Dynamical control of solitons in a parity-time-symmetric coupler by periodic management

TL;DR

This paper investigates the dynamical control of PT-symmetric solitons in a dual-core nonlinear waveguide with periodic management of gain, loss, and coupling, revealing stability conditions and soliton interactions.

Contribution

It introduces a novel periodic management scheme for PT-symmetric couplers, enabling exact solutions and stability analysis of solitons with potential optical applications.

Findings

Identified stability regions depending on management parameters.

Derived analytical predictions for long-period soliton evolution.

Analyzed collisions between moving solitons in the system.

Abstract

We consider a dual-core nonlinear waveguide with the parity-time (PT) symmetry, realized in the form of equal gain and loss terms carried by the coupled cores. To expand a previously found stability region for solitons in this system, and explore possibilities for the development of dynamical control of the solitons, we introduce "management" in the form of periodic sinusoidal variation of the loss-gain (LG) coefficients, along with synchronous variation of the inter-core coupling (ICC) constant. This system, which can be realized in optics (in the temporal and spatial domains alike), features strong robustness when amplitudes of the variation of the LG and ICC coefficients keep a ratio equal to that of their constant counterparts, allowing one to find exact solutions for PT-symmetric solitons. A stability region for the solitons is identified in terms of the management amplitude and period, as well as the soliton's amplitude. In the long-period regime, the solitons evolve adiabatically, making it possible to predict their stability boundaries in an analytical form. The system keeping the Galilean invariance, collisions between moving solitons are considered too.

Equations

$$U(-\mathbf{r})=U^{\ast }(\mathbf{r}),$$

$$i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{{\partial }^{2}u}{\partial t^2}+|u{|}^{2}u-i\left[{\gamma }_0+{\gamma }_1\sin \left(\frac{2\pi }{L}z\right)\right]u+\left[1+\delta \sin \left(\frac{2\pi }{L}z+\phi \right)\right]v$$

$$i\frac{\partial v}{\partial z}+\frac{1}{2}\frac{{\partial }^{2}v}{\partial t^2}+|v{|}^{2}v+i\left[{\gamma }_0+{\gamma }_1\sin \left(\frac{2\pi }{L}z\right)\right]v+\left[1+\delta \sin \left(\frac{2\pi }{L}z+\phi \right)\right]u$$

$${\gamma }_0\ge 0,{\gamma }_1\ge 0,\delta \ge 0$$

$$v\left(z,t\right)=\left(i{\gamma }_0\pm \sqrt{1-{\gamma }_0^2}\right)u\left(z,t\right),$$

$$v\left(z,t\right)=A\exp \left[i\left(A^2/2\pm \sqrt{1-{\gamma }_0^2}\right)z\right]\mathrm{sech}\left(At\right),$$

$${\left|u\left(z,t\right)\right|}^2={\left|v\left(z,t\right)\right|}^2,$$

$$A^2\le A_{\mathrm{crit}}^2\left({\gamma }_0\right)=(4/3)\sqrt{1-{\gamma }_0^2},$$

$$\delta /{\gamma }_1=1/{\gamma }_0.$$

$$v\left(z,t\right)=A\exp \left[i\left(A^2/2\pm \sqrt{1-{\gamma }_0^2}\right)z\mp i\frac{L{\gamma }_1\sqrt{1-{\gamma }_0^2}}{2\pi {\gamma }_0}\cos \left(\frac{2\pi }{L}z\right)\right]\mathrm{sech}\left(At\right),$$

$$L\gg 4\pi /A^2.$$

$$t$$

$$z$$

$$\left(u,v\right)$$

$$i\frac{\partial \tilde{u}}{\partial \tilde{z}}+\frac{1}{2}\frac{{\partial }^2\tilde{u}}{\partial {\tilde{t}}^2}+|\tilde{u}{|}^2\tilde{u}-i{\gamma }_0\tilde{u}+\tilde{v}$$

