A purely Kerr nonlinear model admitting flat-top solitons

TL;DR

This paper introduces a new Kerr nonlinear model with spatially modulated self-repulsive nonlinearity that supports stable flat-top solitons, including fundamental, multipole, and vortex types, without requiring competing nonlinearities.

Contribution

The study presents a purely Kerr nonlinear model capable of supporting flat-top solitons, providing exact solutions and stability analysis, expanding the understanding of soliton formation in nonlinear media.

Findings

Exact analytical solutions for stable flat-top solitons in 1D.

Stable flat-top solitons include fundamental, multipole, and vortex types.

Identification of stability and instability regions for various soliton configurations.

Abstract

We elaborate one- and two-dimensional (1D and 2D) models of media with self-repulsive cubic nonlinearity, whose local strength is subject to spatial modulation that admits the existence of flat-top solitons of various types, including fundamental ones, 1D multipoles, and 2D vortices. Previously, solitons of this type were only produced by models with competing nonlinearities. The present setting may be implemented in optics and Bose-Einstein condensates. The 1D version gives rise to an exact analytical solution for stable flat-top solitons, and generic families may be predicted by means of the Thomas-Fermi approximation. Stability of the obtained flat-top solitons is analyzed by means of linear-stability analysis and direct simulations. Fundamental solitons and 1D multipoles with $k=1$ and $2$ nodes, as well as vortices with winding number $m=1$, are completely stable. For multipoles with $k\geq 3$ and vortices with $m\geq 2$, alternating stripes of stability and instability are identified in their parameter spaces.

Equations

$$i{\psi }_t=-(1/2){\nabla }^2\psi +\sigma (r){\left|\psi \right|}^2\psi ,$$

$$\sigma(r)=\left\{\begin{array}[]{c}\sigma_{\mathrm{int}},r\leq r_{0},\\ \sigma_{\mathrm{ext}}\exp[\alpha(r+\alpha^{-1/2}-r_{0})^{2}-1],r>r_{0},\end{array}\right.$$

$$\mu w=-\left(1/2\right)(w^{\prime \prime }+r^{-1}{w}^{\prime }-m^2{r}^{-2}w)+\sigma (r)w^3$$

$$w(|x|>x_0)=(\alpha /\sqrt{2{\sigma }_{\mathrm{ext}}})(|x|+{\alpha }^{-1/2}-x_0)$$

$$\times \exp (-(1/2)[\alpha (|x|+{\alpha }^{-1/2}-x_0{)}^2-1]).$$

$$w_{\mathrm{TF}}^2={\left[\sigma (r)\right]}^{-1}[\mu -m^2/(2r^2)]$$

$$N_{2D}=2\pi {\int }_0^{\infty }{w}^2(r)dr\approx \pi {\sigma }_{\mathrm{int}}^{-1}\times$$

$$[\mu r_0(r_0+{\alpha }^{-1/2})-(m^2/2)\ln (2e\mu r_0^2/m^2)]$$

$$i\lambda p_{\pm }=\mp \left(1/2\right)[p_{+}^{\prime \prime }+r^{-1}{p}_{\pm }^{\prime }-(m\pm n{)}^2{r}^{-2}{p}_{\pm }]p$$

$$\mp \mu p_{\pm }+\sigma w^2(2p_{\pm }+p_{\mp }),$$

Full text

The formation of bright spatial solitons in one-dimensional (1D) uniform media is a commonly known result of the balance between the diffraction and self-focusing nonlinearity FPC ; however, making them stable in higher-dimensional media is a challenging issue owning to the presence of the wave collapse, which can, in an usual way, be suppressed with an aid of linear periodic potentials FPC ; NL . The situation may be different in physical settings with inhomogeneous strength of the local nonlinearity NL . In optics, such settings may be engineered by means of properly designed photonic-crystal structures, with voids filled by solid Russell or liquid phot-cryst1 ; phot-cryst2 ; phot-cryst3 materials with different values of the Kerr coefficient. Alternatively, one can use nonuniform distributions of nonlinearity-enhancing dopants dopant ; Gaetano . In Bose-Einstein condensates (BECs), similar nonlinearity landscapes can be created by means of the Feshbach resonance (FR) locally controlled by spatially nonuniform optical nonuniform-Feshbach1 ; nonuniform-Feshbach2 ; nonuniform-Feshbach3 or magnetic Feshbach-magnetic1 ; Feshbach-magnetic2 fields. In particular, it was predicted that self-defocusing nonlinearity, whose local strength grows from the center to periphery in the $D$-dimensional space, with radial coordinate $r$, at any rate faster than $r^{D}$, can support a great variety of robust self-trapped modes, including 1D fundamental, dipole and multipole solitons, 2D solitary vortices with arbitrarily high topological charge $m$ Olga ; Olgaalg , and sophisticated 3D modes, such as soliton gyroscopes Defo9 and skyrmions Defo8 . A characteristic feature of these localized modes is nonlinearizability of the underlying equations for their decaying tails, on the contrary to the usual bright solitons maintained by uniform self-focusing, whose exponential tails are produced by the corresponding linearized equations. The exploration of different kinds of bright solitons supported by this scheme has currently been extended to a variety of other physical settings Defo5 -Defo14 .

