Method for obtaining exact solutions of the nonlinear Schrödinger equation for a double-square-well potential

Both symmetric and symmetry breaking analytic solutions to the one-dimensional nonlinear Schrödinger equation with a double square well potential are known, but not straightforward to obtain numerically. The former generalize solutions to the linear equations, the latter owe their very existence to the nonlinearity. These include, for example, solutions corresponding to the wave function localized almost entirely in one of the wells. Here we propose a systematic method for generating these solutions starting from the linear limit. In particular we find a simple exact formula giving the bifurcation point in terms of the parameters of the symmetric solution. This bifurcation point is then reproduced to a surprising degree of accuracy by a simple variational method.

Figure 1

Symmetric double square well potential.

Figure 2

Examples of symmetric and asymmetric profiles of the wave function $f\left(x\right)$. Here $V_0=300$ throughout, (a) $\eta =-2$ and (b) $\eta =-150$ for the symmetric profiles and (c) $\eta =-2.25$ and (d) $\eta =-54$ for asymmetric ones. Also $2a=1$ and $2b=0.1$.

Figure 3

(Color online) The dependence of the chemical potential $\mu$ on the nonlinear coefficient $\eta$ for the lowest symmetric and asymmetric solutions. Parameters $V_0$, $a$, and $b$ are the same as in Fig. 2. The bifurcation point occurs for $\mid \eta \mid =-2.17$.

Figure 4

(Color online) Dependence of logarithm of $\mid {\eta }_{\mathrm{bif}}\mid$ on $\sqrt{V_0}b$ with $a=0.5$ and three values of $b$. Points correspond to exact solutions given by Eq. (28). The curves are given by Eq. (31).