Tail-free self-accelerating solitons and vortices

TL;DR

This paper demonstrates the existence of robust, tailless self-accelerating solitons and vortices in two-component Bose-Einstein condensates influenced by long-range microwave-mediated interactions, with potential applications in optical media.

Contribution

It introduces a novel mechanism for self-accelerating tailless solitons and vortices in BECs with long-range interactions, including effects of spin-orbit coupling and nonlinearities.

Findings

Existence of 1D and 2D self-accelerating tailless solitons and vortices.

Self-accelerating solitons can follow curved trajectories.

Mechanism applicable to optical media with thermal nonlinearity.

Abstract

Self-accelerating waves in conservative systems, which usually feature slowly decaying tails, such as Airy waves, have drawn great interest in studies of quantum and classical wave dynamics. They typically appear in linear media, while nonlinearities tend to deform and eventually destroy them. We demonstrate, by means of analytical and numerical methods, the existence of robust one- and two-dimensional (1D and 2D) self-accelerating tailless solitons and solitary vortices in a model of two-component Bose-Einstein condensates, dressed by a microwave (MW) field, whose magnetic component mediates long-range interaction between the matter-wave constituents, with the feedback of the matter waves on the MW field taken into account. In particular, self-accelerating 2D solitons may move along a curved trajectory in the coordinate plane. The system may also include the spin-orbit coupling between the components, leading to similar results for the self-acceleration. The effect persists if the contact cubic nonlinearity is included. A similar mechanism may generate 1D and 2D self-accelerating solitons in optical media with thermal nonlinearity.

Equations

$$i\partial_{t}\left(\begin{array}[]{c}\Psi_{\downarrow}\\ \Psi_{\uparrow}\end{array}\right)=\left[-\frac{1}{2}\partial_{x}^{2}+U\left(x\right)+\left(\begin{array}[]{cc}iK\partial_{x}&\Omega-H\\ \Omega-H^{\ast}&-iK\partial_{x}\end{array}\right)-\left(\begin{array}[]{cc}\beta_{1}|\Psi_{\downarrow}|^{2}+\beta_{2}|\Psi_{\uparrow}|^{2}&0\\ 0&\beta_{1}|\Psi_{\uparrow}|^{2}+\beta_{2}|\Psi_{\downarrow}|^{2}\end{array}\right)\right]\left(\begin{array}[]{c}\Psi_{\downarrow}\\ \Psi_{\uparrow}\end{array}\right).$$

$${\int }_{-\infty }^{+\infty }{\Psi }^{†}\Psi dx=1.$$

$${\partial }_x^{2}H=-\gamma {\Psi }_{\downarrow }^{\ast }{\Psi }_{\uparrow },$$

$$H\left(x,t\right)=-\frac{\gamma }{2}{\int }_{-\infty }^{+\infty }\left|x-x^{\prime }\right|{\Psi }_{\downarrow }^{\ast }\left(x^{\prime },t\right){\Psi }_{\uparrow }\left(x^{\prime },t\right)d{x}^{\prime }.$$

$$H(x)\approx -\chi |x|,\chi \equiv \left(\gamma /2\right){\int }_{-\infty }^{+\infty }{\psi }_{\downarrow }^{\ast }\left(x\right){\psi }_{\uparrow }\left(x\right)dx.$$

$$\gamma \equiv Nm{\epsilon }_0{\mu }_0^2{\omega }_{\mathrm{MW}}^2{M}^2/({\hslash }^{2}A{k}^3)$$

$$\Psi\left(x,t\right)=\left(\begin{array}[c]{c}\psi_{\downarrow}(x)\\ \psi_{\uparrow}(x)\end{array}\right)e^{-i\mu t},$$

$$\mu\left(\begin{array}[]{c}\psi_{\downarrow}\\ \psi_{\uparrow}\end{array}\right)=\left[-\frac{1}{2}\partial_{x}^{2}+\left(\begin{array}[]{cc}iK\partial_{x}&\Omega-H\\ \Omega-H^{\ast}&-iK\partial_{x}\end{array}\right)-\left(\begin{array}[]{cc}\beta_{1}|\psi_{\downarrow}|^{2}+\beta_{2}|\psi_{\uparrow}|^{2}&0\\ 0&\beta_{1}|\psi_{\uparrow}|^{2}+\beta_{2}|\psi_{\downarrow}|^{2}\end{array}\right)\right]\left(\begin{array}[]{c}\psi_{\downarrow}\\ \psi_{\uparrow}\end{array}\right).$$

