[PDF] Semiclassical Asymptotics Of The Focusing Nonlinear Schrodinger Equation For Square Barrier Initial Data eBook

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Page : pages
File Size : 41,55 MB
Release : 2002
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Semiclassical asymptotics of the focusing nonlinear Schrodinger equation for square barrier initial data.

Author : Robert M. Jenkins
Publisher :
Page : 314 pages
File Size : 28,70 MB
Release : 2009
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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 : Spyridon Kamvissis
Publisher : Princeton University Press
Page : 281 pages
File Size : 48,17 MB
Release : 2003-09-07
Category : Mathematics
ISBN : 069111482X

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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 : Sergey M. Belov
Publisher :
Page : 320 pages
File Size : 42,84 MB
Release : 2008
Category : Gross-Pitaevskii equations
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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 : Michael Borghese
Publisher :
Page : pages
File Size : 41,97 MB
Release : 2017
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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 → ∞.

Author : Alexander Altland
Publisher : Cambridge University Press
Page : 785 pages
File Size : 21,93 MB
Release : 2010-03-11
Category : Science
ISBN : 0521769752

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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.

Author : Vassili N. Kolokoltsov
Publisher : Cambridge University Press
Page : 394 pages
File Size : 36,6 MB
Release : 2010-07-15
Category : Mathematics
ISBN : 1139489739

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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.