Bogoliubov–de Gennes theory of the snake instability of gray solitons in higher dimensions

Gray solitons are a one-parameter family of solutions to the one-dimensional nonlinear Schrödinger equation (NLSE) with positive cubic nonlinearity, as found in repulsively interacting dilute Bose-Einstein condensates or electromagnetic waves in the visible spectrum in waveguides described by Gross-Pitaevskii mean-field theory. In two dimensions, these solutions to the NLSE appear as a line or plane of depressed condensate density or light intensity, but numerical solutions show that this line is dynamically unstable to “snaking”: the initially straight line of density or intensity minimum undulates with exponentially growing amplitude. To assist future studies of quantum mechanical instability beyond mean-field theory, we here pursue an approximate analytical description of the snake instability within Bogoliubov–de Gennes perturbation theory. Within this linear approximation the two-dimensional result applies trivially to three dimensions as well, describing buckling modes of the low-density plane. We extend the analytical results of Kuznetsov and Turitsyn [Sov. Phys. JETP 67, 1583 (1988)] to shorter wavelengths of the snake modulation and show to what extent the snake mode can be described accurately as a parametric instability, in which the position and grayness parameter of the initial soliton simply become dependent on the transverse dimension(s). We find that the parametric picture remains accurate up to second order in the snaking wave number, if the snaking soliton is also dressed by an outward-propagating sound wave, but that beyond second order in the snaking wave number the parametric description breaks down.

Figure 1

Snake mode. Gross-Pitaevskii evolution of condensate order parameter modulus $\left|\mathrm{\Psi }\right|$ (upper plots) and phase $arg\left(\mathrm{\Psi }\right)$ (lower plots) at different times (indicated in units of the inverse chemical potential $\hslash /\mu$ by the dimensionless parameter $\tau$) from an initial gray soliton with a perturbation that is initially too small to be seen. The well-known snake instability makes the initial density trough undulate like a crawling snake. Ultimately quantized vortices and anti-vortices appear, as seen in the lower right frame, where there are four points around which the phase sweeps through the full color range representing 0 to $2\pi$, respectively being a singularity of the phase. The spatial axes $x$ and $y$ are in units of the condensate healing length, as explained in Sec. 2.

Figure 2

Growth rate $\lambda /k_{\text{max}}$ vs $k/k_{\text{max}}$ plot. The plot shows the numerical calculation (solid, blue line) and the linear approximation (dashed, red line, having slope of $\frac{\kappa }{\sqrt{3}}$) of $\lambda \left(k\right)$ normalized by $k_{\text{max}}$. The linear approximation agrees with the numerical result for small $k$ but deviates quickly for larger $k$.

Figure 3

BdG instability growth rates $\lambda \left(k\right)$. This plot shows the growth rates $\lambda \left(k\right)$ for finite $k$, for different values of $\beta$, scaled depending on $\beta$ in order to show agreement with analytical results. The dashed upper line represents the first order approximation and the dashed line below - the second order. The solid lines represent different $\beta$ values. The $\beta$ values correspond to the solid lines from top to bottom as follows: $\beta =0.8$, 0.7, 0.6, 0.5, and 0.4, respectively.

Figure 4

Growth rate $\lambda /k_{\text{max}}$ vs $k/k_{\text{max}}$ plot for different values of $\beta$. The plot shows the BdG instability growth rate $\lambda \left(k\right)$ normalized by the maximal value $k_{\text{max}}$ for three values of $\beta$, namely $\beta =0.4$, 0.6, and 0.8, respectively. The numerical result is represented by the solid line. Dashed lines represent the first, second, and third order of approximation, in that order from top to bottom. (a) $\beta =0.4$. (b) $\beta =0.6$. (c) $\beta =0.8$.

Figure 5

Comparison of ${\varphi }_k\left(x\right)$ The plot shows pairwise the real (on the left) and the imaginary part (on the right) of ${\varphi }_k\left(x\right)$, for the indicated different values of $k$. In all cases the grayness parameter of the soliton is $\beta =0.5$. The solid blue line shows the numerical result and the dotted red line the analytical approximation up to second order in $k/\kappa$. All solutions go to zero for large of $\left|x\right|$, but at $k$-dependent rates, and so optimizing the horizontal range to show most detail has made the horizontal range in the right panel of (a) wider than for the other five panels. The insets in the real-part plots show the numerical and analytically approximate curves of $\lambda \left(k\right)$, with a vertical gray line marking the position of $k$ for this panel. The maximum $k$ for which the snake mode is an instability at this soliton grayness is $k_{\text{max}}(\beta =0.5)=0.7435$. (a) $k=0.0404$. (b) $k=0.1919$. (c) $k=0.5253$.