$$\mu\psi=\left(\begin{array}[c]{cc}-\frac{1}{2}\partial_{x}^{2}+iK\partial_{x}&\Omega+\chi|x|\\ \Omega+\chi^{\ast}|x|&-\frac{1}{2}\partial_{x}^{2}-iK\partial_{x}\end{array}\right)\psi,$$

$${\psi }^{(\mathrm{VA})}=\left[\cos \alpha \left(\genfrac{}{}{0ex}{}{\cos \theta }{\sin \theta }\right)e^{-i{p}_{0}x}-\sin \alpha \left(\genfrac{}{}{0ex}{}{\sin \theta }{\cos \theta }\right)e^{+i{p}_{0}x}\right]\varphi (x),$$

$$\varphi (x)={\psi }_0\mathrm{Ai}(\left|x\right|/\sigma +{\xi }_0),$$

$$\displaystyle E=\int_{-\infty}^{+\infty}dx\psi^{\dagger}\left[-\frac{1}{2}\frac{d^{2}}{dx^{2}}+\left(\begin{array}[c]{cc}iK\partial_{x}&\Omega\\ \Omega&-iK\partial_{x}\end{array}\right)\right]\psi$$

$$+\frac{\gamma }{4}\int \int dxd{x}^{\prime }\left|x-x^{\prime }\right|\left[{\psi }_{\downarrow }^{\ast }\left(x\right){\psi }_{\uparrow }\left(x\right){\psi }_{\downarrow }^{\ast }\left(x^{\prime }\right){\psi }_{\uparrow }\left(x^{\prime }\right)+c.c.\right].$$

$${\varphi }_{\downarrow }(x)\approx C\left(H^{\ast }-\Omega \right){\varphi }_{\uparrow }(x),$$

$$i{\partial }_t{\varphi }_{\uparrow }=\left[-\frac{1}{2}{\partial }_x^2-i(p_0-1){\partial }_x+2C\Omega H-{\beta }_1{\left|{\varphi }_{\uparrow }\right|}^2\right]{\varphi }_{\uparrow },$$

$${\partial }_x^{2}H=C\gamma \Omega |{\varphi }_{\uparrow }{|}^2.$$

$${\varphi }_{\uparrow }={\varphi }_{\uparrow }^{\prime }\left(x^{\prime },t\right)\exp \left[i\left(axt+\frac{1-p_0}{2}C{t}^2-\frac{a^2{t}^3}{6}\right)\right],$$

$$x^{\prime }=x-\left(a/2\right)t^2,H^{\prime }=H+(2C\Omega {)}^{-1}ax.$$

$${\Psi }_{\uparrow }(x,t)$$

$${\Psi }_{\downarrow }\left(x,t\right)$$

$${\Phi }_{\downarrow }\approx {\mu }_0^{-1}\left(\Omega -H^{\ast }\right)e^{2ix}{\Phi }_{\uparrow }.$$

$$i{\partial }_t{\Phi }_{\uparrow }=\left[-(1/2){\partial }_x^2\mp 2H(x)-\left({\beta }_1+{\beta }_2\right){\left|{\Phi }_{\uparrow }\right|}^2\right]{\Phi }_1,$$

$${\partial }_x^{2}H=\mp \gamma {\left|{\Phi }_{\uparrow }(x^{\prime })\right|}^2,$$

$$\displaystyle i\partial_{t}\left(\begin{array}[]{c}\Psi_{\downarrow}\\ \Psi_{\uparrow}\end{array}\right)$$

$$\left({\partial }_x^2+{\partial }_y^2\right)H=-\gamma {\Psi }_{\downarrow }^{\ast }{\Psi }_{\uparrow }.$$

$$i{\partial }_t{\Phi }_{\uparrow }=-\frac{1}{2}\left({\partial }_{xx}^2+{\partial }_{yy}^2\right){\Phi }_{\uparrow }\mp 2\mathrm{Re}(H){\Phi }_{\uparrow }-\left({\beta }_1+{\beta }_2\right)|{\Phi }_{\uparrow }{|}^2{\Phi }_{\uparrow },$$

$$\left({\partial }_{xx}^2+{\partial }_{yy}^2\right)H=\mp \gamma {\left|{\Phi }_{\uparrow }\right|}^2.$$

