The Nonlinear Schrödinger Equation

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.

Gadi Fibich is a Professor of Applied Mathematics at Tel Aviv University.

"The Nonlinear Schrödinger Equation (NLS) theory was an object of great interest during last decades. ... the present book includes almost all questions connected with theoretical and experimental investigations of the above mentioned matter during the years since 1960 until now. ... the book abounds in recent results, facts and examples that makes it very interesting for the researchers who work actively in this field." (Dimitar A. Kolev, zbMATH, 1351.35001, 2017)

"This monograph is devoted to the analysis of optical collapse modelled by the nonlinear Schrödinger (NLS) equation. ... The book contains everything a modern reader wants to learn about the NLS equation. ... The book is a nice addition to the existing literature on the subject of the NLS equation. New and experienced researchers alike may use this text to get the latest informationabout the state-of-the-art in the field." (Dmitry E. Pelinovsky, Mathematical Reviews, April, 2016)