Spatial control of the competition between self-focusing and self-defocusing nonlinearities in one- and two-dimensional systems

We introduce a system with competing self-focusing (SF) and self-defocusing (SDF) terms, which have the same scaling dimension. In the one-dimensional (1D) system, this setting is provided by a combination of the SF cubic term multiplied by the delta function $\delta \left(x\right)$ and a spatially uniform SDF quintic term. This system gives rise to the most general family of 1D Townes solitons, with the entire family being unstable. However, it is completely stabilized by a finite-width regularization of the $\delta$ function. The results are produced by means of numerical and analytical methods. We also consider the system with a symmetric pair of regularized $\delta$ functions, which gives rise to a wealth of symmetric, antisymmetric, and asymmetric solitons, linked by a bifurcation loop, that accounts for the breaking and restoration of the symmetry. Soliton families in two-dimensional (2D) versions of both the single- and double-$\delta$-functional systems are also studied. The 1D and 2D settings may be realized for spatial solitons in optics and in Bose-Einstein condensates.

Figure 1

Families of stationary soliton solutions to Eq. (31) found numerically for the 1D single-well configuration. The dependence of the total power on the propagation constant $N\left(\mu \right)$ is shown for several values of the quintic coefficient: (a) $\sigma =0$, (b) $\sigma =0.05$, (c) $\sigma =0.075$, (d) $\sigma =0.1$, and (e) $\sigma =0.15$. All these curves represent stable solitons. The dashed curve shows analytical approximation (18) for $\sigma =0.$ The horizontal dotted line corresponds to $N\left(\mu \right)\equiv 1$, $\sigma =0$. The latter family, given by Eqs. (7) and (8), is unstable. Qualitatively, the shape of the $N\left(\mu \right)$ curves is explained by the Thomas-Fermi approximation based on Eqs. (35, 36, 37).

Figure 2

The thin and thick solid curves show an example of a pair of stable 1D solitons pinned to the single regularized $\delta$ function in the model with $\sigma =0.1$. The dashed curve depicts profile (15) of the regularized $\delta$ function. These solitons pertain to a common value of the propagation constant, $\mu =0.1$, and their total powers (norms) are $N_1=11.16$ and $N_2=2.66$, respectively.

Figure 3

(a) Dependences $N\left(\sigma \right)$ for soliton families produced by Eq. (31) with $\mu ={10}^{-1}$ (curve V), $\mu ={10}^{-2}$ (curve IV), $\mu ={10}^{-4}$ (curve III), and $\mu ={10}^{-5}$ (curve II). The dashed curve (curve I) represents the exact result (10) obtained with the ideal $\delta$ function (in this case, $N$ does not depend $\mu$). (b) Dependence $N\left({\sigma }_{\mathrm{cr}}\right)$, where ${\sigma }_{\mathrm{cr}}$ is the largest value of the quintic coefficient $\sigma$ for which the solitons exist in (a).

Figure 4

Examples of the evolution of stable and unstable symmetric solitons in the 1D double-well model for (a) $x_0=5$, $\sigma =0.1$, $\mu =0.14$ and (b) $x_0=5$, $\sigma =0.05$, $\mu =0.008$.

Figure 5

A stable antisymmetric soliton with norm $N=26.394$ (the thick curve in the top frame and the bottom right frame) produced by Eq. (29) with $\sigma =0.01$ compared to an unstable antisymmetric one with $N=18.816$ (the thin curve in the top frame and the bottom left frame) obtained (as in Ref. [20]) for $\sigma =0$. The other parameters are $x_0=0.9$ [the dashed curve in the top frame depicts the profile of the cubic nonlinear regularization in Eq. (32)] and $\mu =3.8688$ [see Fig. 8 in Ref. 20].

Figure 6

Evolution of unstable ($x_0=5$, $\sigma =0.2$, $\mu =0.003$) and stable ($x_0=4$, $\sigma =0.1$, $\mu =0.025$) asymmetric solitons in the 1D double-well model.

Figure 7

(left) Bifurcation diagram for the 1D double-well system, shown in terms of $N\left(\mu \right)$ curves, for $x_0=5$ and $\sigma =0.05$ and (right) the close-up of the symmetry-breaking part for smaller values of $\mu$. Here and in Fig. 8, red, blue, and black curves represent symmetric, antisymmetric, and asymmetric modes, respectively, while solid and dashed segments of the curves refer to stable and unstable solitons.

Figure 8

The asymmetry parameter (30) vs total power $N$ for the same set of soliton branches in the 1D double-well system with $x_0=5$ and $\sigma =0.05$ as shown in Fig. 7.