The objective of this Letter is to demonstrate another natural setting, which gives rise to families of 1D and 2D flat-top solitons. This is a known variety of self-trapped modes, with a potential for applications Porras , which are usually supported by systems with competing focusing and defocusing nonlinearities New , such as cubic-quintic Bulgaria ; Humberto ; Humberto2 ; nine-authors , cubic-quartic Petrov ; we , and quadratic-cubic chi2 ; Grisha1 ; Grisha2 combinations, as well as by cubic terms with an additional logarithmic factor Grisha1 ; NJP (the two former types of the nonlinearity were recently realized experimentally in optics Cid and as “quantum droplets” BEC droplets1 ; droplets2 , respectively). However, stable flat-top solitons were not previously found in physical models with the cubic-only nonlinearity, while here we demonstrate that this is possible, in 1D and 2D geometries alike, for various soliton species (fundamental, multipole, vortical), if the coefficient of the cubic defocusing is subjected to an appropriate spatial modulation. In addition to the systematically produced numerical results, we also obtain particular exact stable solutions for the 1D fundamental flat-top solitons, and develop the Thomas-Fermi (TF) approximation for generic soliton states and 2D vortices. Stability of all the solutions is investigated via the linear-stability analysis and numerical simulations.

The basic model is introduced as the scaled Schrödinger equation governing the evolution of the dimensionless complex amplitude of a light beam propagating in a cubic nonlinear medium, or the mean-field wave function in a Bose-Einstein condensate (BEC), $\psi\left(\mathbf{r},t\right)$:

[TABLE]

which is written in the 2D form, with $\mathbf{r}=\left(x,y\right)$. Here $t$ is the evolution variable, representing time in the BEC version of the model, or the propagation distance in optics. The axially symmetric nonlinearity-modulation profile, $\sigma(r)>0$, is chosen in the form which readily helps to create flat-top modes of radius $r_{0}$,

[TABLE]

with constants $\sigma_{\mathrm{int,ext}}>0$ and $\alpha>0$. Note that Eq. (2) implies that $\sigma(r=r_{0})=\sigma_{\mathrm{ext}}$. The 1D version of the model corresponds to Eq. (1) with single coordinate $x$, while $r$ and $r_{0}$ are replaced by $|x|$ and $x_{0}>0$ in Eq. (2). This profile can be created by means of the above-mentioned methods – for instance, by the application of an FR-controlling magnetic-field profile to the BEC layer, with a constant detuned value at $r<r_{0}$, and one approaching the exact resonant value at $r>r_{0}$. It is relevant to mention that similar cylindrical optical-box potentials are used in current BEC experiments Navon .

Wave functions of stationary states with real chemical potential $\mu$ (alias propagation constant $-\mu$ in terms of optics) and integer vorticity $m$ are sought for, in ploar coordinates $\left(r,\theta\right)$ , as $\psi\left(\mathbf{r},t\right)=w(r)\exp\left(im\theta-i\mu t\right)$, with real $w$ determined by the equation

[TABLE]

in 2D, or its counterpart in 1D. First, the 1D version of the model admits an exact solution, with $\mu=3\alpha/2$, $w\left(x\leq\left|x_{0}\right|\right)=\sqrt{3\alpha/2\sigma_{\mathrm{ext}}}$, and

[TABLE]

Note that Eq. (4) gives $dw/dx\left(x=x_{0}\right)=0$, which is necessary for the continuity at $x=x_{0}$, while the remaining continuity condition for $w(x)$ imposes a relation on constants of modulation profile ( 2): $\sigma_{\mathrm{int}}=3\sigma_{\mathrm{ext}}$. This profile is continuous in the limit case of $x_{0}=0$, for which exact solution (4) remains valid. Numerical solutions are displayed below for the continuous modulation profile with $\sigma_{\mathrm{int}}=\sigma_{\mathrm{ext}}$.

An analytical approximation for generic soliton shapes can be obtained in the Thomas-Fermi (TF) approximation, which neglects derivatives in Eq. (3) Fetter :

[TABLE]

at $r^{2}>m^{2}/2\mu$, and $w_{\mathrm{TF}}^{2}=0$ at $r^{2}<m^{2}/2\mu$. This approximation makes it possible to predict the dependence of the norm of the soliton family on the chemical potential,

[TABLE]

(here $e$ is the base of the natural logarithm), which is valid at $\mu>m^{2}/\left(2r_{0}^{2}\right)$.

