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Lyapunov Stability Of Ground States Of Nonlinear Dispersive Evolution Equations
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Book Synopsis Lyapunov Stability of Ground States of Nonlinear Dispersive Evolution Equations by : Michael I. Weinstein
Download or read book Lyapunov Stability of Ground States of Nonlinear Dispersive Evolution Equations written by Michael I. Weinstein and published by . This book was released on 1986 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt: A solitary wave is a localized, finite energy solution of a nonlinear evolution equation. It results from a balance of dispersion and a focusing nonlinearity. Two fundamental equations in the theory of nonlinear waves that possess such solutions are the nonlinear Schrodinger equation (NLS) and the Korteweg deVries equation (KdV). NLS arises in the mathematical description of electromagnetic wave propagation through nonlinear media. KdV arises in the study of waves in shallow water. This paper presents a new proof of orbital stability of ground state solitary waves of the nonlinear Schrodinger equation for a general class of nonlinearities. The stability of the solitary wave for the generalized Korteweg deVries equation is also shown.
Book Synopsis Invariant Manifolds and Dispersive Hamiltonian Evolution Equations by : Kenji Nakanishi
Download or read book Invariant Manifolds and Dispersive Hamiltonian Evolution Equations written by Kenji Nakanishi and published by European Mathematical Society. This book was released on 2011 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notion of an invariant manifold arises naturally in the asymptotic stability analysis of stationary or standing wave solutions of unstable dispersive Hamiltonian evolution equations such as the focusing semilinear Klein-Gordon and Schrodinger equations. This is due to the fact that the linearized operators about such special solutions typically exhibit negative eigenvalues (a single one for the ground state), which lead to exponential instability of the linearized flow and allows for ideas from hyperbolic dynamics to enter. One of the main results proved here for energy subcritical equations is that the center-stable manifold associated with the ground state appears as a hyper-surface which separates a region of finite-time blowup in forward time from one which exhibits global existence and scattering to zero in forward time. The authors' entire analysis takes place in the energy topology, and the conserved energy can exceed the ground state energy only by a small amount. This monograph is based on recent research by the authors. The proofs rely on an interplay between the variational structure of the ground states and the nonlinear hyperbolic dynamics near these states. A key element in the proof is a virial-type argument excluding almost homoclinic orbits originating near the ground states, and returning to them, possibly after a long excursion. These lectures are suitable for graduate students and researchers in partial differential equations and mathematical physics. For the cubic Klein-Gordon equation in three dimensions all details are provided, including the derivation of Strichartz estimates for the free equation and the concentration-compactness argument leading to scattering due to Kenig and Merle.
Book Synopsis Introduction to Nonlinear Dispersive Equations by : Felipe Linares
Download or read book Introduction to Nonlinear Dispersive Equations written by Felipe Linares and published by Springer. This book was released on 2014-12-15 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Asymptotic Stability of the Ground States of the Nonlinear Schrödinger Equation by : Ozgur Mizrak
Download or read book Asymptotic Stability of the Ground States of the Nonlinear Schrödinger Equation written by Ozgur Mizrak and published by ProQuest. This book was released on 2009 with total page 91 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider a class of nonlinear Schrodinger equations in N = 3, 4, 5 space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L2( RN )) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that "shadows" the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.
Book Synopsis Evolution Equations by : Gisele Ruiz Goldstein
Download or read book Evolution Equations written by Gisele Ruiz Goldstein and published by CRC Press. This book was released on 2019-04-24 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: Celebrating the work of renowned mathematician Jerome A. Goldstein, this reference compiles original research on the theory and application of evolution equations to stochastics, physics, engineering, biology, and finance. The text explores a wide range of topics in linear and nonlinear semigroup theory, operator theory, functional analysis, and li
Book Synopsis Nonlinear Dispersive Equations by : Terence Tao
Download or read book Nonlinear Dispersive Equations written by Terence Tao and published by American Mathematical Soc.. This book was released on with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: "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.".
