Nonlinear Schrödinger equation

A nonlinear partial differential equation for a complex wave envelope, combining dispersion with an amplitude-dependent phase or focusing effect. Its forms describe phenomena including optical pulses, Bose–Einstein condensates, and water waves.

Dispersion: The dependence of a wave’s speed or propagation constant on its frequency or wavenumber. Dispersion spreads wave packets and competes with the equation’s nonlinear tendency to concentrate them.
Partial differential equation: An equation relating an unknown function to its partial derivatives with respect to multiple variables. The nonlinear Schrödinger equation evolves a field across space and time using partial derivatives.
Nonlinear optics: The study of how light interacts with materials whose response depends on the light field. The equation models slowly varying optical pulse envelopes in nonlinear media.
Erwin Schrödinger: A physicist who formulated the wave equation central to nonrelativistic quantum mechanics. His linear equation supplied the mathematical form later extended with nonlinear interactions.
Defocusing nonlinear Schrödinger equation: The nonlinear Schrödinger equation variant in which nonlinearity opposes wave concentration. It supports dark solitons and avoids the focusing equation’s modulational instability of a uniform background.
Self-phase modulation: An intensity-dependent change in a wave’s phase as it propagates through a nonlinear medium. It supplies the nonlinear phase shift in the standard optical equation.
Complex amplitude: A complex-valued quantity encoding the magnitude and phase of an oscillating signal. The equation evolves this envelope rather than the rapidly oscillating physical carrier wave.
Optical fiber: A flexible waveguide that transmits light through a dielectric core by guiding optical modes. Pulse propagation in fibers is a principal setting for the equation’s dispersion–nonlinearity balance.
Gross–Pitaevskii theory: A mean-field theory of dilute interacting Bose gases described by a nonlinear wavefunction equation. It established a major physical setting for nonlinear Schrödinger-type dynamics.
Korteweg–de Vries equation: An integrable nonlinear equation describing long, unidirectional dispersive waves in shallow water and other media. It produces solitons like the nonlinear Schrödinger equation but evolves a real field with different dispersion.

Linked from 5 pages