$$i\frac{\partial \tilde{v}}{\partial \tilde{z}}+\frac{1}{2}\frac{{\partial }^2\tilde{v}}{\partial {\tilde{t}}^2}+|\tilde{v}{|}^2\tilde{v}+i{\gamma }_0\tilde{v}+\tilde{u}$$

$$\delta <1,$$

$$A_0^2<{\left(A_0^2\right)}_{\mathrm{crit}}=\frac{4}{3}\sqrt{1-{\gamma }_0^2}\left(1-\delta \right)\equiv \frac{4}{3}\sqrt{1-{\gamma }_0^2}\left(1-\frac{{\gamma }_1}{{\gamma }_0}\right)$$

$$\left\{u_{\eta },v_{\eta }\right\}=\left\{u_0\left(z,t-cz\right),v_0\left(z,t-cz\right)\right\}\exp \left(-\left(i/2\right)c^{2}z\right),c=\eta .$$

Full text

adiabatic approximation; dual-core waveguides; gain-loss balance; cubic nonlinearity

1.  Introduction

One of the most fundamental tenets in physics is the charge-parity-time ($\mathcal{CPT}$) symmetry, which holds for all Lorentz-invariant systems obeying the causality principle field1 ; field2 . It implies invariance of the system with respect to the combined parity transformation, $\mathcal{P}$, which reverses the coordinate axes; charge conjugation, $\mathcal{C}$, which swaps particles and antiparticles; and time reversal, $\mathcal{T}$. Its reduced forms, such as $\mathcal{PT}$ and $\mathcal{CP}$ symmetries, may be violated in specific situations, but they also play a profoundly important role in many physical theories. The usual proof of the presence of the latter symmetries is performed to Hermitian Hamiltonians, whose eigenvalues are always real.

However, the invariance of the system with respect to the $\mathcal{PT}$ and $\mathcal{CP}$ transformations does not imply that the underlying Hamiltonian must necessarily be Hermitian. Indeed, it was known from some early examples 1 -5 , and was then discovered, in the systematic form, by Bender and Boettcher bender1 ; bender2 (see also review bender3 and book ptqm ) that, in the most general case, Hamiltonians which commute with the $\mathcal{PT}$ operator may include a dissipative (anti-Hermitian) term. Such $\mathcal{PT}$-symmetric non-Hermitian Hamiltonians often include a complex potential, $U(\mathbf{r})$, whose real and imaginary parts must be, respectively, even and odd functions of spatial coordinates ($\mathbf{r}$), i.e.,

[TABLE]

where $\ast$ stands for the complex conjugate. Actually, $\mathcal{PT}$-symmetric Hamiltonians may admit transformation into Hermitian ones Mostafazadeh ; Barash . A well-established fact is that the spectrum of Hamiltonians with complex potentials subject to constraint (1) is real below a critical strength of the imaginary part of the potential, at which the $\mathcal{PT}$ symmetry gets broken, making the system unstable breaking (exceptions in the form of models with unbreakable $\mathcal{PT}$ symmetry are known too unbreakable ).

Thus far, the $\mathcal{PT}$ symmetry was not directly realized in quantum systems with complex potentials. On the other hand, a possibility to realize it was predicted for classical optical media with symmetrically inserted gain and loss theo1 -review . This possibility is based on the commonly known similarity between the quantum-mechanical Schrödinger equation and the propagation equation for optical waveguides, written in the paraxial approximation. Following these ideas, the $\mathcal{PT}$ symmetry was experimentally implemented in various optical and photonic systems exp1 -exci3 . Emulation of the $\mathcal{PT}$ symmetry was also predicted in atomic Bose-Einstein condensates (BECs), assuming that the gain may be provided by elements working as matter-wave lasers Cartarius .

As concerns the emulation of fundamental properties of quantum systems in terms of classical optics and in semi-classical BEC, (quasi-) $\mathcal{CP}$ symmetries may be implemented too, in continuous Flach ; CP1 ; CP2 and discrete Hadi media alike.

