Figure 1

Amplitude of the couplers for the asymmetric solution in the Hamiltonian case of $\gamma =0$ and for $s=\sqrt{63}/32$, $\epsilon =1$ (left panel) and $s=\sqrt{15}/8$, $\epsilon =-1$ (right panel). The dashed lines correspond to the predictions of the RWA theory (see the discussion in the text). It is worth noticing that in the left panel, asymmetric solutions are unstable towards finite-time blowup for frequencies smaller than those shown in the figure. On the contrary, in the right panel, solutions are always stable even for frequencies higher than those shown in the picture.Reuse & Permissions

Figure 2

Plane with curves and regions of solutions that share the same properties when $\epsilon =1$ and $s=\sqrt{15}/8$ (see text). In the nonlabeled region of the top right, neither symmetric nor antisymmetric solutions exist. Here Curve 1 corresponds to the linear limit, Curve 2 denotes the $\mathcal{P}\mathcal{T}$ phase transition curve, Curve 3 indicates the destabilization of the A branch, while Curve 4 is associated with the destabilization of the symmetric branch. The detailed description of the different regions enclosed by the curves is offered in the text. Dashed lines correspond to the RWA predictions. Notice that the colors of the dashed lines are inverted with respect to those of the numerical results for a better visualization, i.e., the blue dashed line is the theoretical prediction corresponding to the red solid one, while the red dashed line represents the prediction corresponding to the blue solid one. This inversion pattern is followed also in all figures comparing theory and numerical computations from here onward. Dots mark the parameters for which simulations are performed in Fig. 5.Reuse & Permissions

Figure 3

Averaged energy (top left) and modulus of the Floquet multipliers (top right) [only the multipliers with moduli higher than one are shown] as a function of the gain or loss parameter $\gamma$ for $\epsilon =1$, ${\omega }_{\mathrm{b}}=0.45$, and $s=\sqrt{15}/8$. Notice the logarithmic scale in the $y$ axis of the latter graph. In the left panel, the blue [lower] (red [upper]) solid line corresponds to the symmetric (antisymmetric) solution, while the red (blue), i.e., reversed colors, dashed lines correspond to the symmetric (antisymmetric) branch RWA predictions. One can observe a reasonable agreement between the numerical energy and its theoretical prediction counterpart, although some discrepancy is clearly visible for this low (i.e., far from the linear limit) value of the frequency. The bottom panel shows similar comparisons but now for the dependence on ${\omega }_{\mathrm{b}}$ for fixed $\gamma =0.4$. Here it is evident that the agreement is good close to the linear limit of larger values of ${\omega }_{\mathrm{b}}$ and progressively worsens as one departs from this limit (by lowering the breather frequency).Reuse & Permissions

Figure 4

Same as Fig. 2 (top panel) and as the top panel of Fig. 3 (bottom panels) but for the soft nonlinearity case of lower coupling $s=\sqrt{63}/32$. Again, the full numerical results are offered by the solid lines for the two branches [blue for symmetric (lower) and red for antisymmetric (upper)], while the dashed lines with reverse colors correspond to the RWA.Reuse & Permissions

Figure 5

Evolution of unstable solutions for the soft potential; the instability is driven only by the numerical truncation errors. (Top left panel) Symmetric solution with ${\omega }_{\mathrm{b}}=0.45$ and $\gamma =0.01$. (Top right panel) Symmetric solution with ${\omega }_{\mathrm{b}}=0.45$ and $\gamma =0.45$. (Bottom left panel) The antisymmetric solution with ${\omega }_{\mathrm{b}}=0.70$ and $\gamma =0.66$. (Bottom right panel) The antisymmetric solution at $\gamma =0.16$ and ${\omega }_{\mathrm{b}}=1.2$ is taken as initial condition for $\gamma =0.51$; notice that for this value of ${\omega }_{\mathrm{b}}$, the saddle-center bifurcation takes place at ${\gamma }_{\mathrm{SC}}=0.168$.Reuse & Permissions

Figure 6

Same as in Fig. 2 but in the hard potential case, i.e., when $\epsilon =-1$. The only change with respect to that figure consists of an interchange of the color and thickness between Curves 3 and 4. Dots mark the parameters for which simulations are performed in Fig. 8.Reuse & Permissions

Figure 7

Similar to Fig. 3 but for ${\omega }_{\mathrm{b}}=1.25$ and $\epsilon =-1$ in the top panels and for $\gamma =0.3$ and $\epsilon =-1$ in the bottom panels. Both the existence (left) and stability (right) diagrams of symmetric [blue (upper in left panels) solid] and antisymmetric [red (lower in left panels) solid] branches and their rotating-wave approximations (the latter with reverse colors and dashed lines) are shown.Reuse & Permissions

Figure 8

Evolution of unstable solutions for the hard potential; the instability is driven only by the numerical truncation errors. (Top left panel) Antisymmetric solution with ${\omega }_{\mathrm{b}}=1.25$ and $\gamma =0.35$. (Top right panel) The antisymmetric solution with ${\omega }_{\mathrm{b}}=1.25$ and $\gamma =0.37$. (Bottom left panel) Symmetric solution with ${\omega }_{\mathrm{b}}=1.25$ and $\gamma =0.38$. (Bottom right panel) The symmetric solution at $\gamma =0.4$ and ${\omega }_{\mathrm{b}}=0.8$ is taken as initial condition for $\gamma =0.45$; notice that for this value of ${\omega }_{\mathrm{b}}$, the saddle-center bifurcation takes place at ${\gamma }_{\mathrm{SC}}=0.404$.Reuse & Permissions