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Invariant Manifolds And Dispersive Hamiltonian Evolution Equations
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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 Attractors of Hamiltonian Nonlinear Partial Differential Equations by : Alexander Komech
Download or read book Attractors of Hamiltonian Nonlinear Partial Differential Equations written by Alexander Komech and published by Cambridge University Press. This book was released on 2021-09-30 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is the first to present the theory of global attractors of Hamiltonian partial differential equations. A particular focus is placed on the results obtained in the last three decades, with chapters on the global attraction to stationary states, to solitons, and to stationary orbits. The text includes many physically relevant examples and will be of interest to graduate students and researchers in both mathematics and physics. The proofs involve novel applications of methods of harmonic analysis, including Tauberian theorems, Titchmarsh's convolution theorem, and the theory of quasimeasures. As well as the underlying theory, the authors discuss the results of numerical simulations and formulate open problems to prompt further research.
Download or read book PDE Dynamics written by Christian Kuehn and published by SIAM. This book was released on 2019-04-10 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an overview of the myriad methods for applying dynamical systems techniques to PDEs and highlights the impact of PDE methods on dynamical systems. Also included are many nonlinear evolution equations, which have been benchmark models across the sciences, and examples and techniques to strengthen preparation for research. PDE Dynamics: An Introduction is intended for senior undergraduate students, beginning graduate students, and researchers in applied mathematics, theoretical physics, and adjacent disciplines. Structured as a textbook or seminar reference, it can be used in courses titled Dynamics of PDEs, PDEs 2, Dynamical Systems 2, Evolution Equations, or Infinite-Dimensional Dynamics.
Book Synopsis Geometric Numerical Integration and Schrödinger Equations by : Erwan Faou
Download or read book Geometric Numerical Integration and Schrödinger Equations written by Erwan Faou and published by European Mathematical Society. This book was released on 2012 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of geometric numerical integration is the simulation of evolution equations possessing geometric properties over long periods of time. Of particular importance are Hamiltonian partial differential equations typically arising in application fields such as quantum mechanics or wave propagation phenomena. They exhibit many important dynamical features such as energy preservation and conservation of adiabatic invariants over long periods of time. In this setting, a natural question is how and to which extent the reproduction of such long-time qualitative behavior can be ensured by numerical schemes. Starting from numerical examples, these notes provide a detailed analysis of the Schrodinger equation in a simple setting (periodic boundary conditions, polynomial nonlinearities) approximated by symplectic splitting methods. Analysis of stability and instability phenomena induced by space and time discretization are given, and rigorous mathematical explanations are provided for them. The book grew out of a graduate-level course and is of interest to researchers and students seeking an introduction to the subject matter.
Book Synopsis XVIIth International Congress on Mathematical Physics by : Arne Jensen
Download or read book XVIIth International Congress on Mathematical Physics written by Arne Jensen and published by World Scientific. This book was released on 2014 with total page 743 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an in-depth study of not just about Tan Kah-kee, but also the making of a legend through his deeds, self-sacrifices, fortitude and foresight. This revised edition sheds new light on his political agonies in Mao's China over campaigns against capitalists and intellectuals.
Book Synopsis Strichartz Estimates for Wave Equations with Charge Transfer Hamiltonians by : Gong Chen
Download or read book Strichartz Estimates for Wave Equations with Charge Transfer Hamiltonians written by Gong Chen and published by American Mathematical Society. This book was released on 2021-12-09 with total page 84 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
Book Synopsis Type II blow up solutions with optimal stability properties for the critical focussing nonlinear wave equation on $mathbb {R}^{3+1}$ by : Stefano Burzio
Download or read book Type II blow up solutions with optimal stability properties for the critical focussing nonlinear wave equation on $mathbb {R}^{3+1}$ written by Stefano Burzio and published by American Mathematical Society. This book was released on 2022-07-18 with total page 88 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
Book Synopsis On Stability of Type II Blow Up for the Critical Nonlinear Wave Equation in $mathbb {R}^{3+1}$ by : Joachim K Krieger
Download or read book On Stability of Type II Blow Up for the Critical Nonlinear Wave Equation in $mathbb {R}^{3+1}$ written by Joachim K Krieger and published by American Mathematical Society. This book was released on 2021-02-10 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author shows that the finite time type II blow up solutions for the energy critical nonlinear wave equation $ Box u = -u^5 $ on $mathbb R^3+1$ constructed in Krieger, Schlag, and Tataru (2009) and Krieger and Schlag (2014) are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter $lambda (t) = t^-1-nu $ is sufficiently close to the self-similar rate, i. e. $nu >0$ is sufficiently small. Our method is based on Fourier techniques adapted to time dependent wave operators of the form $ -partial _t^2 + partial _r^2 + frac 2rpartial _r +V(lambda (t)r) $ for suitable monotone scaling parameters $lambda (t)$ and potentials $V(r)$ with a resonance at zero.