$$x^{\prime }=x-V_{x}t-(1/2)a_x{t}^2,y^{\prime }=y-V_{y}t-(1/2)a_y{t}^2,$$

$$H=H^{\prime }\pm (a_{x}x+a_{y}y)/2,$$

$${\Phi }_{\uparrow }={\Phi }_{\uparrow }^{\prime }\left(x^{\prime },y^{\prime },t\right)\exp \{i[a_{x}t+V_x)x+(a_{y}t+V_y)y-\varphi (t)]\},$$

$$\varphi (t)=\frac{{\left(a_{x}t+V_x\right)}^3-V_x^3}{6a_x}+\frac{{\left(a_{y}t+V_{x}t\right)}^3-V_y^3}{6a_y}.$$

$$H\approx \mp \frac{\gamma }{2\pi }\left(\int \int {\left|{\Phi }_{\uparrow }\left(x,y\right)\right|}^{2}dxdy\right)\ln r.$$

$$i{E}_z+(1/2)\left(E_{xx}+E_{yy}\right)+nE=0,$$

$$n-l^2\left(n_{xx}+n_{yy}\right)=|E{|}^2.$$

Full text

I Introduction

Self-accelerating Airy waves were predicted in 1979 in the context of quantum mechanics berry . Then, this concept was transferred to optics siv , plasmonics plasmonics , acoustics acc , gas discharge gas-discharge , and hydrodynamics water , using the similarity of the linear Schrödinger equation to the paraxial wave-propagation equation in classical physics. These wave modes offer applications to plasma guiding 10 , signal transmission 12 ; 13 , laser-beam filamentation 14 , optical micromanipulation 15 ; 16 ; 17 ; 18 ; 19 ; 20 , generation of ”light bullets” 21 ; 22 ; 23 ; 24 , and so on 26 ; 27 ; 28 .

Quantum self-accelerating waves have been experimentally demonstrated in electron optics electron . Self-accelerating Dirac waves have also been predicted in relativistic quantum mechanics dirac . Coherent Bose-Einstein condensates (BEC) may be appropriate for the realization of the self-acceleration in matter waves. The latter effect has not yet been demonstrated experimentally, although it has been theoretically elaborated, assuming the use of laser beams to imprint appropriate phase modulation onto the BEC kli , or the use of a trapping potential moving with acceleration yuc .

Ideal Airy waves with slowly decaying oscillatory tails carry an infinite norm, therefore truncated Airy waves with a finite norm were used in the theory and experiments siv ; pan ; however, the truncation causes gradual decay of the self-accelerating wave packets. Furthermore, the study of the evolution of the Airy waves, which are eigenmodes of the linear propagation, in various nonlinear media ell ; jia ; abd ; huy ; ka ; lot ; fat ; rud ; pan ; zhang ; ka2 ; do ; dri ; all ; Thawatchai shows that the nonlinearity causes deformation and, often, destruction of the self-accelerating waves. Another type of self-accelerating solitary-wave pairs was predicted Peschel and experimentally demonstrated Peschel-exper in nonlinear photonic crystals with opposite signs of the dispersion (effective mass) for the paired modes.

The above-mentioned settings were implemented neglecting dissipation in the medium. On the other hand, robust optical tail-free self-accelerating pulses have been predicted and experimentally demonstrated under the action of various nonconservative effects, such as the sliding-frequency filtering sliding , ionization of the dielectric medium ionization , diffusion in photorefractive crystals diffusion , and intra-pulse stimulated Raman scattering Raman . In the latter case, the number of photons (integral norm of the pulse) is conserved, but the Raman effect breaks the conservation of the momentum and Hamiltonian.

The present work shows that the long-range nonlinear interaction between constituents of a binary BEC, mediated by a microwave (MW) field (this interaction was elaborated in Refs. we and we2 ), supports spatially symmetric tail-free self-accelerating hybrid solitons, both one- and two-dimensional (1D and 2D), built of matter-wave and MW components (in that sense, they resemble exciton-polariton solitons, which are also matter-field hybrids Exc , although the present model is a strictly conservative one, while exciton-polaritons states exist in dissipative semiconductor cavities, therefore they should be supported by pump fields). It is relevant to stress that both the self-trapping and self-acceleration are induced by the same interaction, while in previously studied nonlinear systems the acceleration was driven by terms such as the induced-Raman-scattering one, while the self-trapping was provided by the Kerr nonlinearity. Further, we demonstrate that the self-acceleration mechanism works equally well in the presence of the spin-orbit coupling (SOC) between the two components of the BEC wave function, as well as in the presence of the usual nonlinear contact interactions. In addition to the results for the matter-wave solitons, it is demonstrated that similar self-accelerating optical solitons can be produced in conservative optical media with strongly nonlocal nonlinearity.

