The nonlinear Schrödinger equation : singular solutions and optical collapse
著者
書誌事項
The nonlinear Schrödinger equation : singular solutions and optical collapse
（Applied mathematical sciences, v. 192）
Springer, c2015
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注記
Includes bibliographical references (p. 847-858) and index
内容説明・目次
内容説明
This book is an interdisciplinary introduction to optical collapse of laser beams, which is modelled by singular (blow-up) solutions of the nonlinear Schroedinger equation. With great care and detail, it develops the subject including the mathematical and physical background and the history of the subject. It combines rigorous analysis, asymptotic analysis, informal arguments, numerical simulations, physical modelling, and physical experiments. It repeatedly emphasizes the relations between these approaches, and the intuition behind the results.
The Nonlinear Schroedinger Equation will be useful to graduate students and researchers in applied mathematics who are interested in singular solutions of partial differential equations, nonlinear optics and nonlinear waves, and to graduate students and researchers in physics and engineering who are interested in nonlinear optics and Bose-Einstein condensates. It can be used for courses on partial differential equations, nonlinear waves, and nonlinear optics.
Gadi Fibich is a Professor of Applied Mathematics at Tel Aviv University.
"This book provides a clear presentation of the nonlinear Schrodinger equation and its applications from various perspectives (rigorous analysis, informal analysis, and physics). It will be extremely useful for students and researchers who enter this field."
Frank Merle, Universite de Cergy-Pontoise and Institut des Hautes Etudes Scientifiques, France
目次
Derivation of the NLS.- Linear propagation.- Early self-focusing research.- NLS models.- Existence of NLS solutions.- Solitary waves.- Variance identity.- Symmetries and the lens transformation.- Stability of solitary waves.- The explicit critical singular peak-type solution.- The explicit critical singular ring-type solution.- The explicit supercritical singular peak-type solution.- Blowup rate, blowup profile, and power concentration.- The peak-type blowup profile.- Vortex solutions.- NLS on a bounded domain.- Derivation of reduced equations.- Loglog law and adiabatic collapse.- Singular H1 ring-type solutions.- Singular H1 vortex solutions.- Singular H1 peak-type solutions.- Singular standing-ring solutions.- Singular shrinking-ring solutions.- Critical and threshold powers for collapse.- Multiple filamentation.- Nonlinear Geometrical Optics (NGO) method.- Location of singularity.- Computation of solitary waves.- Numerical methods for the NLS.- Effects of spatial discretization.- Modulation theory.- Cubic-quintic and saturated nonlinearities.- Linear and nonlinear damping.- Nonparaxiality and backscattering (nonlinear Helmholtz equation).- Ultrashort pulses.- Normal and anomalous dispersion.- NGO method for ultrashort pulses with anomalous dispersion.- Continuations beyond the singularity.- Loss of phase and chaotic interactions.
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