Systematic solitary waves by linear limit continuation from two anisotropic traps in two-dimensional Bose-Einstein condensates

Abstract

Linear limit continuation was recently developed as a systematic and effective method for constructing numerically exact solitary waves from their respective linear limits. In this work, we apply the technique to two typical anisotropic harmonic traps in two-dimensional Bose-Einstein condensates to further establish the method and also to find more solitary waves. Many wave patterns are identified in the near-linear regime and they are subsequently continued into the Thomas-Fermi regime, and then they are further continued into the isotropic trap if possible. Finally, the parametric connectivity of the pertinent solitary waves is also discussed.

Figures (12)

• Figure 1: Linear degenerate sets can be geometrically represented by lattice planes, focusing only on the $n_x, n_y \geq 0$ region. Here, the low-lying degenerate sets of two prototypical sets of lattice planes $[3, 1]$ (left panel) and $[3, 2]$ (right panel) are illustrated, and the two scenarios correspond to trap aspect ratios $\kappa=1/3$ and $2/3$, respectively. In general, the lattice plane $[p, q]$ corresponds to a trap aspect ratio $\kappa=q/p \leq 1$.
• Figure 2: Solitary waves continued from the lowest four $(\mu_0=2, 3, 4, 5)$ linear degenerate sets of $\kappa=1/3$. The first set shows the ground state at $\mu=2.5, 3, 8, 10, 12, 16$ of $\omega_x=3$, followed by a further continuation in $\kappa$ at fixed $\mu=16$. Here, the states at $\omega_x=2.5, 2, 1.7, 1.5, 1.3, 1$ are depicted, and the dagger symbol highlights the onset of the trap continuation. The second set depicts the DS01 state at $\mu=4, 8, 16$ and $\omega_x=2, 1.5, 1$, and the DS02 state at $\mu=5, 9, 16$ and $\omega_x=2, 1.5, 1$. The third set illustrates the DS03 and DS10 states at $\mu=6, 10, 16$ and $\omega_x=2, 1.5, 1$. The fourth set sketches the S soliton and the VX3 state at $\mu=6, 10, 16$ and $\omega_x=2.7, 2.4, 2.3, 1$, and $\omega_x=2, 1$, respectively.
• Figure 3: Solitary waves continued from the $\mu_0=6, 7$ linear degenerate sets of the $\kappa=1/3$ trap. The first set shows the DS04 state at $\mu=7, 11, 16$ and $\omega_x=2, 1.7, 1.3, 1$, and the DS11 state at $\mu=7, 11, 16$ and $\omega_x=2, 1$. The second set depicts the U2 soliton and the VX4 state at $\mu=7, 11, 16$ and $\omega_x=2.7, 2.4, 1$, and $\omega_x=2, 1.5, 1$, respectively. The third set illustrates the DS05 and the DS12 states at $\mu=8, 11, 16$ and $\omega_x=2, 1.5, 1$. The fourth set sketches the U2I soliton and the VX5 state at $\mu=8, 11, 16$ and $\omega_x=2.7, 2.4, 1$, and $\omega_x=2.5, 2, 1$, respectively.
• Figure 4: Solitary waves continued from the $\mu_0=8$ linear degenerate set of the $\kappa=1/3$ trap, part one. The first set shows the DS06 state at $\mu=9, 12, 16$ and $\omega_x=2.5, 2, 1.5, 1$, and the DS13 state at $\mu=9, 12, 16$ and $\omega_x=2, 1$. The second set depicts the DS20 state at $\mu=9, 11, 13, 16$ and $\omega_x=2.5, 1.92, 1.91, 1.88, 1.335, 1.333, 1.1, 1$. The third set illustrates the O3 and U2O2 states at $\mu=9, 12, 16$ and $\omega_x=2.6, 1.91, 1$, and $\omega_x=2.2, 1.91, 1$, respectively. The fourth set displays the U28 and W2 states at $\mu=9, 12, 16$ and $\omega_x=2.8, 2.5, 1$, and $\omega_x=2.9, 2.8, 2.757$, respectively. Here, the W2 state reaches an existence boundary highlighted by the exists symbol. However, the state can be continued into the isotropic trap in the near-linear regime, as shown in the final set at $\mu=9$. The states of $\omega_x=3, 2.9, 2.8, 2.7, 2.64, 2.5395, 2.5390, 2.45, 1.4526, 1.4524, 1.3, 1$ are sketched.
• Figure 5: Solitary waves continued from the $\mu_0=8$ linear degenerate set of the $\kappa=1/3$ trap, part two. The first set shows the VX8a and VX8b states at $\mu=9, 13, 16$ and $\omega_x=2, 1.2, 1$, and $\omega_x=2.9, 2, 1$, respectively. The second set depicts the VX10a state at $\mu=9, 11, 13, 16$ and $\omega_x=2.9, 2.8, 2.7, 2.6, 2.5, 2, 1.5, 1$. The third set illustrates the VX10b and VX12a states at $\mu=9, 13, 16$ and $\omega_x=2.5, 2, 1$, and $\omega_x=1.77, 1.7657, 1$, respectively. The fourth set sketches the VX12b state at $\mu=9, 9.4, 16$ and $\omega_x=2, 1.5, 1$, and the VX12c state at $\mu=9, 13, 16$ and $\omega_x=2, 1.5, 1$.