The $\mathcal{PT}$ symmetry in an optical waveguide (as well as its $\mathcal{CP}$  counterpart) may naturally combine with the material Kerr nonlinearity, giving rise to propagation models based on cubic nonlinear Schrödinger equations (NLSEs) with the complex potentials subject to condition (1). These models may generate $\mathcal{PT}$-symmetric solitons, which were addressed in many theoretical works soliton , Konotop -Barash-discr , unbreakable (see also reviews review1 ; review2 ), and experimentally demonstrated too exp7 . Although the presence of the gain and loss makes $\mathcal{PT}$-symmetric media dissipative, solitons exist in them in continuous families, similar to the commonly known situation in conservative models families , while usual dissipative solitons exist as isolated solutions (attractors, if they are stable) diss2 .

One of basic settings for the realization of the $\mathcal{PT}$ structure is provided by dual-core waveguides (couplers), with the gain and loss separately placed in parallel cores, which are coupled by tunnelling of the field (light, in optics, or matter waves, in BEC). Stable solitons in conservative couplers with the Kerr nonlinearity were predicted decades ago. These solitons may be symmetric or asymmetric with respect to the identical cores, the symmetry-breaking bifurcation happening at a critical value of the soliton’s total energy/norm (in terms of optics/BEC) Wabnitz ; Pare ; Maimistov ; Wabnitz2 , see also a review in Ref. Peng .

A remarkable property of the model of the coupler which includes the cubic nonlinearity in each core, and the above-mentioned $\mathcal{PT}$-symmetric terms, in the form of the linear gain and loss in the two cores, is that $\mathcal{PT}$-symmetric and antisymmetric solitons not only can be found in an analytical form, but also their stability region can be identified in an exact form Driben ; Sukho (this region is finite for the symmetric solitons, while antisymmetric ones are completely unstable, although their instability may be weak). Unlike the conservative counterpart of the system, asymmetric solitons cannot exist in the presence of the gain and loss, because asymmetry between components of the soliton in the amplified and damped cores does not admit establishment of the balance between the gain and loss.

Expansion of the stability region for $\mathcal{PT}$-symmetric solitons and, more generally, developing methods for dynamical control of the solitons is a relevant problem. One potential possibility is suggested by the use of the “management” technique, i.e., periodic modulations of the loss-gain (LG) and inter-core-coupling (ICC) coefficients. In terms of the conservative model of the nonlinear coupler, the management scheme, which corresponds to $\gamma_{0}=\gamma_{1}=0$ and $\delta>0$ in Eq. (LABEL:uv), see below, was introduced in Ref. Skinner , where effects of the management on symmetric and asymmetric solitons and the transition between them were studied. Similar management schemes are well known to stabilize otherwise unstable or fragile solitons in other settings, such as the* dispersion management* applied to solitons in single-core waveguides (the local dispersion coefficient with a periodically flipping sign book ; Turitsyn ), or the stabilization of two-dimensional solitons (which are otherwise unstable against the critical collapse Gadi ) by means of periodic nonlinearity management Towers ; Kraenkel ; Ueda . In Ref. DR a particular realization of the above-mentioned management format, implemented as periodic sign change of the LG and ICC coefficients, was applied to the stabilization of symmetric solitons in the $\mathcal{PT}$-supersymmetric coupler with the cubic intra-core nonlinearity and equal LG and ICC coefficients (the supersymmetry implies setting $\gamma_{0}=1$ in terms of Eq. (LABEL:uv) with $\gamma_{1}=\delta=0$, see below). In the supersymmetric coupler with constant parameters, all solitons are unstable (see Eq. (7) below), while the application of the management creates a stability region for them in the corresponding parameter space. Another application of the management to $\mathcal{PT}$-symmetric solitons was recently elaborated in terms of the single NLSE, with a localized complex potential, satisfying condition (1) and subject to cosinusoidal modulation AF .

