By Ryogo Kubo

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**Example text**

95) In various subfields of physics, this equation is known under the generic name of nonlinear Schr¨odinger equation (NLSE). In the theory of quantum gases—closest to the subject of this book—this equation is called the time-dependent Gross–Pitaevskii equation (GPE), with the reservation that GPE is traditionally associated with the Bose-Einstein condensate (BEC), and thus superfluidity. 95) do feature superfluidity, there is a normal-fluid regime as well. 11) is conserved by GPE. For a given N , there is a ground state, ψ0 , corresponding to the minimal possible energy.

R0 →ri . 122). ∗ That is, the freedom of replacing a → a + ∇f , where f is an arbitrary smooth scalar field; for ∇·a 0, choose f such that Δf = −∇ · a. 119) is∗ Evort = w(r1 ) · w(r2 ) 3 d r1 d 3 r2 . 119). 72)], associated with the flow pattern. 130). This brings us to Mi Pvort = πn0 ri × dli . 72). In two dimensions, an extra simplification arises due to the analogy with 2D electrostatics. 121), the Laplace equation Δu = 0, everywhere except for the vortex centers, at which u is singular and zˆ is a unit 3D vector perpendicular to the xy-plane in which the vector r lives.

By the same argument, vortices in 3D are lines with no free ends in the bulk: They either form closed loops or terminate at the box boundaries. Vortex defects in 3D are often referred to as vortex lines or filaments. In 2D, elementary vortex defects have point singularities. 113) where ϕ is the azimuthal angle and ρ is the distance from the vortex line. The solution is not diverging at ρ = 0, implying n(ρ) ∝ ρ2 at ρ → 0. 114) By dimensional argument, it is clear that the size of the core—the region where the density n is significantly diﬀerent from its asymptotic uniform value n0 —is on the order of the healing length l0 .

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