from other viewpoints.

Nonlinear Waves and Solitons on Contours and Closed Surfaces

Author: Andrei Ludu

Publisher: Springer Science & Business Media

ISBN: 3642228941

Category: Mathematics

Page: 489

View: 7875

This volume is an introduction to nonlinear waves and soliton theory in the special environment of compact spaces such a closed curves and surfaces and other domain contours. It assumes familiarity with basic soliton theory and nonlinear dynamical systems. The first part of the book introduces the mathematical concept required for treating the manifolds considered, providing relevant notions from topology and differential geometry. An introduction to the theory of motion of curves and surfaces - as part of the emerging field of contour dynamics - is given. The second and third parts discuss the modeling of various physical solitons on compact systems, such as filaments, loops and drops made of almost incompressible materials thereby intersecting with a large number of physical disciplines from hydrodynamics to compact object astrophysics. This book is intended for graduate students and researchers in mathematics, physics and engineering. This new edition has been thoroughly revised, expanded and updated.

Nonlinear Evolution Equations and Soliton Solutions

Author: Yucui Guo

Publisher: N.A

ISBN: 9781634827690


Page: 530

View: 7326

This book studies the methods for solving non-linear, partial differential equations that have physical meaning, and soliton theory with applications. Specific descriptions on the formation mechanism of soliton solutions of non-linear, partial differential equations are given, and some methods for solving this kind of solution such as the Inverse Scattering Transform method, Backlund Transformation method, Similarity Reduction method and several kinds of function transformation methods are introduced. Integrability of non-linear, partial differential equations is also discussed. This book is suitable for graduate students whose research fields are in applied mathematics, applied physics and non-linear science-related directions as a textbook or a research reference book. This book is also useful for non-linear science researchers and teachers as a reference book. The characteristics of this book are: 1. The author provides clear concepts, rigorous derivation, thorough reasoning, and rigorous logic in the book. Since the research boom of non-linear, partial differential equations was rising in the 1960s, the research on non-linear, partial differential equations and soliton theory has only been several decades, which can be described as a very young discipline compared to the other branches in mathematics. Although there are a few related books, they are mostly in highly specialised interdisciplinary areas. There is no book which is suitable for cross-disciplines and for people with college level mathematics and college physics background. This book fills that gap; 2. The book is easy to be understood by readers since it provides step-by-step approaches. All results in the book have been deduced and collated by the author to make sure that they are correct and perfect; 3. The derivation from the physical models to mathematical models is emphasised in the book. In mathematical physics, we cannot just simply consider the mathematical problems without a physical image, which often plays the key role for understanding the mathematical problems; 4. Mathematical transformation methods are provided. The basic idea of various methods for solving non-linear, partial differential equations is to simplify the complex equations into simple ones through some transformations or decompositions. However, we cannot find any patterns for using such transformations or decompositions, and certain conjectures and assumptions have to be used. However, the skill and the logic of using the transformations and decompositions are very important to researchers in this field.

Painleve Equations in the Differential Geometry of Surfaces

Author: Alexander I. Bobenko,Alexander I. Bobenko TU Berlin,Ulrich Eitner

Publisher: Springer Science & Business Media

ISBN: 3540414142

Category: Mathematics

Page: 120

View: 8350

This book brings together two different branches of mathematics: the theory of Painlevé and the theory of surfaces. Self-contained introductions to both these fields are presented. It is shown how some classical problems in surface theory can be solved using the modern theory of Painlevé equations. In particular, an essential part of the book is devoted to Bonnet surfaces, i.e. to surfaces possessing families of isometries preserving the mean curvature function. A global classification of Bonnet surfaces is given using both ingredients of the theory of Painlevé equations: the theory of isomonodromic deformation and the Painlevé property. The book is illustrated by plots of surfaces. It is intended to be used by mathematicians and graduate students interested in differential geometry and Painlevé equations. Researchers working in one of these areas can become familiar with another relevant branch of mathematics.

Complex Hamiltonian Dynamics

Author: Tassos Bountis,Haris Skokos

Publisher: Springer Science & Business Media

ISBN: 3642273041

Category: Language Arts & Disciplines

Page: 255

View: 7611

This book introduces and explores modern developments in the well established field of Hamiltonian dynamical systems. It focuses on high degree-of-freedom systems and the transitional regimes between regular and chaotic motion. The role of nonlinear normal modes is highlighted and the importance of low-dimensional tori in the resolution of the famous FPU paradox is emphasized. Novel powerful numerical methods are used to study localization phenomena and distinguish order from strongly and weakly chaotic regimes. The emerging hierarchy of complex structures in such regimes gives rise to particularly long-lived patterns and phenomena called quasi-stationary states, which are explored in particular in the concrete setting of one-dimensional Hamiltonian lattices and physical applications in condensed matter systems. The self-contained and pedagogical approach is blended with a unique balance between mathematical rigor, physics insights and concrete applications. End of chapter exercises and (more demanding) research oriented problems provide many opportunities to deepen the reader’s insights into specific aspects of the subject matter. Addressing a broad audience of graduate students, theoretical physicists and applied mathematicians, this text combines the benefits of a reference work with those of a self-study guide for newcomers to the field.