Bistability and instability of dark-antidark solitons in the cubic-quintic nonlinear Schrödinger equation

We characterize the full family of soliton solutions sitting over a background plane wave and ruled by the cubic-quintic nonlinear Schrödinger equation in the regime where a quintic focusing term represents a saturation of the cubic defocusing nonlinearity. We discuss the existence and properties of solitons in terms of catastrophe theory and fully characterize bistability and instabilities of the dark-antidark pairs, revealing mechanisms of decay of antidark solitons into dispersive shock waves.

Figure 1

Dark-antidark pair sitting on the unit background ${\rho }_0=1$ for $\alpha =0.6$, $v=0.3$. Panel (a) shows the potential $V\left(\rho \right)-E$ vs $\rho$ and panel (b) shows the phase-space picture (contour lines of $\mathcal{H}$). Panels (c) and (d) show the relative intensity (thick black solid line) and phase (thin red solid line) profiles of dark and antidark solitons, respectively.

Figure 2

Cusp-catastrophe picture for dark-antidark soliton pairs. Each curve shows the evolution of the parameters $a$ and $b$ of the normal-form potential $V\left(y\right)=y^4/4+a{y}^2/2+by$, calculated for a dark-antidark soliton pair with fixed internal parameters $v$ and ${\rho }_0$, with $\alpha$ changing from zero up to its critical value ${\alpha }_c$, where all the curves arrive tangentially on the cusp curve [Eq. (8), red solid line] that bounds the soliton existence domain.

Figure 3

Potential and phase-space picture for $v=0$, ${\rho }_0=1$, and different values of $\alpha$: (a), (b) $\alpha =0$ (ideal Kerr case); (c), (d) $\alpha =0.5$; (e), (f) $\alpha =0.9$ (close to critical value ${\alpha }_c=1$, where the right branch of the separatrix becomes vanishingly small).

Figure 4

(a) Dip intensity (darkness) of dark solitons and (b) maximum intensity of antidark solitons as a function of velocity $v$ for different values of $\alpha$. Here the background is ${\rho }_0=1$. The dashed curves represent the existence threshold set by Eq. (9). Note the vertical log scale in (b).

Figure 5

Color-level plot of FWHM [${\theta }_{\text{FWHM}}$ in Eq. (10)] in the parameter plane $(\alpha ,{\rho }_0)$ for $v=0.1$, dark (a) and antidark (b) solitons. Panels (c) and (d) show dark solitons with $v=0.4$ and $v=0.8$, respectively. The dashed lines give the critical condition ${\alpha }_c({\rho }_0,v)$.

Figure 6

(a) Renormalized momentum $M$ vs soliton velocity $v$ and (b) Hamiltonian $H\left(v\right)$ vs $M\left(v\right)$ for dark and antidark solitons with fixed ${\rho }_0=1$ and $\alpha =0.7$. The red bullets mark the marginal condition ${\partial }_{v}M=0$ for antidark solitons.

Figure 7

Stability maps for antidark solitons in the parameter plane $(v,\alpha )$ for (a) ${\rho }_0=1$ and (b) ${\rho }_0=2$. The light-shaded (cyan) and dark-shaded (red) domains correspond to unstable and stable solutions, respectively. The dot-dashed line sets the value ${\alpha }_0$ below which antidark solitons have finite brightness at cutoff. The dynamics illustrated in Fig. 8 and Figs. 9 and 10 are relative to the sampled values marked by bullets and stars in (a), respectively.

Figure 8

Dynamics of antidark solitons with ${\rho }_0=1$, $\alpha =0.7$, exhibiting different behavior depending on the initial velocity $v$ [see bullets in Fig. 7a]: (a) $v=0.5$, stable propagation; (b) $v=0.4$, decay into a nearby stable antidark soliton; (c) $v=0$, decay into a pair of antidark shallow solitons with opposite velocities.

Figure 9

Decay of an unstable antidark solitons (${\rho }_0=1$, $v=0$, $\alpha =0.1$) into two symmetric dispersive shock fans (trains of dark solitons): (a) color level plot of the intensity; (b) snapshot of intensity (black solid line) and phase (red solid line) at $z=5$. For comparison, the input intensity (renormalized to the maximum of the plot) is shown (solid blue line).

Figure 10

As in Fig. 7 for $v=0.8$.