Exact solutions to the four Goldstone modes around a dark soliton of the nonlinear Schrödinger equation

@article{Sykes2011ExactST,
  title={Exact solutions to the four Goldstone modes around a dark soliton of the nonlinear Schr{\"o}dinger equation},
  author={Andrew G. Sykes},
  journal={Journal of Physics A},
  year={2011},
  volume={44},
  pages={135206}
}
  • A. Sykes
  • Published 2 February 2011
  • Physics, Mathematics
  • Journal of Physics A
This paper is concerned with the linearization around a dark soliton solution of the nonlinear Schr?dinger equation. Crucially, we present analytic expressions for the four linearly independent zero eigenvalue solutions (also known as Goldstone modes) to the linearized problem. These solutions are then used to construct a Green matrix which gives the first-order spatial response due to some perturbation. Finally, we apply this Green matrix to find the correction to the dark-soliton wavefunction… 
3 Citations

Figures from this paper

Spatial shifts of colliding dark solitons in deformed non-linear Schrödinger models
We derive a closed expression for the spatial shift experienced by a black soliton colliding with a shallow dark soliton in the context of deformed nonlinear Schrodinger (NLS) models. A perturbative
Scattering of a dark–bright soliton by an impurity
  • M. Alotaibi, L. Carr
  • Physics
    Journal of Physics B: Atomic, Molecular and Optical Physics
  • 2019
We study the dynamics of a dark-bright soliton interacting with a fixed impurity using a mean-field approach. The system is described by a vector nonlinear Schrodinger equation (NLSE) appropriate to
Quantum squeezing of slow-light dark solitons via electromagnetically induced transparency
We consider the quantum effect of slow light dark soliton (SLDS) in a cold atomic gas with defocuing Kerr nonlinearity via electromagnetically induced transparency (EIT). We calculate the quantum

References

SHOWING 1-10 OF 46 REFERENCES
Direct perturbation theory for the dark soliton solution to the nonlinear Schrödinger equation with normal dispersion.
TLDR
The perturbation theory for the dark soliton solution is constructed by linear Green's function theory and in application to the self-induced Raman scattering, the adiabatic corrections to the soliton's parameters are obtained.
A perturbation method for dark solitons based on a complete set of the squared Jost solutions
A perturbation method for dark solitons is developed, which is based on the construction and the rigorous proof of the complete set of squared Jost solutions. The general procedure solving the
Dark Multi-Soliton Solution of the Nonlinear Schrödinger Equation with Non-Vanishing Boundary
The inverse scattering transform for the nonlinear Schrödinger equation in normal dispersion with non-vanishing boundary values is re-examined using an affine parameter to avoid double-valued
Direct perturbation theory for dark solitons.
  • Konotop, Vekslerchik
  • Physics, Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1994
TLDR
Orthogonality and closure of the basis of squared Jost functions are proved and the general form of a correction to the one-soliton solution up to first order is obtained.
A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions
Because of the essential difficulty caused by the non-vanishing boundary condition, a systematic perturbation approach for dark solitons has not yet been found. Based on a rigorous proof of the
A direct approach to studying soliton perturbations
Starting with an integrable nonlinear evolution equation, the author investigates perturbations about a one-soliton solution, through the inversion of a linear equation for the first-order correction
Quantum dark soliton: Nonperturbative diffusion of phase and position (6 pages)
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
Quantum decay of dark solitons.
TLDR
At low enough temperatures T background fluctuations should be considered as being quantized which enables us to calculate finite lifetime of the solitons τ∼T(-4), and it is found that the coherent nature of the quantum fluctuations leads to long-range interactions between the soliton mediated by the superradiation.
Foundation of direct perturbation method for dark solitons
The foundation of the direct perturbation theory for solitons is a complete set of the squared Jost functions. With a suitable definition of the adjoint functions and inner products which yields
Quantum and thermal effects of dark solitons in a one-dimensional Bose gas.
TLDR
It is found that the phase fluctuations lower the classically predicted soliton speed and seed instabilities.
...
1
2
3
4
5
...