Gross-Pitaevskii 1D — notch defect propagating through a condensate
A dark soliton in a Bose-Einstein condensate (BEC) is a density notch (dip to zero for a stationary soliton) accompanied by a phase jump of π. Governed by the 1D Gross-Pitaevskii equation (nonlinear Schrödinger): iℏ ∂ψ/∂t = −(ℏ²/2m)∂²ψ/∂x² + g|ψ|²ψ. The exact soliton solution: ψ(x,t) = n₀[iv/c_s + √(1−v²/c_s²) tanh(√(1−v²/c_s²)(x−vt)/ξ)].