[PDF] Breaking In The Semiclassical Solution Of The Focusing Nonlinear Schrodinger Equation eBook

Author : Sergey M. Belov
Publisher :
Page : 320 pages
File Size : 35,58 MB
Release : 2008
Category : Gross-Pitaevskii equations
ISBN :

GET BOOK

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.

Author : Spyridon Kamvissis
Publisher : Princeton University Press
Page : 281 pages
File Size : 20,44 MB
Release : 2003-09-07
Category : Mathematics
ISBN : 069111482X

GET BOOK

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.

Author : Robert M. Jenkins
Publisher :
Page : 314 pages
File Size : 41,26 MB
Release : 2009
Category :
ISBN :

GET BOOK

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.

Author : Remi Carles
Publisher : World Scientific
Page : 256 pages
File Size : 23,50 MB
Release : 2008-03-04
Category : Mathematics
ISBN : 9814471747

GET BOOK

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.

Author : Christopher W. Curtis
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 46,47 MB
Release : 2015-03-26
Category : Nonlinear wave equations
ISBN : 1470410508

GET BOOK

This volume contains the proceedings of the AMS Special Session on Nonlinear Waves and Integrable Systems, held on April 13-14, 2013, at the University of Colorado, Boulder, Colorado. The field of nonlinear waves is an exciting area of modern mathematical research that also plays a major role in many application areas from physics and fluids. The articles in this volume present a diverse cross section of topics from this field including work on the Inverse Scattering Transform, scattering theory, inverse problems, numerical methods for dispersive wave equations, and analytic and computational methods for free boundary problems. Significant attention to applications is also given throughout the articles with an extensive presentation on new results in the free surface problem in fluids. This volume will be useful to students and researchers interested in learning current techniques in studying nonlinear dispersive systems from both the integrable systems and computational points of view.

Author : Nalan Antar
Publisher : BoD – Books on Demand
Page : 164 pages
File Size : 42,16 MB
Release : 2022-07-06
Category : Mathematics
ISBN : 1839699787

GET BOOK

The nonlinear Schrödinger equation is a prototypical dispersive nonlinear partial differential equation that has been derived in many areas of physics and analyzed mathematically for many years. With this book, we aim to capture different perspectives of researchers on the nonlinear Schrödinger equation arising from theoretical, numerical, and experimental aspects. The eight chapters cover a variety of topics related to nonlinear optics, quantum mechanics, and physics. This book provides scientists, researchers, and engineers as well as graduate and post-graduate students working on or interested in the nonlinear Schrödinger equation with an in-depth discussion of the latest advances in nonlinear optics and quantum physics.

Author :
Publisher : ScholarlyEditions
Page : 1326 pages
File Size : 40,91 MB
Release : 2012-01-09
Category : Mathematics
ISBN : 1464964920

GET BOOK

Issues in General and Specialized Mathematics Research: 2011 Edition is a ScholarlyEditions™ eBook that delivers timely, authoritative, and comprehensive information about General and Specialized Mathematics Research. The editors have built Issues in General and Specialized Mathematics Research: 2011 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about General and Specialized Mathematics Research in this eBook to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in General and Specialized Mathematics Research: 2011 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.