The rest of the paper is organized as follows. The 1D model is formulated in Sec. II, which includes the linear SOC effect in the two-component BEC. Systematic results, both analytical and numerical ones, for the self-acceleration for 1D solitons are reported in Sec. III. The 2D extension of the model is presented in Sec. IV, where emphasis is made on the stable self-acceleration of vortices. This section also includes a brief consideration of a nonlocal optical system which may be represented by a similar model, thus predicting similar results for 1D and 2D self-accelerating spatial optical solitons. The paper is concluded by Sec. V.

II The one-dimensional system

Because some essential results reported below are obtained for the binary BEC including the SOC effect, it is relevant to outline its implementation in the relevant setting. It may be realized using the scheme shown in Fig. 1 Spielman : counterpropagating Raman-laser beams $L_{1}$ and $L_{2}$ drive the atomic gas, adiabatically eliminating level 2 and creating the SOC system with pseudospin 1/2, whose components represent atoms in states 0 and 1. The emulation of various aspects of gauge physics in ultracold gases by means of SOC has drawn a great deal of interest lin -liyun1 . In particular, SOC solitons have been predicted in 1D 1D ; wubiao , 2D Ben-Li ; chuanwei ; 2D-SOC-solitons , and 3D 3D geometries, see also a review in EPL . However, SOC breaks the Galilean invariance, which makes the creation of moving solitons a nontrivial issue wubiao ; Ben-Li ; chuanwei . It was found that 2D solitons in the system with SOC of the Rashba type feature mobility only in one direction, up to a critical value of the velocity, beyond which delocalization occurs Ben-Li . Self-accelerating solitons have not been found in previous works dealing with SOC models.

We here consider the pseudo-spinor BEC with two components coupled by the interaction mediated by the magnetic component of the MW field, as schematically shown in Fig. 1 we . It has been found that the interaction gives rise to quiescent hybrid MW-matter-wave solitons in 1D we , as well as to giant solitary vortices in 2D, which are stable, at least, up to topological charge $S=5$ we2 . The solitons persist in the presence of additional contact interactions of either sign, corresponding to self- and cross-attraction or repulsion of the pseudo-spinor’s components.

In the 1D setting, components of the pseudo-spinor wave function, $\Psi\equiv\binom{\Psi_{\downarrow}}{\Psi_{\uparrow}}$, which correspond to states 0 and 1 in Fig. 1, coupled by MW magnetic-field potential $H$, obey coupled Gross-Pitaevskii equations (GPEs), which may or may not include the SOC terms with strength $K$, represented by the first spatial derivatives, which are combined with the Rabi-coupling frequency, $\Omega$:

[TABLE]

Real coefficients $\beta_{1}$ and $\beta_{2}$ represent here, severally, the self- and cross-component contact interactions, $\beta_{1,2}>0$ ($<0$) corresponding, respectively, to the attractive (repulsive) sign of the interactions. The wave function is subject to the normalization,

[TABLE]

The feedback of the matter-wave components on the MW potential (a specific manifestation of the general local field effect LFE ) is accounted for by the Poisson equation we ,

[TABLE]

whose solution can be written with the help of the 1D Green’s function:

[TABLE]

Note that the asymptotic form of the potential, produced by Eq. (4) at $|x|\rightarrow\infty$, is a linear function of the coordinate, which is a commonly known property of solutions to the 1D Poisson equation with a localized source of the field:

[TABLE]

In the presence of SOC, Eq. (1), the position, energy and time are scaled, respectively, by the inverse SOC wavenumber, $k^{-1}$, recoil energy, $E_{R}\equiv\hbar^{2}k^{2}/(2m)$, and $\hbar/E_{R}$, so that $K\equiv 1$ is fixed in Eq. (1), unless we consider the system without SOC, by setting $K=0$. Further, $U\left(x\right)$ in Eq. (1) is a trapping potential (actually, we aim to consider free-space solitons, with $U=0$), and

