Nonlinear Dispersive Equations

Author: Terence Tao
Editor: American Mathematical Soc.
ISBN: 0821841432
File Size: 16,32 MB
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"Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.".

Nonlinear Dispersive Equations

Author: Jaime Angulo Pava
Editor: American Mathematical Soc.
ISBN: 0821848976
File Size: 80,70 MB
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This book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied includes Korteweg-de Vries, Benjamin-Ono, and Schrodinger equations. Many special Jacobian elliptic functions play a role in these examples. The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena.

Introduction To Nonlinear Dispersive Equations

Author: Felipe Linares
Editor: Springer
ISBN: 1493921819
File Size: 77,71 MB
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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.

Mathematical Aspects Of Nonlinear Dispersive Equations Am 163

Author: Jean Bourgain
Editor: Princeton University Press
ISBN: 1400827795
File Size: 53,10 MB
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This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions. The technical papers concentrate on a specific problem and the related analysis and are addressed to active researchers. The book deals with many topics that have been the focus of intensive research and, in several cases, significant progress in recent years, including hyperbolic conservation laws, Schrödinger operators, nonlinear Schrödinger and wave equations, and the Euler and Navier-Stokes equations.

Nonlinear Dispersive Equations

Author: Terence Tao
Editor: American Mathematical Soc.
ISBN: 0821841432
File Size: 35,39 MB
Format: PDF, Docs
Read: 4737
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"Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.".

Dispersive Equations And Nonlinear Waves

Author: Herbert Koch
Editor: Springer
ISBN: 3034807368
File Size: 41,57 MB
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The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation. The energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part. Using this concrete example, it walks the reader through the induction on energy technique, which has become the essential methodology for tackling large data critical problems. This includes refined/inverse Strichartz estimates, the existence and almost periodicity of minimal blow up solutions, and the development of long-time Strichartz inequalities. The third part describes wave and Schrödinger maps. Starting by building heuristics about multilinear estimates, it provides a detailed outline of this very active area of geometric/dispersive PDE. It focuses on concepts and ideas and should provide graduate students with a stepping stone to this exciting direction of research.​

Nonlinear Dispersive Waves And Fluids

Author: Avy Soffer
Editor: American Mathematical Soc.
ISBN: 1470441098
File Size: 25,44 MB
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This volume contains the proceedings of the AMS Special Session on Spectral Calculus and Quasilinear Partial Differential Equations and the AMS Special Session on PDE Analysis on Fluid Flows, which were held in January 2017 in Atlanta, Georgia. These two sessions shared the underlying theme of the analysis aspect of evolutionary PDEs and mathematical physics. The articles address the latest trends and perspectives in the area of nonlinear dispersive equations and fluid flows. The topics mainly focus on using state-of-the-art methods and techniques to investigate problems of depth and richness arising in quantum mechanics, general relativity, and fluid dynamics.

Nonlinear Dispersive Equations Describing Boson Stars

Author: Enno Lenzmann
Editor:
ISBN:
File Size: 12,65 MB
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Nonlinear Dispersive Equations

Author: Jaime Angulo Pava
Editor: American Mathematical Soc.
ISBN: 0821848976
File Size: 13,34 MB
Format: PDF, ePub, Docs
Read: 7852
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This book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied includes Korteweg-de Vries, Benjamin-Ono, and Schrodinger equations. Many special Jacobian elliptic functions play a role in these examples. The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena.

Mathematical Aspects Of Nonlinear Dispersive Equations

Author: Jean Bourgain
Editor:
ISBN: 9780691128603
File Size: 21,47 MB
Format: PDF, Kindle
Read: 6327
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This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions. The technical papers concentrate on a specific problem and the related analysis and are addressed to active researchers. The book deals with many topics that have been the focus of intensive research and, in several cases, significant progress in recent years, including hyperbolic conservation laws, Schrödinger operators, nonlinear Schrödinger and wave equations, and the Euler and Navier-Stokes equations.

Some Results For Nonlinear Dispersive Wave Equations With Applications To Control

Author: Bingyu Zhang
Editor:
ISBN:
File Size: 23,84 MB
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On Long Time Behavior Of Solutions To Nonlinear Dispersive Equations

Author: Stefano Francesco Burzio
Editor:
ISBN:
File Size: 51,10 MB
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Mots-clés de l'auteur: critical wave equation ; blowup ; Yang-Mills equation ; nonlinear waves ; null structures ; space-time compactification ; Penrose transform.

Dispersive Partial Differential Equations

Author: M. Burak Erdoğan
Editor: Cambridge University Press
ISBN: 1107149045
File Size: 41,21 MB
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The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, engineering, and the physical sciences. Both classical and modern methods used in the field are described in detail, concentrating on the model cases that simplify the presentation without compromising the deep technical aspects of the theory, thus allowing students to learn the material in a short period of time. This book is appropriate both for self-study by students with a background in analysis, and for teaching a semester-long introductory graduate course in nonlinear dispersive PDEs. Copious exercises are included, and applications of the theory are also presented to connect dispersive PDEs with the more general areas of dynamical systems and mathematical physics.

