Chladni Solitons and the Onset of the Snaking Instability for Dark Solitons in Confined Superfluids

Complex solitary waves composed of intersecting vortex lines are predicted in a channeled superfluid. Their shapes in a cylindrical trap include a cross, spoke wheels, and Greek $\mathrm{\Phi }$, and trace the nodal lines of unstable vibration modes of a planar dark soliton in analogy to Chladni’s figures of membrane vibrations. The stationary solitary waves extend a family of solutions that include the previously known solitonic vortex and vortex rings. Their bifurcation points from the dark soliton indicating the onset of new unstable modes of the snaking instability are predicted from scale separation for Bose-Einstein condensates (BECs) and superfluid Fermi gases across the BEC–BCS crossover, and confirmed by full numerical calculations. Chladni solitons could be observed in ultracold gas experiments by seeded decay of dark solitons.

Figure 1

Superfluid order parameter ${\psi }_0(z/\xi )$ of kink solutions of Eq. (4) in the BEC-BCS crossover for different values of the coupling constant $\eta =1/k_F{a}_F$, where the axial coordinate $z$ is measured in units of the healing length $\xi =\hslash /\sqrt{4m{\mu }^B}$. The inset shows the (bound) ground state eigenvalue $\lambda$ of Eq. (6), which modulates the energy available for the excitation of transverse modes. The dotted line indicates the change of sign of the chemical potential.

Figure 2

Length scale $r_0$ (full line) of the snaking instability given by Eq. (8) and healing length $\xi$ of Eq. (3) (dashed line) as functions of the interaction parameter $1/k_F{a}_F$ of the BEC-BCS crossover. For comparison, the results from time-dependent Bogoliubov–de Gennes simulations and RPA of Ref. [35] are shown.

Figure 3

Bifurcation points for the emergence of Chladni solitons with quantum numbers ($p,l$) from the planar kink in a cylindrically confined BEC. Numerical data (labeled open symbols) obtained from solving the Gross-Pitaevskii equation are compared with the analytical prediction (dashed line) of Eq. (10) with $\lambda =-\frac{1}{4}$. The insets display 3D density isocontours of dark soliton states at 5% of maximum density in the weak (upper left) and strong (lower right) nonlinearity regimes.

Figure 4

Free excitation energy $F_{pl}$ of stationary solitary waves vs chemical potential in a cylindrically trapped BEC (central panel). The full (red) line corresponds to the dark soliton [see insets in Fig. 3] and lines with symbols correspond to Chladni solitons ($p,l$) that originated from the bifurcation points of Fig. 3. Units of the transverse harmonic trap and axial density $n_1$ are used as shown. The insets show density isocontours at 5% of maximum density for the different Chladni solitons with ${\mu }^B=10\hslash {\omega }_{\perp }$.