Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrodinger Equation (eBook)

Spyridon Kamvissis, Kenneth D.T-R McLaughlin, Peter D. Miller (Autoren)

eBook Download: PDF

2003
312 Seiten
Princeton University Press (Verlag)
978-1-4008-3718-2 (ISBN)

Infos

Spyridon Kamvissis is a researcher at the Max Planck Institute of Mathematics in Bonn, Germany, and a Professor of Mathematics at the National Technical University in Athens, Greece. Kenneth D. T-R McLaughlin is Associate Professor of Mathematics at the University of North Carolina, Chapel Hill. Peter D. Miller is Assistant Professor of Mathematics at the University of Michigan, Ann Arbor.

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This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrodinger 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 Holder 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.

Produkt-Details

• Erscheint lt. Verlag: 18.8.2003
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 50 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Technik
• Schlagworte: Abelian integral • analytic continuation • analytic function • Ansatz • Approximation • Asymptote • asymptotic analysis • Asymptotic Distribution • asymptotic expansion • Banach Algebra • Basis (linear algebra) • Boundary (topology) • Boundary value problem • bounded operator • Calculation • Cauchy–Riemann equations • Cauchy's Integral Formula • Cauchy's Integral Theorem • Cauchy's theorem (geometry) • Change of variables • coefficient • complex plane • Cramer's Rule • Degeneracy (mathematics) • Derivative • Diagram (category theory) • Differentiable function • differential equation • Differential operator • Dirac equation • Disjoint union • Divisor • Eigenfunction • Eigenvalues and Eigenvectors • Elliptic Integral • Energy Minimization • Equation • Euler–Lagrange equation • Euler's formula • existential quantification • Explicit formulae (L-function) • Fourier transform • Fredholm Theory • Function (mathematics) • Gauge Theory • Heteroclinic orbit • Hilbert transform • Identity matrix • Implicit function • implicit function theorem • Infimum and supremum • Initial Value Problem • integrable system • Integral curve • integral equation • inverse problem • Jacobian matrix and determinant • Kerr effect • Laurent Series • Limit point • linear equation • Linear space (geometry) • Line (geometry) • Logarithmic derivative • Lp space • Minor (linear algebra) • Monotonic Function • Neumann series • Normalization property (abstract rewriting) • Numerical Integration • ordinary differential equation • orthogonal polynomials • Parameter • Parametrix • paraxial approximation • Parity (mathematics) • partial derivative • partial differential equation • perturbation theory • Perturbation theory (quantum mechanics) • Pole (complex analysis) • polynomial • probability measure • Quadratic differential • quadratic programming • Radon–Nikodym theorem • Reflection coefficient • Riemann surface • Simultaneous Equations • Sobolev Space • Soliton • Special case • Taylor series • Theorem • theory • Trace (linear algebra) • Upper half-plane • Variational method (quantum mechanics) • Variational Principle • WKB approximation
• ISBN-10: 1-4008-3718-9 / 1400837189
• ISBN-13: 978-1-4008-3718-2 / 9781400837182