[TABLE]

is the effective strength of the MW-mediated long-range interaction, with $\varepsilon_{0}$ and $\mu_{0}$ being the vacuum permittivity and permeability, $N$ the number of atoms, $m$ the atomic mass, $\omega_{\mathrm{MW}}$ the MW frequency, $M$ the atomic magnetic moment, and $A$ the confinement area in the transverse plane. Considering, for instance, BEC of 87Rb atoms, transversely confined in area $\sim 1$ $\mathrm{\mu}$m2, coupled to MW with wavelength of $\sim 1$ mm, and the action of SOC with wavenumber $k\sim 1\mathrm{\mu}m^{-1}$, Eq. (6) yields $\gamma\sim 10^{-9}N$. Thus, for BEC of $10^{7}$ atoms (actually, condensates made of up to $10^{8}$ atoms are available, according to current experimental results dis ), one obtains $\gamma\sim 10^{-2}$. Following this estimate, we fix $\gamma=0.02$ in numerical simulations following below. Furthermore, by using even tighter transverse confinement to reduce the transverse-localization area, $A$, the necessary number of atoms can be made essentially smaller than $10^{7}$, which is used here as the estimate.

III Self-accelerating one-dimensional solitons

First, it is necessary to produce stationary solitons with real chemical potential $\mu$, in the form of

[TABLE]

with the “pseudo-spin-up” and “down” components of the complex spinor wave function, $\psi_{\uparrow,\downarrow}(x)$, obeying the stationary free-space version of Eq. (1), with $U(x)=0$:

[TABLE]

Actually, it is more convenient to produce the stationary wave function not through Eq. (8), but rather solving Eqs. (1)-(4) by dint of the imaginary-time-integration method. The simulations were carried out in domain $|x|\leq 80$, or in a smaller one, if it was sufficient for a particular situation, with zero boundary conditions at edges of the domain. As a result, three distinct species of the 1D solitons have been identified, viz., ones of the regular, stripe, and plane-wave types (solitons of the latter type are carried by the plane-wave phase, see below). In the most fundamental case, when the contact interactions are absent, i.e., $\beta_{1,2}=0$ in Eq. (1), and solitons may only be supported by the MW-mediated interaction, typical examples of the three species are displayed, respectively, in the left top ($\Omega=1.5$), middle ($\Omega=0.7$), and bottom ($\Omega=0.7$) panels of Fig. 2 for $\gamma=0.02$, along with the respective profiles in the momentum space, in the right panels. In particular, the stripe and plane-wave-type solitons coexist at the same values of parameters, being almost mutually degenerate under normalization condition (2), with chemical potentials $\mu={-0.7300}$ and ${-0.7298}$, respectively. A distinctive peculiarity of the soliton of the plane-wave type is asymmetry between its components.

The soliton of the regular type, presented in Fig. 2(a1,a2) for $\Omega^{2}>1$, resembles those in the usual 1D SOC model (which does not include the MW-mediated coupling between the components) review , with zero carrier wavenumber, $p_{x}=0$, while at $\Omega^{2}<1$ the formal linearization of Eq. (1) for tails of solitons [which implies setting $H=0$, according to Eq. (4)] demonstrates that they may develop oscillations with wavenumbers $p_{x}=\pm\sqrt{\sqrt{\mu^{2}-\Omega^{2}}+1-|\mu|}$ (the solitons exist at $\mu<-1$). Solitons carried by $p_{x}$ with the single sign are categorized as modes of the plane-wave type, while the superposition of the two wavenumbers with opposite signs gives rise to stripe solitons liyun1 . However, MW potential $H$, dressing the condensate, completely changes the asymptotic shape (tails) of localized modes at large $|x|$, where, taking Eq. (5) into regard, Eq. (8) reduces to

[TABLE]

with $\chi$ defined as per Eq. (5).

The presence of the effective linear potential $\sim|x|$ in the asymptotic equation (9) suggests to approximate solutions by Airy functions berry . Accordingly, the variational approximation (VA) for solutions to Eq. (8) may be based on the following ansatz:

[TABLE]

where the shape function is chosen as

[TABLE]

with $\xi_{0}\approx-1.019$ being the first local maximum of the Airy function, ${\mathrm{Ai}}(\xi)$, hence $x=0$ is the center of the adopted profile. Variational parameters are $\alpha$, $\theta$, $p_{0}$, and $\sigma$, while $\psi_{0}$ is a normalization constant. In contrast, in the usual 1D SOC model, which does not include the MW field, $\phi(x)$ is approximated by a sech or Gaussian ansatz liyun1 . At $\alpha=0$ or $\pi/2$, ansatz (10) reduces to an envelope multiplying the plane wave, while at $\alpha=\pi/4$, depending on $\Omega$, the ansatz may represent either a stripe soliton, if wavelength $2\pi/p_{0}$ is small in comparison with the envelope’s width, or a regular single-peak soliton otherwise. Values of the variational parameters are numerically determined by minimizing the system’s energy,

[TABLE]

In Eq. (15), the contact interactions are again disregarded, setting $\beta_{1,2}=0$ in Eq. (1), aiming to address the fundamental setting, with the two components of the pseudo-spinor wave function interacting solely via the MW magnetic field. The VA-predicted soliton profiles are displayed in Fig. 2 along with their numerically found counterparts, demonstrating that the VA is reasonably accurate, providing, in particular, an accurate approximation for the regular solitons.

