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On the Gross-Pitaevskii evolution linearized around the degree-one vortex
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abstract
We study the evolution of the Gross-Pitaevskii equation linearized around the Ginzburg-Landau vortex of degree one under equivariant symmetry. Among the main results of this work, we determine the spectrum of the linearized operator, uncover a remarkable $L^2$-norm growth phenomenon related to a zero-energy resonance, and provide a complete construction of the distorted Fourier transform at small energies. The latter hinges upon a meticulous analysis of the behavior of the resolvent in the upper and lower half-planes in a small disk around zero-energy.
Forward citations
Cited by 3 Pith papers
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Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model
Small equivariant perturbations of the degree-one vortex decay with radiation rate, while the internal mode damps like epsilon squared over one plus Gamma epsilon squared t, proving asymptotic stability.
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Vortex dynamics for the Gross-Pitaevskii equation
For the planar Gross-Pitaevskii equation, explicit n-vortex solutions are constructed whose dynamics follow the Helmholtz-Kirchhoff system at leading order, with the first correction given by a linear wave equation.
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Nondegeneracy and Morse Index of Ginzburg--Landau Vortices
For Ginzburg-Landau vortices of degree 2 and 3, the only bounded zero modes are the three geometric symmetries, and the Morse indices are 2 and 6.
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