Symmetry breaking in symmetric and asymmetric double-well potentials

Motivated by recent experimental studies of matter waves and optical beams in double-well potentials, we study the corresponding solutions of the nonlinear Schrödinger equation. Using a Galerkin-type approach, we obtain a detailed handle on the nonlinear solution branches of the problem, starting from the corresponding linear ones, and we predict the relevant bifurcations for both attractive and repulsive nonlinearities. The dynamics of the ensuing unstable solutions is also examined. The results illustrate the differences that arise between the steady states and the bifurcations emerging in symmetric and asymmetric double wells.

Figure 1

(Color online) Left panel: The symmetric $(x_0=0)$ double-well potential $V\left(x\right)$ [see Eq. (2)] with parameters $\Omega =0.1$, $V_0=1$, and $w=0.5$. The horizontal solid lines $({\omega }_0,{\omega }_1)$ indicate the eigenvalues corresponding to the eigenstates $u_0$, $u_1$ of the linear Schrödinger equation $(s=0)$. Right panel: The wave functions of the ground state $u_0$ (blue solid line) and the first excited state $u_1$ (red dashed-dotted line) for the above double-well potential.

Figure 2

(Color online) Top panel: The norm of the solutions of Eq. (1) for attractive nonlinearity $(s=-1)$ as a function of $\mu$ for a symmetric $(x_0=0)$ double well. The potential parameters are $\Omega =0.1$, $V_0=1$, and $w=0.5$. The blue solid line denotes the symmetric solution, the red dashed-dotted line denotes the antisymmetric one, while the green dashed line denotes the asymmetric solution that is generated from the bifurcation at ${\mu }_c\approx 0.122$. Bottom panels: The profiles of the wave functions corresponding to the symmetric (left), asymmetric (middle), and antisymmetric (right) branches for $\mu =0.05$. The black dotted line shows the double-well potential.

Figure 3

(Color online) Top panel: The real part ${\lambda }_r$ of the most unstable eigenvalue of the symmetric solution as a function of $\mu$; note that the symmetry-breaking bifurcation destabilizes the symmetric solution for $\mu <{\mu }_c=0.122$. Bottom panel: The result of the linear stability analysis around the symmetric solution for $\mu =0.05<{\mu }_c$ in the complex plane $({\lambda }_r,{\lambda }_i)$. The existence of an eigenvalue with a positive real part implies instability of the solution.

Figure 4

(Color online) Top panel: Same as in Fig. 2 but for the repulsive nonlinearity $(s=+1)$. The red dashed-dotted line denotes the symmetric solution, the blue solid the antisymmetric one, while the green dashed line corresponds to the asymmetric solution generated from the pitchfork bifurcation occurring at ${\mu }_c\approx 0.168$. Contrary to the case $s=-1$, the bifurcation originates from the antisymmetric branch. Bottom panels: The profiles of the wave functions corresponding to the symmetric (left), antisymmetric (middle), and asymmetric (right) branches for $\mu =0.22$.

Figure 5

(Color online) Top panel: The maximal real eigenvalue associated with the linear stability analysis of the antisymmetric-solution branch. Bottom panel: The respective result of the linear stability analysis around the antisymmetric solution when $\mu =0.22>{\mu }_{\mathrm{cr}}$ in the complex plane. An eigenvalue with a positive real part implies instability of the solution.

Figure 6

(Color online) Left panel: The asymmetric double-well potential $V\left(x\right)$ [see Eq. (2)] with parameters $\Omega =0.1$, $V_0=1$, $w=0.5$, and $x_0=0.5$. The horizontal solid lines $({\omega }_0,{\omega }_1)$ indicate the eigenvalues corresponding to the first two eigenstates $u_0,u_1$ of the linear Schrödinger equation $(s=0)$. Right panel: The wave functions of the ground state $u_0$ (green dashed line) and the first excited state $u_1$ (red dashed-dotted line) for the above double-well potential.

Figure 7

(Color online) Top panel: The norm of the solutions of Eq. (1) for attractive nonlinearity $(s=-1)$ as a function of $\mu$ for an asymmetric double well with $x_0=0.5$. The potential parameters are $\Omega =0.1$, $V_0=1$, and $w=0.5$. The lower (green dashed) and upper (red dashed-dotted) branches are the nonlinear continuations (for decreasing $\mu$) of the ground state and the first excited state of the linear problem, respectively; both of these branches end at the linear limit $(N=0)$. On the contrary, the remaining two branches in the middle (blue solid line and cyan dashed line) exist up to the critical point ${\mu }_c\approx 0.09$, where they collide and disappear through a saddle-node bifurcation. Bottom panel: The profiles of the wave functions corresponding to the above-mentioned branches for $\mu =0.05$. The top right (red dashed-dotted) and bottom left (green dashed) solutions are the nonlinear counterparts of the first excited state and the ground state shown in Fig. 6; the top left (blue solid) and bottom right (cyan dashed) solutions are, respectively, the unstable and stable solutions emerging from the saddle-node bifurcation. The black dotted line shows the double-well potential.

Figure 8

(Color online) Top panel: The real part ${\lambda }_r$ of the most unstable eigenvalue of the “more symmetric” solution as a function of $\mu$; bottom panel: the result of the linear stability analysis around that solution for $\mu =0.05<{\mu }_c$ in the complex plane $({\lambda }_r,{\lambda }_i)$. The existence of an eigenvalue with a positive real part indicates the instability of the solution.

Figure 9

(Color online) Top panel: Same as in Fig. 7 but for the repulsive nonlinearity $(s=+1)$. The upper (red dashed-dotted) and lower (green dashed) branches are the nonlinear continuations (for increasing $\mu$) of the ground state and the first excited state of the linear problem, respectively; both of these branches end at the linear limit $(N=0)$. On the contrary, the remaining two branches in the middle (blue solid line and cyan dashed line) exist up to the critical point ${\mu }_c\approx 0.207$, where they collide and disappear through a saddle-node bifurcation. Bottom panels: The profiles of the wave functions corresponding to the above-mentioned branches for $\mu =0.22$. The black dotted line shows the double-well potential.

Figure 10

(Color online) Same as in Fig. 8 but for the repulsive nonlinearity $(s=+1)$. Top panel: The real part ${\lambda }_r$ of the most unstable eigenvalue of the more “symmetric” solution (as compared to the anti-phase ones—see the middle right panel of Fig. 9) as a function of $\mu$. Bottom panel: The result of the linear stability analysis around this solution for $\mu =0.22>{\mu }_c$ in the complex plane $({\lambda }_r,{\lambda }_i)$.

Figure 11

(Color online) Spatio-temporal countour plot of the density of the unstable solutions in the symmetric double-well potential $(x_0=0)$. Top panel shows the evolution of the symmetric solution for the attractive nonlinearity $(s=-1)$ and the bottom panel the evolution of the antisymmetric solution for the repulsive nonlinearity $(s=+1)$. The potential parameters are the same as in Fig. 1.

Figure 12

(Color online) Spatio-temporal countour plot of the density of the unstable solutions in the asymmetric double-well potential $(x_0=0.5)$. Top and bottom panels show, respectively, the evolution of the solutions for the attractive $(s=-1)$ and the repulsive $(s=+1)$ nonlinearity. The potential parameters are the same as in Fig. 6.