The MW-mediated long-range interaction, which is the underlying ingredient of the present system, determines not only the shape of the solitons, but also their dynamics, as shown by systematic simulations of Eq. (1). It was found that the addition of small random perturbations to the regular and stripe solitons does not produce any conspicuous effect (which implies that they are completely stable modes in the quiescent state), while perturbed solitons of the plane-wave type start self-accelerated motion, keeping their integrity, as shown in Fig. 3 for the system without the contact interactions ($\beta_{1,2}=0$).

Stable self-accelerating solitons persist in the presence of the contact nonlinearity with the repulsive sign, as shown in Figs. 4, as well as under the action of relatively weak local attraction, see Fig. 5. Strong attraction may change the situation, as it tends to transform the solitons considered here, characterized by the ansatz based on Eqs. (10) and (11), into usual sech-shaped solitons, for which the interaction with the MW field becomes negligible. Naturally, the strong self-repulsion makes the moving soliton much broader, as seen in Fig. 4. As concerns the gradual decrease of the acceleration, observed in Fig. 4, starting from $t\simeq 4500$, and in Fig. 5, starting from $t\simeq 7000$, detailed consideration of the numerical data demonstrates that this effect is explained by a brake force, which is applied to the soliton by radiation emitted by it at the initial stage of the evolution, and eventually reflected from the edge of the integration domain.

These results suggest existence of a family of stable self-accelerating solitons of the plane-wave type, including the quiescent and moving solitons displayed in Figs. 2(c1,c2) and (3), respectively, the acceleration being an internal parameter of the family. In the system dominated by the SOC terms, the family can be constructed in an analytical form. To this end, we look for the corresponding solution to Eq. (1) as $\Psi_{\uparrow\downarrow}\left(x,t\right)=\operatorname{exp}\left(ip_{0}x-i\mu t\right)\phi_{\uparrow\downarrow}\left(x,t\right)$, with slowly varying amplitudes $\phi_{\uparrow\downarrow}\left(x,t\right)$, carrier momentum $p_{0}$, and chemical potential $\mu$. Using Eq. (1), the small component [$\phi_{\downarrow}$, see Fig. 2(c1)] is eliminated in favor of the larger one:

[TABLE]

with $C\equiv\left(p_{0}-\mu+p_{0}^{2}/2\right)^{-1}$ [recall we set $K\equiv 1$ in Eq. (1), if the SOC terms are present]. This approximation for $\phi_{\downarrow}(x)$ is plotted in Fig. 2(c1) by means of the yellow dashed-dotted curve, showing very good agreement with its numerically found counterpart. Further, substituting Eq. (16) in the remaining equation for $\phi_{\uparrow}$ in system (1) leads, in the first approximation note0 , to a single GPE, in which term $CH^{\ast}\phi_{\uparrow}$ from Eq. (16) couples wave function $\phi_{\uparrow}$ to potential $H$:

[TABLE]

Further, the substitution of relation (16) in Eq. (3) leads to the Poisson equation, which gives rise to real MW potential $H$ [unlike the complex field, produced by Eqs. (3) and (4) in the general case]:

[TABLE]

Then, the existence of self-accelerating solitons is explained by the fact that Eqs. (17) and (18) are invariant with respect to an exact transformation from the laboratory reference frame to one moving at an arbitrary constant acceleration, $a$ (while the usual GPE, as well as nonlinearly coupled GPE systems, are not invariant with respect to this transformation Quebec ; Parker ):

[TABLE]

Thus, any quiescent soliton produced by Eqs. (17) and (18) generates a family of solitons moving with arbitrary acceleration ($a$), which is indeed the intrinsic parameter of the family, as conjectured above. In particular, term $(2C\Omega)^{-1}ax$, added to the MW potential by the transformation, naturally breaks the symmetry of the linear in $x$ asymptotic field (5) with respect to to $x>0$ and $x<0$.

