The Gross–Pitaevskii equation[1][2] is a nonlinear model equation for the order parameter or wavefunction of a Bose–Einstein condensate. It is similar in form to the Ginzburg–Landau equation and is sometimes referred to as a nonlinear Schrödinger equation.
A Bose–Einstein condensate (BEC) is a gas of bosons that are in the same quantum state, and thus can be described by the same wavefunction. A free quantum particle is described by a single-particle Schrödinger equation. Interaction between particles in a real gas is taken into account by a pertinent many-body Schrödinger equation. If the average spacing between the particles in a gas is greater than the scattering length (that is, in the so-called dilute limit), then one can approximate the true interaction potential that features in this equation by a pseudopotential. The non-linearity of the Gross–Pitaevskii equation has its origin in the interaction between the particles. This becomes evident by equating the coupling constant of interaction in the Gross–Pitaevskii equation with zero (see the following section), on which the single-particle Schrödinger equation describing a particle inside a trapping potential is recovered.
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