137 research outputs found
Dynamics of overlapping vortices in complex scalar fields
We investigate dynamics of overlapping vortices in the nonlinear
Schr\"{o}dinger equation, the nonlinear heat equation and in the equation with
an intermediate Schr\"{o}dinger-diffusion dynamics. Because of formal
similarity on a perturbative level we discuss also the nonlinear wave equation
(Goldstone model). Special solutions are found like vortex helices,
double-helices and braids, breather states and vortex mouths. A pair of
vortices in the Goldstone model scatters by the right angle in the head-on
collision. It is found that in a dissipative system there is a characteristic
lenght scale above which vortices can be entangled but below which the
entanglement is unstable.Comment: detailed analysis of the continuation in the complex time plane;
latest revision: the paper is made more readable. 9 pages in Late
Quantum Dark Soliton: Non-Perturbative Diffusion of Phase and Position
The dark soliton solution of the Gross-Pitaevskii equation in one dimension
has two parameters that do not change the energy of the solution: the global
phase of the condensate wave function and the position of the soliton. These
degeneracies appear in the Bogoliubov theory as Bogoliubov modes with zero
frequencies and zero norms. These ``zero modes'' cannot be quantized as the
usual Bogoliubov quasiparticle harmonic oscillators. They must be treated in a
non-perturbative way. In this paper I develop non-perturbative theory of zero
modes. This theory provides non-perturbative description of quantum phase
diffusion and quantum diffusion of soliton position. An initially well
localized wave packet for soliton position is predicted to disperse beyond the
width of the soliton.Comment: 6 page
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