Download or read book Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data written by and published by . This book was released on 2002 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Semiclassical asymptotics of the focusing nonlinear Schrodinger equation for square barrier initial data.
Download or read book Semiclassical Asymptotics of the Focusing Nonlinear Schrodinger Equation for Square Barrier Initial Data written by Robert M. Jenkins and published by . This book was released on 2009 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: The small dispersion limit of the focusing nonlinear Schroodinger equation (fNLS) exhibits a rich structure with rapid oscillations at microscopic scales. Due to the non self-adjoint scattering problem associated to fNLS, very few rigorous results exist in the semiclassical limit. The first such results were for for reflectionless WKB-like initial data, which generalizes the well known sech solutions. Soon after another generalization of the sech potential, adding a complex phase, was discovered. In both studies the authors observed sharp breaking curves in the space-time separating regions with disparate asymptotic behaviors. In this paper we consider another exactly solvable family of initial data, specifically the family of centered square pulses, q(x,0) = q & chi[-L, L]for real amplitudes q. Using Riemann-Hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of fNLS globally in space and up to an order one (O(1)) maximal time. In particular, we find breaking curves emerging in accord with the previous studies. Finally, we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data. This is the first case in which the genus structure of the semiclassical asymptotics for fNLS have been calculated for non-analytic initial data.
Download or read book Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrodinger Equation AM 154 written by Spyridon Kamvissis and published by Princeton University Press. This book was released on 2003-09-07 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: Providing an asymptotic analysis via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrodinger equation in the semiclassical asymptotic regime, this text exploits complete integrability to establish pointwise asymptotics for this problem's solution.
Download or read book On the Semiclassical Limit for the Focusing Nonlinear Schr dinger Equation written by S. R. Clarke and published by . This book was released on 2000 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Breaking in the Semiclassical Solution of the Focusing Nonlinear Schrodinger Equation written by Sergey M. Belov and published by . This book was released on 2008 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the one dimensional semiclassical focusing cubic nonlinear Schrodinger equation with a one parameter family of decaying initial conditions using the Lax pair and the Riemann-Hilbert approach to inverse scattering. In previous studies the solution was found to develop fast oscillations in modulus passed some curves in the space-time plane (breaking curves or nonlinear caustics). We carried out a detailed asymptotic analysis of the solution as we approach a catastrophic break of the our analytic procedure. We developed numerical integration on a Riemann surface to compute the relevant quantities numerically near the catastrophic break and providing new insights to the first break.
Download or read book A Proof of the Soliton Resolution Conjecture for the Focusing Nonlinear Schr dinger Equation written by Michael Borghese and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We give a proof of the long-time asymptotic behavior of the focusing nonlinear Schrödinger equation for generic initial condition in which we have simple discrete spectral data and an absence of spectral singularities. The proof relies upon the theory of Riemann-Hilbert problems and the ∂ ̄ method for nonlinear steepest descent. To leading order, the solution will appear as a multi-soliton solution as t → ∞.
Download or read book Mathematical Reviews written by and published by . This book was released on 2004 with total page 1770 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Semiclassical Analysis written by Maciej Zworski and published by American Mathematical Soc.. This book was released on 2012 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: "...A graduate level text introducing readers to semiclassical and microlocal methods in PDE." -- from xi.
Download or read book Theory of Solitons written by S. Novikov and published by Springer Science & Business Media. This book was released on 1984-05-31 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mathematical Optics written by Vasudevan Lakshminarayanan and published by CRC Press. This book was released on 2012-12-14 with total page 632 pages. Available in PDF, EPUB and Kindle. Book excerpt: Going beyond standard introductory texts, Mathematical Optics: Classical, Quantum, and Computational Methods brings together many new mathematical techniques from optical science and engineering research. Profusely illustrated, the book makes the material accessible to students and newcomers to the field. Divided into six parts, the text presents state-of-the-art mathematical methods and applications in classical optics, quantum optics, and image processing. Part I describes the use of phase space concepts to characterize optical beams and the application of dynamic programming in optical waveguides. Part II explores solutions to paraxial, linear, and nonlinear wave equations. Part III discusses cutting-edge areas in transformation optics (such as invisibility cloaks) and computational plasmonics. Part IV uses Lorentz groups, dihedral group symmetry, Lie algebras, and Liouville space to analyze problems in polarization, ray optics, visual optics, and quantum optics. Part V examines the role of coherence functions in modern laser physics and explains how to apply quantum memory channel models in quantum computers. Part VI introduces super-resolution imaging and differential geometric methods in image processing. As numerical/symbolic computation is an important tool for solving numerous real-life problems in optical science, many chapters include Mathematica® code in their appendices. The software codes and notebooks as well as color versions of the book’s figures are available at www.crcpress.com.
Download or read book Condensed Matter Field Theory written by Alexander Altland and published by Cambridge University Press. This book was released on 2010-03-11 with total page 785 pages. Available in PDF, EPUB and Kindle. Book excerpt: This primer is aimed at elevating graduate students of condensed matter theory to a level where they can engage in independent research. Topics covered include second quantisation, path and functional field integration, mean-field theory and collective phenomena.
Download or read book Instantons and Large N written by Marcos Mariño and published by Cambridge University Press. This book was released on 2015-09-03 with total page 381 pages. Available in PDF, EPUB and Kindle. Book excerpt: This highly pedagogical textbook for graduate students in particle, theoretical and mathematical physics, explores advanced topics of quantum field theory. Clearly divided into two parts; the first focuses on instantons with a detailed exposition of instantons in quantum mechanics, supersymmetric quantum mechanics, the large order behavior of perturbation theory, and Yang–Mills theories, before moving on to examine the large N expansion in quantum field theory. The organised presentation style, in addition to detailed mathematical derivations, worked examples and applications throughout, enables students to gain practical experience with the tools necessary to start research. The author includes recent developments on the large order behavior of perturbation theory and on large N instantons, and updates existing treatments of classic topics, to ensure that this is a practical and contemporary guide for students developing their understanding of the intricacies of quantum field theory.
