BookDOI
The nonlinear Schrödinger equation : self-focusing and wave collapse
TLDR
In this article, the authors present a basic framework to understand structural properties and long-time behavior of standing wave solutions and their relationship to a mean field generation and acoustic wave coupling.Abstract:
Basic Framework.- The Physical Context.- Structural Properties.- Rigorous Theory.- Existence and Long-Time Behavior.- Standing Wave Solutions.- Blowup Solutions.- Asymptotic Analysis near Collapse.- Numerical Observations.- Supercritical Collapse.- Critical Collapse.- Perturbations of Focusing NLS.- Coupling to a Mean Field.- Mean Field Generation.- Gravity-Capillary Surface Waves.- The Davey-Stewartson System.- Coupling to Acoustic Waves.- Langmuir Oscillations.- The Scalar Model.- Progressive Waves in Plasmas.read more
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MonographDOI
Nonlinear dispersive equations : local and global analysis
TL;DR: In this paper, the Korteweg de Vries equation was used for ground state construction in the context of semilinear dispersive equations and wave maps from harmonic analysis.
Journal ArticleDOI
Nonlinear waves in PT -symmetric systems
TL;DR: The concept of parity-time symmetric systems is rooted in non-Hermitian quantum mechanics where complex potentials obeying this symmetry could exhibit real spectra as discussed by the authors, which has applications in many fields of physics, e.g., in optics, metamaterials, acoustics, Bose-Einstein condensation, electronic circuitry, etc.
Journal ArticleDOI
Ultrashort filaments of light in weakly ionized, optically transparent media
TL;DR: In this article, the authors present the landmarks of the 10-odd-year progress in this field, focusing on the theoretical modeling of the propagation equations, whose physical ingredients are discussed from numerical simulations.
Journal ArticleDOI
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations
TL;DR: In this paper, the authors study defocusing analogues of these equations, namely defocusing nonlinear Schrodinger, defocusing modified Korteweg-de Vries (mKdV), and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable.
Journal ArticleDOI
Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R^3
TL;DR: In this article, the authors obtained global well-posedness, scattering, and global L 10 spacetime bounds for energy-class solutions to the quintic defocusing Schrodinger equa- tion in R 1+3, which is energy-critical.