Note that, when $p_{0}=1$ [i.e., the SOC term vanishes in Eq. (17)], transformation (19) remain valid, and Eqs. (17) and (18) are tantamount to the model introduced in Ref. we without SOC (however, moving solitons were not considered in that work). Thus, the existence of the family of robust self-accelerating solitons does not depend on the presence of the SOC terms in Eq. (1), and it is not broken either by the inclusion of the contact-interaction term with coefficient $-\beta_{1}$. The invariance with respect to the self-accelerating transformation cannot be produced in an exact form for the full system of Eqs. (1) and (3), but its ability to maintain the self-acceleration is clearly demonstrated by numerical results (in particular, by Figs. 3-5).

The GPE system may also be reduced to the single equation in the case opposite to that considered above, namely, if the Rabi coupling dominates over SOC in Eq. (1), i.e., $\Omega$ is a large parameter. In this case, the substitution of

[TABLE]

with $\mu_{0}=\pm\Omega$ and slowly varying amplitudes $\Phi_{\uparrow\downarrow}\left(x,t\right)$, yields a relation between them,

[TABLE]

On the contrary to the above case, when component $\phi_{\downarrow}$ was small in comparison with $\phi_{\uparrow}$, as per Eq. (16), Eq. (21) implies that the absolute values of the two components are nearly equal. Eventually, the substitution of expressions (20) and (21) in Eqs. (37) and (3) leads to the following equations, which differ from Eq. (17), with $p_{0}=1$ (without the SOC term) and Eq. (18) only by the notation for coefficients:

[TABLE]

i.e., the single GPE limit is a *universal *one, being equally relevant in the cases of strongly unequal and nearly equal components of the pseudo-spinor wave function.

IV Two-dimensional systems

IV.1 The pseudo-spinor condensate coupled by the microwave field

In the 2D setting, Eqs. (1) and (3) are replaced by equations which combine the 2D version of SOC Ben-Li ; chuanwei ; 2D-SOC-solitons and the interaction of the pseudo-spinor wave function with the MW field in two dimensions we2 :

[TABLE]

[TABLE]

To provide straightforward insight into dynamics of the 2D system, we again resort to the limit case of the strong Rabi coupling between the two components in Eq. (33), which dominates over SOC. Then, two components of the pseudo-spinor may be reduced to one, cf. Eq. (21), and the system of Eqs. (33) and (34) amounts to the single-component GPE coupled to the 2D Poisson equation, i.e., as a matter of fact, the 2D extension of Eqs. (17) and (18):

[TABLE]

[TABLE]

A straightforward but crucially important fact is that, similar to what was found above for the 1D system [see Eq. (19)], Eqs. (35) and (36) are invariant with respect to the transformation to the reference frame which moves in the 2D space with arbitrary initial velocities $V_{x}$,$~{}V_{y}$ and arbitrary constant accelerations $a_{x}$,$~{}a_{y}$:

[TABLE]

Accordingly, coordinates $\left(x_{c},y_{c}\right)$ of the center of the stable 2D soliton (which may carry embedded vorticity we2 ) move as $x_{c}=V_{x}t+(1/2)a_{x}t^{2},~{}y_{c}=V_{y}t+(1/2)a_{y}t^{2}$, which may be a curvilinear trajectory in the 2D plane: at small $t$, it is close to a straight line with slope $x/y=V_{x}/V_{y}$, while at $t\rightarrow\infty$ it becomes asymptotically close to a line with $x/y=a_{x}/a_{y}$. In particular, in the case of $a_{x}=V_{y}=0$ the trajectory is a parabola: $y_{c}=\left(a_{y}/2V_{x}^{2}\right)x_{c}^{2}$.

Note that the solution of the 2D Poisson equation (36) has the standard logarithmic asymptotic form far from the region where the source of the field is located:

[TABLE]

The difference of the field component of the self-accelerating 2D solitons from that of their quiescent counterparts is more essential than in the 1D case, as the potential terms linear in $x$ and $y$ [see Eq. (37)] are qualitatively different from the logarithmic term in Eq. (38).

