Cubic NLS as a nonlinear dispersive wave model

PDF p. 1: §1.1, paragraph following (1.1)

The cubic nonlinear Schrödinger equation $i{u}_t+\mathrm{\Delta }u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{2}u=0$, with real nonzero $\lambda$, is adopted as a model for propagation of nonlinear dispersive waves. Here a wave is represented by a complex-valued amplitude $u(x,t)$; nonlinear means that the equation contains the cubic term $\mathrm{\mid }u{\mathrm{\mid }}^{2}u$, and dispersive refers to propagation in which different spatial frequencies can evolve at different rates. The article names nonlinear optics, the study of light interacting with matter in regimes with nonlinear response, and Bose–Einstein condensation, a regime in which a macroscopic population occupies a common quantum state, as applications.

This is an adopted model interpretation. It does not assert that the boundary controls in the mathematical formulation have been physically realized, that a particular experiment measured this equation, or that the equation applies without approximation to every optical or condensate system.

Cubic NLS and boundary control formulations

The mathematical formulation Cubic NLS and boundary control formulations specifies the equation and its Dirichlet and Neumann boundary-control alternatives. It does not itself assert their physical realization.

As independent model context, Pitaevskii's zero-order condensate amplitude $a_0$ satisfies $i\mathrm{\hslash }{\mathrm{\partial }}_t{a}_0=-({\mathrm{\hslash }}^2\mathrm{/}2m)\mathrm{\Delta }{a}_0+g\mathrm{\mid }{a}_0{\mathrm{\mid }}^2{a}_0$ in the source's approximation. Here $\mathrm{\hslash }$ is the reduced Planck constant, $m$ the particle mass, and $g$ the effective interaction coefficient; the condensate amplitude is the mean-field amplitude for the condensed component. The stated context is absolute zero and a rarefied gas with weak short-range repulsion, small noncondensate correction and effective scattering amplitude small compared with interparticle spacing. Small operator corrections are discarded. This input does not establish the article's obstacle geometry or its boundary controls.

The frozen article is cited for attribution of its introductory interpretation. The Pitaevskii reference supplies the separately qualified condensate example, not a derivation of the article's controlled model. The ideal periodic Bose-gas result in the physics library and the library's autonomous linear Schrödinger flow concern different hypotheses and do not supply this nonlinear controlled model.