Harmonic Analysis Method For Nonlinear Evolution Equations I

Author: Baoxiang Wang
Editor: World Scientific
ISBN: 9814360740
File Size: 30,41 MB
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This monograph provides a comprehensive overview on a class of nonlinear evolution equations, such as nonlinear SchrAdinger equations, nonlinear KleinOCoGordon equations, KdV equations as well as NavierOCoStokes equations and Boltzmann equations. The global wellposedness to the Cauchy problem for those equations is systematically studied by using the harmonic analysis methods. This book is self-contained and may also be used as an advanced textbook by graduate students in analysis and PDE subjects and even ambitious undergraduate students.

Global Solutions Of Nonlinear Schrodinger Equations

Author: Jean Bourgain
Editor: American Mathematical Soc.
ISBN: 0821819194
File Size: 51,95 MB
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The Colloquium series offers classics in the mathematical literature, and can be favorably compared to books now being published in the AMS Chelsea series. This book has a good affiliation (Princeton University) and is covering an important topic in Partial Differential Equations, which is a very hot area in mathematics at present. Good price and colorful hardcover.

Deriving The New Traveling Wave Solutions For The Nonlinear Dispersive Equation Kdv Zk Equation And Complex Coupled Kdv System Using Extended Simplest Equation Method

Author:
Editor:
ISBN:
File Size: 30,88 MB
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Abstract: In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov–Kuznetsov (KdV-ZK) equation and complex coupled KdV system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations (NLEEs) in mathematical physics.

Rogue And Shock Waves In Nonlinear Dispersive Media

Author: Miguel Onorato
Editor: Springer
ISBN: 331939214X
File Size: 16,98 MB
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This self-contained set of lectures addresses a gap in the literature by providing a systematic link between the theoretical foundations of the subject matter and cutting-edge applications in both geophysical fluid dynamics and nonlinear optics. Rogue and shock waves are phenomena that may occur in the propagation of waves in any nonlinear dispersive medium. Accordingly, they have been observed in disparate settings – as ocean waves, in nonlinear optics, in Bose-Einstein condensates, and in plasmas. Rogue and dispersive shock waves are both characterized by the development of extremes: for the former, the wave amplitude becomes unusually large, while for the latter, gradients reach extreme values. Both aspects strongly influence the statistical properties of the wave propagation and are thus considered together here in terms of their underlying theoretical treatment. This book offers a self-contained graduate-level text intended as both an introduction and reference guide for a new generation of scientists working on rogue and shock wave phenomena across a broad range of fields in applied physics and geophysics.

Nonlinear Dispersive Wave Systems

Author: Debnath Lokenath
Editor: World Scientific
ISBN: 9814554960
File Size: 59,66 MB
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The book includes all the subject matter covered in a typical undergraduate course in engineering thermodynamics. It includes a series of worked examples in each chapter, carefully chosen to expose students to diverse applications of engineering thermodynamics. Each worked example is designed to be representative of a class of physical problems. At the end of each chapter, there are an additional 10 to 15 problems for which numerical answers are provided.

Lectures On The Energy Critical Nonlinear Wave Equation

Author: Carlos E. Kenig
Editor: American Mathematical Soc.
ISBN: 1470420147
File Size: 32,97 MB
Format: PDF
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This monograph deals with recent advances in the study of the long-time asymptotics of large solutions to critical nonlinear dispersive equations. The first part of the monograph describes, in the context of the energy critical wave equation, the "concentration-compactness/rigidity theorem method" introduced by C. Kenig and F. Merle. This approach has become the canonical method for the study of the "global regularity and well-posedness" conjecture (defocusing case) and the "ground-state" conjecture (focusing case) in critical dispersive problems. The second part of the monograph describes the "channel of energy" method, introduced by T. Duyckaerts, C. Kenig, and F. Merle, to study soliton resolution for nonlinear wave equations. This culminates in a presentation of the proof of the soliton resolution conjecture, for the three-dimensional radial focusing energy critical wave equation. It is the intent that the results described in this book will be a model for what to strive for in the study of other nonlinear dispersive equations. A co-publication of the AMS and CBMS.

Nonlinear Dispersive Waves

Author: Mark J. Ablowitz
Editor: Cambridge University Press
ISBN: 1139503480
File Size: 73,24 MB
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The field of nonlinear dispersive waves has developed enormously since the work of Stokes, Boussinesq and Korteweg–de Vries (KdV) in the nineteenth century. In the 1960s, researchers developed effective asymptotic methods for deriving nonlinear wave equations, such as the KdV equation, governing a broad class of physical phenomena that admit special solutions including those commonly known as solitons. This book describes the underlying approximation techniques and methods for finding solutions to these and other equations. The concepts and methods covered include wave dispersion, asymptotic analysis, perturbation theory, the method of multiple scales, deep and shallow water waves, nonlinear optics including fiber optic communications, mode-locked lasers and dispersion-managed wave phenomena. Most chapters feature exercise sets, making the book suitable for advanced courses or for self-directed learning. Graduate students and researchers will find this an excellent entry to a thriving area at the intersection of applied mathematics, engineering and physical science.