Figure 1

(a) and (b) The lowest branches of the nonlinear modes of Eq. (6) for the potential $V\left(x\right)=x^2$ for repulsive $\left(\sigma =-1\right)$ (a) and attractive $\left(\sigma =1\right)$ (b) nonlinearities, respectively. (c) and (d) The same branches but for $V\left(x\right)=x^4$. All the modes bifurcate from the linear harmonic oscillator modes corresponding to the limit $N\to 0$. The fragments of curves corresponding to stable solutions are shown in bold.Reuse & Permissions

Figure 2

Branches ${\Gamma }_n^{\left(a,r\right)}$ and plots of real and imaginary parts of eigenvalues $\Lambda$ of the operator $L_n^{-}{L}_n^{+}$ vs $\omega$ for $V\left(x\right)=x^2$ and $n=2$. (a) $\mathit{N}$ vs $\omega$ for the branch ${\Gamma }_2^{\left(a\right)}$ (the part of the branch corresponding to stable solutions is shown in bold). (b) Real and (c) imaginary parts of the eigenvalues $\Lambda$ for the attractive case. The splitting of the lowest eigenvalues $\Lambda =4$ and 16 of the linear problem is highlighted by the vertical dashed line. The nonzero imaginary part in panel (c) corresponds to the eigenvalues originated by $\Lambda =16$ of the linear problem for ${\omega }^{\ast }<\omega <5$ (see the text). Plots (d)–(f) are analogous to (a)–(c) but for the repulsive case. The branch studied is ${\Gamma }_2^{\left(r\right)}$ and the eigenvalues originated by $\Lambda =16$ remain complex for the whole interval of $\omega$ studied [see panel (f)], and therefore the mode is unstable.Reuse & Permissions

Figure 3

Branches ${\Gamma }_n^{\left(a,r\right)}$ and plots of the real and imaginary parts of the eigenvalues $\Lambda$ vs $\omega$ for the potential $V\left(x\right)=x^2$ and $n=3$. All the plots are organized in the same way as in Fig. 2.Reuse & Permissions

Figure 4

Branches ${\Gamma }_n^{\left(a,r\right)}$ and plots of the real and imaginary parts of eigenvalues $\Lambda$ of the operator $L_n^{-}{L}_n^{+}$ vs$\omega$ for $V\left(x\right)=x^2+0.01x^4$ and $n=1$. Plots (a)–(f) are organized as in Fig. 2. In the attractive case the dashed line on the plots (a)–(c) marks the upper boundary of the instability window. The lower boundary of this instability window is very close to $N=0$, and not visible on the scale of the plots (a)–(f). Plots (A)–(C) show the branch ${\Gamma }_1^{\left(a\right)}$ and the real and imaginary parts of eigenvalues ${\Lambda }_{1,0}$ and ${\Lambda }_{1,2}$ vs $\omega$ close to the linear limit $\omega ={\tilde{\omega }}_1$ (i.e., $N=0$) with magnification. In the plots (A)–(C) the lower boundary of the instability window is marked with a dashed line.Reuse & Permissions

Figure 5

Branches ${\Gamma }_n^{\left(a,r\right)}$ and plots of the real and imaginary parts of eigenvalues $\Lambda$ of the operator $L_n^{-}{L}_n^{+}$ vs $\omega$ for $V\left(x\right)=x^2+0.01x^4$ and $n=2$. In the attractive case the lower boundary is very close to $N=0$ and not visible on the scale of the figure. All the plots are organized as in Fig. 2.Reuse & Permissions

Figure 6

Branches ${\Gamma }_n^{\left(a,r\right)}$ and plots of the real and imaginary parts of eigenvalues $\Lambda$ of the operator $L_n^{-}{L}_n^{+}$ vs $\omega$ for $V\left(x\right)=x^2+0.01x^4$ and $n=3$. In the attractive case the lower boundary is very close to $N=0$, and not visible on the scale of the figure. All the plots are organized as in Fig. 2.Reuse & Permissions

Figure 7

Branches ${\Gamma }_n^{\left(a,r\right)}$ and plots of the real and imaginary parts of eigenvalues $\Lambda$ of the operator $L_n^{-}{L}_n^{+}$ vs $\omega$ for $V\left(x\right)=x^2-0.01x^4$ and $n=1$. All the plots are organized as in Fig. 2.Reuse & Permissions

Figure 8

Branches ${\Gamma }_n^{\left(a,r\right)}$ and the plots of the real and imaginary parts of eigenvalues $\Lambda$ of the operator $L_n^{-}{L}_n^{+}$ vs $\omega$ for $V\left(x\right)=x^4$ and $n=1$. All the plots are organized in the same manner as in Fig. 2.Reuse & Permissions

Figure 9

Branches ${\Gamma }_n^{\left(a,r\right)}$ and the plots of the real and imaginary parts of eigenvalues $\Lambda$ of the operator $L_n^{-}{L}_n^{+}$ vs $\omega$ for $V\left(x\right)=x^4$ and $n=2$. All the plots are organized in the same manner as in Fig. 2.Reuse & Permissions

Figure 10

Branches ${\Gamma }_n^{\left(a,r\right)}$ and the plots of the real and imaginary parts of eigenvalues $\Lambda$ of the operator $L_n^{-}{L}_n^{+}$ vs $\omega$ for $V\left(x\right)=x^4$ and $n=3$. All the plots are organized in the same manner as in Fig. 2. The bold circle in the plot (e) highlights the collision of the eigenvalues of opposite Krein signatures which does not lead to instability.Reuse & Permissions