The predictions are corroborated, in Figs. 6 and 7, by numerical solutions of Eqs. (35) and (36) (in the absence and presence of the contact interaction, severally) for stable 2D solitons with embedded vorticity $S=1$ (vortex rings), which move at a constant acceleration in the $x$ direction, in exact agreement with Eq. (37). A remarkable fact is that, in the case shown in Fig. 7, the accelerating vortex soliton remains stable in the presence of a relatively strong contact self-attraction with $\beta_{1}+\beta_{2}=10$ in Eq. (35), in spite of the well-known propensity of the cubic self-attraction to destabilize 2D vortex-ring solitons against the collapse and ring splitting 2D . Note also that the action of the self-attraction naturally leads to compression of the vortex ring. Finally, it is relevant to stress that the acceleration observed in Figs. 6 and 7 is much larger than in Figs. 3-5, because in the latter case it was induced by small random perturbations initially added to the 1D solitons, while in the situation displayed in Figs. 6 and 7 the acceleration was explicitly added to the input generating the 2D vortex solitons, as per Eq. (37).

IV.2 The nonlocal optical system

The mechanism of the formation of the tailless self-accelerating solitons, elaborated above in terms of the two-component BEC coupled by the MW field, can be realized as well in a nonlocal optical model with amplitude $E\left(x,y,z\right)$ of the electromagnetic wave and local perturbation $n\left(x,y,z\right)$ of the refractive index, which are governed by the coupled system of the paraxial propagation equation and an equation which determines how the index perturbation is created by the field distribution nonlocal :

[TABLE]

Here $z$ is the propagation distance, $x$ and $y$ are transverse coordinates, and $l$ is the correlation length of the nonlocality. Rescaling $E\equiv l{\mathcal{E}}$ and taking the limit of a strong nonlocality, $n/l\rightarrow 0$, Eqs. (39) and (40) are reduced to a form tantamount to Eqs. (35) and (36):

[TABLE]

The 1D reduction of the 2D system (41) is obviously possible too. Thus, stable 1D and 2D self-accelerating solitons may be predicted in this optical setting too.

V Conclusion

The objective of this work is to investigate the dynamics of the binary BEC whose components, representing different hyperfine atomic states, are coupled by the magnetic component of the MW (microwave) field. In the general case, the SOC (spin-orbit coupling) is included too. The effective interaction between the two components via the feedback of the atomic states on the MW supports self-trapped modes (solitons), whose asymptotic form is the same as that of the Airy function, in the 1D case. In the presence of SOC, we have found stable 1D solitons of the regular (single-peak) and stripe types, and solitons in the form of envelopes carried by plane waves. The most essential finding is the existence of stable self-accelerating solitons of the plane-wave type and their 2D counterparts, including vortex solitons. In contrast to the previously studied self-accelerating Airy waves siv , in the present system the solitons keep simple self-trapped shapes, without oscillatory tails attached to them, hence their integral norm is well defined and convergent, unlike the divergent norm of the exact Airy waves. The present system, being conservative, is also different from previously studied models which admit self-acceleration of localized modes in optical media featuring nonconservative effects, such as the sliding filtering, diffusion, ionization, and stimulated Raman scattering. The existence of the family of the self-accelerating solitons is demonstrated analytically, by reducing the two-component systems, in the 1D and 2D setting alike, to a single GPE, coupled to the Poisson equation for the potential of the MW field. These reduced systems admit the exact transformation to a reference frame moving with arbitrary acceleration, thus generating self-accelerating solitons from quiescent ones in the exact form. In the 2D geometry, the transformation generates 2D solitons which may move along curved trajectories, due to the interplay between the 2D velocity and acceleration. The self-acceleration mechanism persists if the contact nonlinearity is included. It can also be realized in strongly nonlocal optical media, with the 1D or 2D transverse geometry.

Acknowledgments G.J. Dong acknowledges the support by the National Science Foundation of China (grants No. 11574085 and 91536218), and 111 Project ( B12024), the National Key Research and Development Program of China (Grant No. 2017YFA0304201)，as well as Innovation Program of Shanghai Municipal Education Commission. Z.X.Liang acknowledges the support of the National Science Foundation of China (Grant No. 11374125), the key projects of the Natural Science Foundation of China (Grant No. 11835011) and Youth Innovation Promotion Association of the Chinese Academy of Sciences (Grant No. 2013125). J. L. Qin acknowledges the support by the National Science Foundation of China (No. 11847059). The work of B.A.M. was supported, in part, by the joint program in physics between NSF and Binational (US-Israel) Science Foundation through project No. 2015616, and by the Israel Science Foundation through grant No. 1286/17. This author appreciates hospitality of the State Key Laboratory of Precision Spectroscopy at East China Normal University (Shanghai).