Figure 9

Stability areas for symmetric, antisymmetric, and asymmetric solitons [red (light gray) vertical, blue (dark gray), and black lines, respectively] in the plane of $(\sigma ,N)$ in the 1D double-well system based on Eq. (29). The half distance between the wells is $x_0=5$ and $1.25$ in the left and right panels, respectively.

Figure 10

Numerical results for the 2D single-well model represented by Eq. (40). The top panel shows a set of $N\left(\mu \right)$ curves for the following values of the SDF quintic coefficient: (a) $\sigma =0$, (b) $\sigma =0.01$, (c) $\sigma =0.015$, (d) $\sigma =0.02$, and (e) $\sigma =0.025$. The bottom left panel shows a close-up of the region of small $\mu$, where the transition from stable (solid) branches to unstable (dashed) ones occurs. In the bottom right panel, the domain where the system supports stable solitons is shown by oblique lines.

Figure 11

(left) An example of an unstable 2D soliton in the single-well model based on Eq. (40) with $\sigma =0.02$. The propagation constant and total power of the soliton are $\mu =0.25$ and $N=5.2517$, respectively. (middle) The linear spectrum of small perturbations around the soliton. (right) The transformation of the unstable soliton into a robust breather, which features regular oscillations of the amplitude.

Figure 12

A stable symmetric double-peak soliton of Eq. (41) with $x_0=0.83$, $\sigma =0.0288$, and $\mu =5.76$, $N=49.02$. Thick blue (dark gray) circles in the left panel and in similar figures below depict the shape of the regularization of the $\delta$ functions per Eq. (42). Here and in similar figures below, the right panel shows the spectrum of stability eigenvalues for small perturbations around the soliton.

Figure 13

An unstable symmetric single-peak soliton of Eq. (41) with $x_0=0.45$, $\sigma =0.0484$, and $\mu =1.452$, $N=6.1614$. Here and in similar figures below, the right panel shows the cross section $y=0$ of the wave field in the course of its evolution.

Figure 14

An unstable symmetric double-peak soliton of Eq. (41) with $x_0=0.83$, $\sigma =0.0288$, and $\mu =2.88$, $N=11.95$.

Figure 15

A stable asymmetric soliton of Eq. (41) with $x_0=0.45$, $\sigma =0.0484$, and $\mu =3.267$, $N=17.246$; the asymmetry parameter is $\nu =0.474$.

Figure 16

An unstable asymmetric soliton of Eq. (41) with $x_0=0.45$, $\sigma =0.0484$, and $\mu =0.557$, $N=5.324$; the asymmetry parameter is $\nu =0.131$.

Figure 17

An unstable antisymmetric soliton of Eq. (41) with $x_0=0.45$, $\sigma =0.0484$, and $\mu =1.21$, $N=13.7$.

Figure 18

A stable antisymmetric soliton of Eq. (41) with $x_0=0.83$, $\sigma =0.0288$, and $\mu =5.04$, $N=16.96$.

Figure 19

(left) The total power vs the propagation constant $N\left(\mu \right)$ for solitons in the 2D double-well system with $\sigma =0.0288$ and $x_0=0.83$ [i.e., with nonoverlapping circles in Eq. (41)]. (right) Close-up of a region at small $\mu$ where the asymmetric branch splits off from the symmetric one. Here and in similar figures below, the solid and dashed curves (or chains of symbols) represent stable and unstable families of solitons, respectively. Red (light gray), blue (dark gray), and black designate symmetric, antisymmetric, and asymmetric modes, respectively.

Figure 20

The symmetry-breaking diagram $\nu \left(N\right)$ in the 2D double-well model (41) with $\sigma =0.0288$ and $x_0=0.83$, i.e., nonoverlapping circles.

Figure 21

(left) $N\left(\mu \right)$ curves for the 2D double-well system described by Eq. (41) with partly overlapping circles, with $x_0=0.45$ and $\sigma =0.0484$. (right) Close-up of the region of small $\mu$, where the symmetry-breaking transition from symmetric to asymmetric solitons occurs.

Figure 22

The symmetry-breaking diagram $\nu \left(N\right)$ [the asymmetry measure $\nu$ is defined as per Eq. (44)] for the 2D double-well system based on Eq. (41) with partly overlapping circles, with $x_0=0.45$ and $\sigma =0.0484$. The two bottom panels are close-ups of regions where the symmetry-breaking and -restoring bifurcations (opening up and closing down of the bifurcation loop) take place.