Book Synopsis Spectral and Dynamical Stability of Nonlinear Waves by : Todd Kapitula
Download or read book Spectral and Dynamical Stability of Nonlinear Waves written by Todd Kapitula and published by Springer Science & Business Media. This book was released on 2013-06-06 with total page 369 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book unifies the dynamical systems and functional analysis approaches to the linear and nonlinear stability of waves. It synthesizes fundamental ideas of the past 20+ years of research, carefully balancing theory and application. The book isolates and methodically develops key ideas by working through illustrative examples that are subsequently synthesized into general principles. Many of the seminal examples of stability theory, including orbital stability of the KdV solitary wave, and asymptotic stability of viscous shocks for scalar conservation laws, are treated in a textbook fashion for the first time. It presents spectral theory from a dynamical systems and functional analytic point of view, including essential and absolute spectra, and develops general nonlinear stability results for dissipative and Hamiltonian systems. The structure of the linear eigenvalue problem for Hamiltonian systems is carefully developed, including the Krein signature and related stability indices. The Evans function for the detection of point spectra is carefully developed through a series of frameworks of increasing complexity. Applications of the Evans function to the Orientation index, edge bifurcations, and large domain limits are developed through illustrative examples. The book is intended for first or second year graduate students in mathematics, or those with equivalent mathematical maturity. It is highly illustrated and there are many exercises scattered throughout the text that highlight and emphasize the key concepts. Upon completion of the book, the reader will be in an excellent position to understand and contribute to current research in nonlinear stability.
Book Synopsis Nonlinear Dirac Equation: Spectral Stability of Solitary Waves by : Nabile Boussaïd
Download or read book Nonlinear Dirac Equation: Spectral Stability of Solitary Waves written by Nabile Boussaïd and published by American Mathematical Soc.. This book was released on 2019-11-21 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves. The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrödinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.
Book Synopsis Equadiff 95 - Proceedings Of The International Conference On Differential Equations by : L Magalhaes
Download or read book Equadiff 95 - Proceedings Of The International Conference On Differential Equations written by L Magalhaes and published by World Scientific. This book was released on 1998-04-30 with total page 578 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume, leading experts on differential equations address recent advances in the fields of ordinary differential equations and dynamical systems, partial differential equations and calculus of variations, and their related applications.
Book Synopsis Partially Integrable Evolution Equations in Physics by : R. Conte
Download or read book Partially Integrable Evolution Equations in Physics written by R. Conte and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 609 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the many physical phenomena ruled by partial differential equations, two extreme fields are currently overcrowded due to recent considerable developments: 1) the field of completely integrable equations, whose recent advances are the inverse spectral transform, the recursion operator, underlying Hamiltonian structures, Lax pairs, etc 2) the field of dynamical systems, often built as models of observed physical phenomena: turbulence, intermittency, Poincare sections, transition to chaos, etc. In between there is a very large region where systems are neither integrable nor nonintegrable, but partially integrable, and people working in the latter domain often know methods from either 1) or 2). Due to the growing interest in partially integrable systems, we decided to organize a meeting for physicists active or about to undertake research in this field, and we thought that an appropriate form would be a school. Indeed, some of the above mentioned methods are often adaptable outside their original domain and therefore worth to be taught in an interdisciplinary school. One of the main concerns was to keep a correct balance between physics and mathematics, and this is reflected in the list of courses.
Book Synopsis Handbook of Dynamical Systems by : A. Katok
Download or read book Handbook of Dynamical Systems written by A. Katok and published by Elsevier. This book was released on 2005-12-17 with total page 1235 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey “Principal Structures of Volume 1A.The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations). . Written by experts in the field.. The coverage of ergodic theory in these two parts of Volume 1 is considerably more broad and thorough than that provided in other existing sources. . The final cluster of chapters discusses partial differential equations from the point of view of dynamical systems.
Book Synopsis Evolution Equations by : David Ellwood
Download or read book Evolution Equations written by David Ellwood and published by American Mathematical Soc.. This book was released on 2013-06-26 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is a collection of notes from lectures given at the 2008 Clay Mathematics Institute Summer School, held in Zürich, Switzerland. The lectures were designed for graduate students and mathematicians within five years of the Ph.D., and the main focus of the program was on recent progress in the theory of evolution equations. Such equations lie at the heart of many areas of mathematical physics and arise not only in situations with a manifest time evolution (such as linear and nonlinear wave and Schrödinger equations) but also in the high energy or semi-classical limits of elliptic problems. The three main courses focused primarily on microlocal analysis and spectral and scattering theory, the theory of the nonlinear Schrödinger and wave equations, and evolution problems in general relativity. These major topics were supplemented by several mini-courses reporting on the derivation of effective evolution equations from microscopic quantum dynamics; on wave maps with and without symmetries; on quantum N-body scattering, diffraction of waves, and symmetric spaces; and on nonlinear Schrödinger equations at critical regularity. Although highly detailed treatments of some of these topics are now available in the published literature, in this collection the reader can learn the fundamental ideas and tools with a minimum of technical machinery. Moreover, the treatment in this volume emphasizes common themes and techniques in the field, including exact and approximate conservation laws, energy methods, and positive commutator arguments. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).