Book Synopsis Topics in Occupation Times and Gaussian Free Fields by : Alain-Sol Sznitman
Download or read book Topics in Occupation Times and Gaussian Free Fields written by Alain-Sol Sznitman and published by European Mathematical Society. This book was released on 2012 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book grew out of a graduate course at ETH Zurich during the spring 2011 term. It explores various links between such notions as occupation times of Markov chains, Gaussian free fields, Poisson point processes of Markovian loops, and random interlacements, which have been the object of intensive research over the last few years. These notions are developed in the convenient setup of finite weighted graphs endowed with killing measures. This book first discusses elements of continuous-time Markov chains, Dirichlet forms, potential theory, together with some consequences for Gaussian free fields. Next, isomorphism theorems and generalized Ray-Knight theorems, which relate occupation times of Markov chains to Gaussian free fields, are presented. Markovian loops are constructed and some of their key properties derived. The field of occupation times of Poisson point processes of Markovian loops is investigated. Of special interest are its connection to the Gaussian free field, and a formula of Symanzik. Finally, links between random interlacements and Markovian loops are discussed, and some further connections with Gaussian free fields are mentioned.
Book Synopsis Classical and Multilinear Harmonic Analysis by : Camil Muscalu
Download or read book Classical and Multilinear Harmonic Analysis written by Camil Muscalu and published by Cambridge University Press. This book was released on 2013-01-31 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: This contemporary graduate-level text in harmonic analysis introduces the reader to a wide array of analytical results and techniques.
Book Synopsis Classical and Multilinear Harmonic Analysis: Volume 1 by : Camil Muscalu
Download or read book Classical and Multilinear Harmonic Analysis: Volume 1 written by Camil Muscalu and published by Cambridge University Press. This book was released on 2013-01-31 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and will be useful to graduate students and researchers in both pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. This first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderón–Zygmund and Littlewood–Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman–Meyer theory; Carleson's resolution of the Lusin conjecture; Calderón's commutators and the Cauchy integral on Lipschitz curves. The material in this volume has not previously appeared together in book form.
Book Synopsis Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations by : Charles Li
Download or read book Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations written by Charles Li and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this monograph the authors present detailed and pedagogic proofs of persistence theorems for normally hyperbolic invariant manifolds and their stable and unstable manifolds for classes of perturbations of the NLS equation, as well as for the existence and persistence of fibrations of these invariant manifolds. Their techniques are based on an infinite dimensional generalisation of the graph transform and can be viewed as an infinite dimensional generalisation of Fenichels results. As such, they may be applied to a broad class of infinite dimensional dynamical systems.
Book Synopsis Smooth Invariant Manifolds And Normal Forms by : Alexander Kopanskii
Download or read book Smooth Invariant Manifolds And Normal Forms written by Alexander Kopanskii and published by World Scientific. This book was released on 1994-12-22 with total page 398 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with the qualitative theory of dynamical systems and is devoted to the study of flows and cascades in the vicinity of a smooth invariant manifold. Its main purpose is to present, as completely as possible, the basic results concerning the existence of stable and unstable local manifolds and the recent advancements in the theory of finitely smooth normal forms of vector fields and diffeomorphisms in the vicinity of a rest point and a periodic trajectory. A summary of the results obtained so far in the investigation of dynamical systems near an arbitrary invariant submanifold is also given.
Book Synopsis The Parameterization Method for Invariant Manifolds by : Àlex Haro
Download or read book The Parameterization Method for Invariant Manifolds written by Àlex Haro and published by Springer. This book was released on 2016-04-18 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.
Book Synopsis Evolution Equations by : Bogdan Bojarski
Download or read book Evolution Equations written by Bogdan Bojarski and published by . This book was released on 2000 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Invariant Manifolds for Flows in Banach Spaces by : Kening Lu
Download or read book Invariant Manifolds for Flows in Banach Spaces written by Kening Lu and published by . This book was released on 1988 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Invariant Manifolds by : M.W. Hirsch
Download or read book Invariant Manifolds written by M.W. Hirsch and published by Springer. This book was released on 2006-11-15 with total page 153 pages. Available in PDF, EPUB and Kindle. Book excerpt:
