Onset of transverse instabilities of confined dark solitons

We investigate propagating dark soliton solutions of the two-dimensional defocusing nonlinear Schrödinger or Gross-Pitaevskii (NLS-GP) equation that are transversely confined to propagate in an infinitely long channel. Families of single, vortex, and multilobed solitons are computed using a spectrally accurate numerical scheme. The multilobed solitons are unstable to small transverse perturbations. However, the single-lobed solitons are stable if they are sufficiently confined along the transverse direction, which explains their effective one-dimensional dynamics. The emergence of a transverse modulational instability is characterized in terms of a spectral bifurcation. The critical confinement width for this bifurcation is found to coincide with the existence of a propagating vortex solution and the onset of a “snaking” instability in the dark soliton dynamics that, in turn, give rise to vortex or multivortex excitations. These results shed light on the superfluidic hydrodynamics of dispersive shock waves in Bose-Einstein condensates and nonlinear optics.

Figure 1

Normalized background states [Eq. (15)] and their salient features. (a) Multiple transverse profiles for different widths $L_y$. (b) Elliptic parameter $m$ [Eq. (16)]. (c) Frequency $\mu$ [Eq. (15)]. (d) Maximum approximate CDS speed [Eq. (25)].

Figure 2

Comparison of approximate (dashed) and exact (solid) CDS cross sections for $c=0.5$. (a),(b) Longitudinal cross sections along $y=0$ with transverse confinement widths $L_y=4$ and $L_y=8$, respectively. (c),(d) Transverse cross sections along $\xi =0$ with transverse confinement widths $L_y=4$ and $L_y=8$, respectively.

Figure 3

Maximum absolute error in density ${\left|\psi \right|}^2$ between approximate and numerical CDSs. (a) Fixed $c=0.25$, variable width $L_y$. (b) Fixed $L_y=6$, variable speed $c$. The dotted vertical line separates unstable and stable CDS solutions (see Sec. 5).

Figure 4

Dependence of the phase jump across the CDS solution as (a) the confinement width $L_y$ is varied with fixed CDS speed $c=0.25$ and (b) as the speed $c$ is varied with fixed width $L_y=6$. These plots show deviation from the approximate CDS phase jump (dashed) for large $L_y$. The dotted vertical line separates unstable and stable CDS solutions (see Sec. 5).

Figure 5

Cross sections along (a) the longitudinal direction for $y=0$ and (b) along the transverse direction with $\xi =0$ of single-lobed (solid black), double-lobed (dashed red), and triple-lobed (dash-dotted green) CDSs with propagation speed $c=0.5$ and a range of confinement widths $L_y$.

Figure 6

Phase diagram depicting the conserved quantity $\mathcal{N}$ and contour plots of one, two, and three-lobed channel CDS and solitonic vortex solutions with fixed $c=0.25$ as $L_y$ is varied. The dotted vertical line corresponds to the critical confinement width $L_{\mathrm{cr}}\approx 5.68$ below which CDS solutions are stable (see Sec. 5). The critical width corresponds to the minimal $L_y$ such that the solitonic vortex exists. The vector fields overlaying some contour plots represent the phase gradient $\mathbf{\nabla }(arg\psi )$, a hydrodynamic velocity.

Figure 7

Phase diagram depicting the conserved quantity $\mathcal{N}$ and contour plots of one-lobed CDS and vortex solutions with variable speeds $c$ and several confinement widths $L_y$. Filled circles correspond to the crossover from unstable (dashed curves) to stable (solid curves) CDS solutions (see Sec. 5). The vector fields overlaying some contour plots represent the phase gradient $\mathbf{\nabla }(arg\psi )$, a hydrodynamic velocity.

Figure 8

A portion around the complex origin of the linearized spectra of a CDS with speed $c=0.5$ and confinement widths (a) $L_y=6.4$ and (b) $L_y=6.6$. Two stable (purely imaginary) eigenvalues [denoted by $\odot$ in (a) and almost indistinguishable] coalesce at the origin and emerge as unstable (real) eigenvalues [denoted by $\times$ in (b)]. Contour plots of the intensity (c) and phase (d) of one of the unstable eigenfunctions.

Figure 9

Critical confinement widths as functions of propagation speed, $L_{\mathrm{cr}}\left(c\right)$, for line dark solitons with constant-flux walls (dashed) and CDSs with impenetrable walls (solid). For $L_y>L_{\mathrm{cr}}\left(c\right)$ the solutions are unstable.

Figure 10

Stable and unstable dynamics for initially perturbed one-lobed CDS solutions of speed $c=0.25$ evolved according to NLS equation (1). (a) Stable dynamics for $L_y=5$. (b) Unstable dynamics leading to a single vortex for $L_y=6$. (c) One-lobe CDS decay into two vortices resulting from the wider channel $L_y=10$. The mean zero noise variance is $\sigma ={10}^{-4}$. Note that $L_{\mathrm{cr}}\left(0.25\right)\approx 5.68$ so the instability transition is accurately resolved. The vector fields overlaying some contour plots represent the phase gradient $\mathbf{\nabla }(arg\psi )$, a hydrodynamic velocity.

Figure 11

Typical convergence of Newton calculations with $c=0.5, L_y=7$ with varying grid spacing (${\mathrm{\Delta }}_{\xi ,y}$ in the legend) and longitudinal length $L_x$.