Book Synopsis An Introduction to Semilinear Evolution Equations by : Thierry Cazenave
Download or read book An Introduction to Semilinear Evolution Equations written by Thierry Cazenave and published by Oxford University Press. This book was released on 1998 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents in a self-contained form the typical basic properties of solutions to semilinear evolutionary partial differential equations, with special emphasis on global properties. It has a didactic ambition and will be useful for an applied readership as well as theoretical researchers.
Book Synopsis Semilinear Schrodinger Equations by : Thierry Cazenave
Download or read book Semilinear Schrodinger Equations written by Thierry Cazenave and published by American Mathematical Soc.. This book was released on 2003 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: The nonlinear Schrodinger equation has received a great deal of attention from mathematicians, particularly because of its applications to nonlinear optics. This book presents various mathematical aspects of the nonlinear Schrodinger equation. It studies both problems of local nature and problems of global nature.
Book Synopsis The Nonlinear Schrödinger Equation by : Catherine Sulem
Download or read book The Nonlinear Schrödinger Equation written by Catherine Sulem and published by Springer Science & Business Media. This book was released on 2007-06-30 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: Filling the gap between the mathematical literature and applications to domains, the authors have chosen to address the problem of wave collapse by several methods ranging from rigorous mathematical analysis to formal aymptotic expansions and numerical simulations.
Book Synopsis The Nonlinear Schrödinger Equation by : Gadi Fibich
Download or read book The Nonlinear Schrödinger Equation written by Gadi Fibich and published by Springer. This book was released on 2015-03-06 with total page 870 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an interdisciplinary introduction to optical collapse of laser beams, which is modelled by singular (blow-up) solutions of the nonlinear Schrödinger equation. With great care and detail, it develops the subject including the mathematical and physical background and the history of the subject. It combines rigorous analysis, asymptotic analysis, informal arguments, numerical simulations, physical modelling, and physical experiments. It repeatedly emphasizes the relations between these approaches, and the intuition behind the results. The Nonlinear Schrödinger Equation will be useful to graduate students and researchers in applied mathematics who are interested in singular solutions of partial differential equations, nonlinear optics and nonlinear waves, and to graduate students and researchers in physics and engineering who are interested in nonlinear optics and Bose-Einstein condensates. It can be used for courses on partial differential equations, nonlinear waves, and nonlinear optics. Gadi Fibich is a Professor of Applied Mathematics at Tel Aviv University. “This book provides a clear presentation of the nonlinear Schrodinger equation and its applications from various perspectives (rigorous analysis, informal analysis, and physics). It will be extremely useful for students and researchers who enter this field.” Frank Merle, Université de Cergy-Pontoise and Institut des Hautes Études Scientifiques, France
Book Synopsis Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory by : Peter E. Zhidkov
Download or read book Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory written by Peter E. Zhidkov and published by Springer Science & Business Media. This book was released on 2001-04-24 with total page 153 pages. Available in PDF, EPUB and Kindle. Book excerpt: The emphasis of this book is on questions typical of nonlinear analysis and qualitative theory of PDEs. The selection of the material is related to the author's attempt to illuminate those particularly interesting questions not yet covered in other monographs though they have been the subject of published articles. One chapter, for example, is devoted to the construction of invariant measures for dynamical systems generated by certain equations and a result from a recent paper on basic properties of a system of eigenfunctions of a stationary problem. Also considered is an application of the method of qualitative theory of ODes to proving the existence of radial solutions of stationary problems and stability of solutions of NLSE nonvanishing as the spatial variable tends to infinity. Finally a recent result on the existence of an infinite sequence of invariant measures for the inegrable KdV equation is presented.
