Adiabatic Invariant Approach to Transverse Instability: Landau Dynamics of Soliton Filaments

Consider a lower-dimensional solitonic structure embedded in a higher-dimensional space, e.g., a 1D dark soliton embedded in 2D space, a ring dark soliton in 2D space, a spherical shell soliton in 3D space, etc. By extending the Landau dynamics approach [Phys. Rev. Lett. 93, 240403 (2004)], we show that it is possible to capture the transverse dynamical modes (the “Kelvin modes”) of the undulation of this “soliton filament” within the higher-dimensional space. These are the transverse stability or instability modes and are the ones potentially responsible for the breakup of the soliton into structures such as vortices, vortex rings, etc. We present the theory and case examples in 2D and 3D, corroborating the results by numerical stability and dynamical computations.

Figure 1

Imaginary (left) and real (right) parts of the BdG excitation spectrum for the RDS. The former correspond to the instability modes, while the latter to the vibration modes. The thin horizontal solid lines in both panels correspond to undulation modes of the RDS in the Thomas-Fermi (large nonlinearity or large chemical potential $\mu$) limit, as predicted by Eq. (11). The thin horizontal dashed lines in the right panel correspond to the asymptotic predictions for the ground state spectrum for $k=0$ and $0\le \ell \le 9$ [cf. Eq. (12)]. To avoid clogging the relevant diagram, only the lowest modes of instability are shown.

Figure 2

Comparison of the full NLS dynamics (1) and the adiabatic invariant PDE obtained from Hamilton’s principle applied to Eq. (8) for a RDS for $\mu =24$. Top series of panels: RDS with initial azimuthal position given by $R(\theta )=2.8+0.1\cos (\theta )+3\times {10}^{-7}\cos (8\theta )$. Bottom series of panels: Another RDS, but now with initial azimuthal position given by $R(\theta )=2.4+0.1\cos (2\theta )$. In both series of panels, the top (bottom) subpanels depict the density (phase) of the solution on a $6\times 6$ square. The dynamics of the ensuing PDE is depicted by a solid green line. Note how the derived PDE closely follows the RDS dynamics before the breakup of the latter into a series of vortex pairs.

Figure 3

The BdG excitation spectrum for the dark spherical shell soliton. The layout and meaning are the same as in the case of the ring dark soliton of Fig. 1. The thin horizontal solid lines correspond to the undulation modes in the TF limit, as predicted by Eq. (17), and the thin horizontal dashed lines to the asymptotic prediction for the ground state for $k=0$ and $0\le \ell \le 9$ [cf. Eq. (18)]. Once again, only the dominant instability modes are shown.