In the numerical form, stationary profiles of both 1D and 2D modes were found by means of the Newton’s method applied to Eq. (3). The subsequent stability analysis was based on the usual ansatz, $\psi=[w(r)+p_{+}(r)\mathrm{exp}(in\theta+\lambda t)+p_{-}^{\ast}(r)\mathrm{exp}(-in\theta+\lambda^{\ast}t)]\mathrm{exp}(im\theta-i\mu t)$, where $p_{\pm}(r)$ represent perturbation eigenmodes with eigenvalue $\lambda$, $\ast$ stands for the complex conjugate, and $n$ is an integer azimuthal index. Then, the eigenvalue problem amounts to the solution of linear equations,

[TABLE]

or their 1D counterparts. In particular, the exact solution given by Eq. ( 4) is found to be always stable.

Typical profiles of 1D flat-top solitons with the number of nodes $k=0$, $1$ , and $2$ (fundamental, dipole, and tripole solitons, respectively) at different values of $\mu$ are displayed in Fig. 1, which clearly shows that the solitons’ shape gets flatter with the increase of $\mu$. The functional form of soliton changes considerably as $\mu$ increases. For instance, solitons with large number of nodes $k$ at small values of $\mu$ resemble trigonometric functions, while at large $\mu$ values they can be viewed as complexes of several well-localized dark solitons. Families of all solitons with $k\leq 2$ are completely stable (at least, up to $\mu=40$), while instability domains appear at $k\geq 3$. To illustrate this feature, Fig. 2(a) represents soliton families for $k=6$ by showing their norm vs. $\mu$ at several values of $x_{0}$. It is seen that even this high value of $k$, corresponding to “hashed” flat-top patterns, admits large stability segments, whose share increases with the growth of width $x_{0}$ of the modulation profile. Note also that all the $N(\mu)$ curves satisfy the “anti-Vakhitov-Kolokolov” criterion, $d\mu/dN>0$, which is a necessary condition for the stability of solitons in models with self-repulsive nonlinearities antiVK . The alternating stability and instability domains for $k=6$ are charted in the ($x_{0}$,$\mu$) plane in Fig. 2(b). Figure 2 shows that flat-top states are generally more stable than their more localized counterparts with the same number of nodes $k$.

Typical examples of the evolution of stable 1D flat-top solitons are displayed in Figs. 3(a)-(c), while evolution of their unstable counterparts is shown in Figs. 3(d-f). Unstable multipoles spontaneously develop oscillations, keeping the number of nodes.

The profiles of 2D solitons with vorticities $m=0$ (fundamental solitons), $m=1$, and $m=2$ for different values of $r_{0}$ are displayed in Fig. 4(a), which shows that the width of the flat-top solitons increases with $r_{0}$, similar to their 1D counterparts. Further, Fig. 4(b) displays profiles of the vortices with $m=2$ and different values of $\mu$, demonstrating that 2D solitons also get flatter with the increase of $\mu$.

Typical dependencies $N(\mu)$ for the vortex families with $m=2$ are displayed in Fig. 5(a), featuring a nearly linear form for all values of $r_{0}$, and considerable growth of $N$ with the increase of $r_{0}$. These features are well predicted by the TF approximation, as seen in the figure. The 2D modes with $m=0$ and $1$ are completely stable, at least up to $\mu=40$, while the vortices with $m\geq 2$ demonstrate alternation of stability and instability domains in the ($r_{0}$, $\mu$) plane in Fig. 5(b). Finally, the evolution of the flat-top 2D vortices is displayed in Fig. 6, demonstrating that those with $m=2$ or $3$, which are unstable, split into persistently rotating pairs or triplets of unitary vortices.

In conclusion, we have demonstrated that families of stable flat-top solitons, including 1D multipoles and 2D vortices, can be created in media with cubic self-repulsive nonlinearity whose local strength is subject to an appropriate spatial modulation. To our knowledge, this model predicts the first stable flat-top solitons realized with the cubic-only nonlinearity, in contrast to previous results which demonstrated such modes solely in systems with competing attractive and repulsive nonlinearities, thus providing an alternative way to create and stabilize the flat-top modes with free intrinsic parameters. We have checked the stability of all the obtained flat-top solitons by means of the linear-stability analysis and direct simulations. Both the 1D and 2D solitons become flatter with the increase of their chemical potential. 1D multipoles with $k=0,1,$ and $2$ nodes, as well as 2D solitons with vorticities $m=0$ and $1$ are completely stable, while higher-order modes, with $k\geq 3$ and $m\geq 2$, respectively, feature alternating stability and instability domains. Such self-trapped modes can be created in BEC and optics by means of available experimental techniques.

The work of LZ and JZ was supported by the NSFC (Nos. 61690224, 61690222), and by the Youth Innovation Promotion Association of the Chinese Academy of Sciences (No. 2016357). The work of B.A.M. was partly supported by the Israel Science Foundation through grant No. 1287/17.