Download or read book Nonlinear Markov Processes and Kinetic Equations written by Vassili N. Kolokoltsov and published by Cambridge University Press. This book was released on 2010-07-15 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: A nonlinear Markov evolution is a dynamical system generated by a measure-valued ordinary differential equation with the specific feature of preserving positivity. This feature distinguishes it from general vector-valued differential equations and yields a natural link with probability, both in interpreting results and in the tools of analysis. This brilliant book, the first devoted to the area, develops this interplay between probability and analysis. After systematically presenting both analytic and probabilistic techniques, the author uses probability to obtain deeper insight into nonlinear dynamics, and analysis to tackle difficult problems in the description of random and chaotic behavior. The book addresses the most fundamental questions in the theory of nonlinear Markov processes: existence, uniqueness, constructions, approximation schemes, regularity, law of large numbers and probabilistic interpretations. Its careful exposition makes the book accessible to researchers and graduate students in stochastic and functional analysis with applications to mathematical physics and systems biology.
Download or read book Mathematics for Physics written by Michael Stone and published by Cambridge University Press. This book was released on 2009-07-09 with total page 821 pages. Available in PDF, EPUB and Kindle. Book excerpt: An engagingly-written account of mathematical tools and ideas, this book provides a graduate-level introduction to the mathematics used in research in physics. The first half of the book focuses on the traditional mathematical methods of physics – differential and integral equations, Fourier series and the calculus of variations. The second half contains an introduction to more advanced subjects, including differential geometry, topology and complex variables. The authors' exposition avoids excess rigor whilst explaining subtle but important points often glossed over in more elementary texts. The topics are illustrated at every stage by carefully chosen examples, exercises and problems drawn from realistic physics settings. These make it useful both as a textbook in advanced courses and for self-study. Password-protected solutions to the exercises are available to instructors at www.cambridge.org/9780521854030.
Download or read book Mathematical Theory of Scattering Resonances written by Semyon Dyatlov and published by American Mathematical Soc.. This book was released on 2019-09-10 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: Scattering resonances generalize bound states/eigenvalues for systems in which energy can scatter to infinity. A typical resonance has a rate of oscillation (just as a bound state does) and a rate of decay. Although the notion is intrinsically dynamical, an elegant mathematical formulation comes from considering meromorphic continuations of Green's functions. The poles of these meromorphic continuations capture physical information by identifying the rate of oscillation with the real part of a pole and the rate of decay with its imaginary part. An example from mathematics is given by the zeros of the Riemann zeta function: they are, essentially, the resonances of the Laplacian on the modular surface. The Riemann hypothesis then states that the decay rates for the modular surface are all either or . An example from physics is given by quasi-normal modes of black holes which appear in long-time asymptotics of gravitational waves. This book concentrates mostly on the simplest case of scattering by compactly supported potentials but provides pointers to modern literature where more general cases are studied. It also presents a recent approach to the study of resonances on asymptotically hyperbolic manifolds. The last two chapters are devoted to semiclassical methods in the study of resonances.
Download or read book Shaping Light in Nonlinear Optical Fibers written by Sonia Boscolo and published by John Wiley & Sons. This book was released on 2017-05-30 with total page 486 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a contemporary overview of selected topics in fiber optics. It focuses on the latest research results on light wave manipulation using nonlinear optical fibers, with the aim of capturing some of the most innovative developments on this topic. The book’s scope covers both fundamentals and applications from both theoretical and experimental perspectives, with topics including linear and nonlinear effects, pulse propagation phenomena and pulse shaping, solitons and rogue waves, novel optical fibers, supercontinuum generation, polarization management, optical signal processing, fiber lasers, optical wave turbulence, light propagation in disordered fiber media, and slow and fast light. With contributions from leading-edge scientists in the field of nonlinear photonics and fiber optics, they offer an overview of the latest advances in their own research area. The listing of recent research papers at the end of each chapter is useful for researchers using the book as a reference. As the book addresses fundamental and practical photonics problems, it will also be of interest to, and benefit, broader academic communities, including areas such as nonlinear science, applied mathematics and physics, and optical engineering. It offers the reader a wide and critical overview of the state-of-the-art within this practical – as well as fundamentally important and interesting – area of modern science, providing a useful reference which will encourage further research and advances in the field.
Download or read book Quantum Mechanics written by Thomas Banks and published by CRC Press. This book was released on 2018-12-07 with total page 760 pages. Available in PDF, EPUB and Kindle. Book excerpt: This authoritative, advanced introduction provides a complete, modern perspective on quantum mechanics. It clarifies many common misconceptions regarding wave/particle duality and the correct interpretation of measurements. The author develops the text from the ground up, starting from the fundamentals and presenting information at an elementary level, avoiding unnecessarily detailed and complex derivations in favor of simple, clear explanations. He begins in the simplest context of a two-state system and shows why quantum mechanics is inevitable, and what its relationship is to classical mechanics. He also outlines the decoherence approach to interpreting quantum mechanics. Distinguishing features: Provides a thorough grounding in the principles and practice of quantum mechanics, including a core understanding of the behavior of atoms, molecules, solids, and light. Utilizes easy-to-follow examples and analogies to illustrate important concepts. Helps develop an intuitive sense for the field, by guiding the reader to understand how the correct formulas reduce to the non-relativistic ones. Includes numerous worked examples and problems for each chapter.
