The Nonlinear Schrödinger Equation : Singular Solutions and Optical Collapse / by Gadi Fibich.

Format:

Book

Author/Creator:

Fibich, Gadi, Author.

Series:

Applied Mathematical Sciences, 0066-5452 ; 192

Language:

English

Subjects (All):

Differential equations, Partial.

Atoms.

Physics.

Optics.

Electrodynamics.

Statistical physics.

Partial Differential Equations.

Atomic, Molecular, Optical and Plasma Physics.

Classical Electrodynamics.

Applications of Nonlinear Dynamics and Chaos Theory.

Local Subjects:

Partial Differential Equations.

Atomic, Molecular, Optical and Plasma Physics.

Classical Electrodynamics.

Applications of Nonlinear Dynamics and Chaos Theory.

Physical Description:

1 online resource (XXXI, 862 p. 202 illus., 100 illus. in color.)

Edition:

1st ed. 2015.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2015.

Language Note:

English

Summary:

This book is an interdisciplinary introduction to optical collapse of laser beams, which is modelled by singular (blow-up) solutions of the nonlinear Schrödinger 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 Schrödinger 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, Université de Cergy-Pontoise and Institut des Hautes Études Scientifiques, France.

Contents:

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.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Description based on publisher supplied metadata and other sources.

ISBN:

3-319-12748-9

OCLC:

904797130