Download or read book Semi classical Analysis For Nonlinear Schrodinger Equations Wkb Analysis Focal Points Coherent States Second Edition written by Remi Carles and published by World Scientific. This book was released on 2020-10-05 with total page 367 pages. Available in PDF, EPUB and Kindle. Book excerpt: The second edition of this book consists of three parts. The first one is dedicated to the WKB methods and the semi-classical limit before the formation of caustics. The second part treats the semi-classical limit in the presence of caustics, in the special geometric case where the caustic is reduced to a point (or to several isolated points). The third part is new in this edition, and addresses the nonlinear propagation of coherent states. The three parts are essentially independent.Compared with the first edition, the first part is enriched by a new section on multiphase expansions in the case of weakly nonlinear geometric optics, and an application related to this study, concerning instability results for nonlinear Schrödinger equations in negative order Sobolev spaces.The third part is an overview of results concerning nonlinear effects in the propagation of coherent states, in the case of a power nonlinearity, and in the richer case of Hartree-like nonlinearities. It includes explicit formulas of an independent interest, such as generalized Mehler's formula, generalized lens transform.
Download or read book Semi classical Analysis for Nonlinear Schr dinger Equations written by Rémi Carles and published by World Scientific Publishing Company Incorporated. This book was released on 2008 with total page 243 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Semi classical Analysis for Nonlinear Schr dinger Equations written by Rémi Carles and published by World Scientific Publishing Company. This book was released on 2020-09-29 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The second edition of this book consists of three parts. The first one is dedicated to the WKB methods and the semi-classical limit before the formation of caustics. The second part treats the semi-classical limit in the presence of caustics, in the special geometric case where the caustic is reduced to a point (or to several isolated points). The third part is new in this edition, and addresses the nonlinear propagation of coherent states. The three parts are essentially independent. Compared with the first edition, the first part is enriched by a new section on multiphase expansions in the case of weakly nonlinear geometric optics, and an application related to this study, concerning instability results for nonlinear Schrdinger equations in negative order Sobolev spaces. The third part is an overview of results concerning nonlinear effects in the propagation of coherent states, in the case of a power nonlinearity, and in the richer case of Hartree-like nonlinearities. It includes explicit formulas of an independent interest, such as generalized Mehler's formula, generalized lens transform.
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 Semi classical Analysis For Nonlinear Schrodinger Equations written by Remi Carles and published by World Scientific. This book was released on 2008-03-04 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lecture notes review recent results on the high-frequency analysis of nonlinear Schrödinger equations in the presence of an external potential. The book consists of two relatively independent parts: WKB analysis, and caustic crossing. In the first part, the basic linear WKB theory is constructed and then extended to the nonlinear framework. The most difficult supercritical case is discussed in detail, together with some of its consequences concerning instability phenomena. Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear Schrödinger equations, are also given. In the second part, caustic crossing is described, especially when the caustic is reduced to a point, and the link with nonlinear scattering operators is investigated.These notes are self-contained and combine selected articles written by the author over the past ten years in a coherent manner, with some simplified proofs. Examples and figures are provided to support the intuition, and comparisons with other equations such as the nonlinear wave equation are provided.
Download or read book Wigner Measure and Semiclassical Limits of Nonlinear Schr dinger Equations written by Ping Zhang and published by American Mathematical Soc.. This book was released on with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This book is based on a course entitled "Wigner measures and semiclassical limits of nonlinear Schrodinger equations," which the author taught at the Courant Institute of Mathematical Sciences at New York University in the spring of 2007. The author's main purpose is to apply the theory of semiclassical pseudodifferential operators to the study of various high-frequency limits of equations from quantum mechanics. In particular, the focus of attention is on Wigner measure and recent progress on how to use it as a tool to study various problems arising from semiclassical limits of Schrodinger-type equations." "At the end of each chapter, the reader will find references and remarks about recent progress on related problems. The book is self-contained and is suitable for an advanced graduate course on the topic."--BOOK JACKET.
Download or read book Semiclassical Soliton Ensembles for the Focusing Nonlinear Schr dinger Equation AM 154 written by Spyridon Kamvissis and published by Princeton University Press. This book was released on 2003-08-18 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Hölder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.
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 Nonlinear Schrodinger Equations with Potentials written by Yong-Geun Oh and published by . This book was released on 1988 with total page 158 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 An Introduction to Semiclassical and Microlocal Analysis written by André Bach and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics in a pedagogical, way and is mainly addressed to non-specialists in the subject. It is based on lectures taught by the author over several years, and includes many exercises providing outlines of useful applications of the semi-classical theory.
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 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).
Download or read book Discrete and Continuous Nonlinear Schr dinger Systems written by M. J. Ablowitz and published by Cambridge University Press. This book was released on 2004 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a detailed mathematical analysis of scattering theory, obtains soliton solutions, and analyzes soliton interactions, both scalar and vector.
Download or read book Handbook of Exact Solutions to the Nonlinear Schr dinger Equations written by Usama Al Khawaja and published by Institute of Physics Publishing. This book was released on 2019-11-15 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects all known solutions to the nonlinear Schrödinger equation (NLSE) in one resource. In addition, the book organizes the solutions by classifying and grouping them based on aspects and symmetries they possess. Although most of the solutions presented in this book have been derived elsewhere using various methods, the authors present a systematic derivation of many solutions and even include new derivations. They have also presented symmetries and reductions that connect different solutions through transformations and enable classifying new solutions into known classes. For the user to verify that the presented solutions do satisfy the NLSE, this monumental work is accompanied by Mathematica Notebooks containing all solutions. This work also features a large number of figures, and animations are included to help visualize solutions and their dynamics.
Download or read book Semiclassical Limit of the Non linear Schroedinger Poisson Equation With Subcritical Initial Data written by and published by . This book was released on 2002 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the semi-classical limit of the nonlinear Schroedinger-Poisson (NLSP) equation for initial data of the WKB type. The semi-classical limit in this case is realized in terms of a density-velocity pair governed by the Euler-Poisson equations. Recently we have shown that the isotropic Euler-Poisson equations admit a critical threshold phenomena, where initial data in the sub-critical regime give rise to globally smooth solutions. Consequently, we justify the semi-classical limit for sub-critical NLSP initial data and confirm the validity of the WKB method.
Download or read book Nonlinear Dynamics and Renormalization Group written by Israel Michael Sigal and published by American Mathematical Soc.. This book was released on 2001 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the proceedings from the workshop, Nonlinear Dynamics and Renormalization Group, held at the Centre de recherches mathématiques (CRM) in Montréal (Canada), as part of the year-long program devoted to mathematical physics. In the book, active researchers in the fields of nonlinear partial differential equations and renormalization group contribute recent results on topics such as Ginzburg-Landau equations and blow-up of solutions of the nonlinear Schroedinger equations, quantum resonances, and renormalization group analysis in constructive quantum field theory. This volume offers the latest research in the rapidly developing fields of nonlinear equations and renormalization group.