The aim of the present work is to explore the stabilization and dynamical control of solitons in the $\mathcal{PT}$-symmetric nonlinear coupler by means of the management applied in a general form, which combines constant and periodically varying terms in the LG and ICC coefficients. Stability regions for $\mathcal{PT}$-symmetric solitons are identified by means of systematic simulations, and also in an analytical form, with the help of the adiabatic approximation, in the case of the long-period management format. While the straightforward addition of the management to the system modeling the $\mathcal{PT}$-symmetric nonlinear coupler leads to shrinkage of the stability area, defined in terms of the soliton’s amplitude (as can be seen below in Figs. 5, 6, and 7(a)), the periodic modulation makes it possible to find stable soliton in new situations – in particular, in those when the average value of the gain and loss is zero ($\gamma_{0}=0$, in terms of Eqs. (LABEL:uv)), as shown below in Figs. 1(a), 4(a), and 5(d), as well as in the case when the ICC coefficient may periodically change its sign, which corresponds to $\delta>1$, in terms of Eqs. (LABEL:uv) (see Figs. 5(a-c), 6, 7(a), and 8 below). Collisions between moving stable solitons are also considered, by dint of direct simulations.

The rest of the paper is arranged as follows. The model is introduced in Section 2. The main results, both numerical and analytical, which determine stability regions for the $\mathcal{PT}$-symmetric solitons, are presented in Section 3. Collision between stable solitons are addressed in Section 4. The paper is concluded by Section 5.

2.  The model

We consider the propagation of optical or matter waves in the dual-core system described by coupled NLSEs for wave amplitudes $u(z,t)$ and $v(z,t)$ in the cores which carry, severally, gain and loss:

[TABLE]

In terms of optics, $z$ is the propagation distance, while $t$ is the reduced time in the optical model realized in the temporal domain, as a dual-core optical fiber Wabnitz2 ; Peng , or the transverse coordinate in a dual-core planar waveguide, which represents the coupler in the spatial domain. The group-velocity-dispersion and Kerr coefficients in Eq. (LABEL:uv) are scaled to be one, assuming that the dispersion has the anomalous sign, which is necessary for maintaining bright solitons; in the spatial domain, the same term represents paraxial diffraction.

The management format, applied to the ICC and LG coefficients, includes constant terms, with the constant part of the former parameter scaled to be $1$, and $\gamma_{0}<1$ being the constant part of the latter one. The ICC and LG modulation amplitudes are, respectively, $\delta$ and $\gamma_{1}$, which may be introduced with a phase shift, $\varphi$, and $L$ is the modulation period. Because swapping the two cores of the coupler represents the spatial reflection in the present setting, and the propagation distance is the evolution variable in guided-wave-propagation models, Eqs. (LABEL:uv) are invariant with respect to the $\mathcal{PT}$ transformation, $\left(u,v\right)\rightarrow\left(v^{\ast},u^{\ast}\right),z\rightarrow L/2-z$, the shift of $z$ by $L/2$ being a specific feature added by the management (in other words, $z-L/4$ plays the role of time in the $\mathcal{T}$ transformation).

The dissipative coefficients and $\delta$ are defined to be non-negative, without the loss of generality:

[TABLE]

(values $\delta<0$ and $\varphi$ are tantamount to $-\delta$ and $\varphi+\pi$). Note, however, that the full local LG and ICC coefficients, i.e., $\gamma_{0}+\gamma_{1}\sin\left(2\pi z/L\right)$ and $1+\delta\sin\left(2\pi z/L\right)$, respectively, may take negative values – in particular, because we will consider, among others, the cases of $\gamma_{0}=0$ and $\delta>1$.

In the absence of the management, $\gamma_{1}=\delta=0$, a family of exact soliton solutions to Eq. (LABEL:uv) can be easily found Driben ; Sukho , provided that $\gamma_{0}\leq 1$:

[TABLE]

[TABLE]

where $A$ is an arbitrary amplitude, and signs $+$ and $-$ correspond to the $\mathcal{PT}$-symmetric and antisymmetric solitons, respectively, which are so named Driben because they correspond, respectively, to the usual symmetric and antisymmetric solitons in the usual coupler’s model in the absence of the gain and loss ($\gamma_{0}=0$; note that more general compound solitons, with an arbitrary phase difference between the two components, are not possible, because the $\mathcal{P}$ transformation has only two eigenvalues, $+1$ and $-1$). Note that solution (4) implies

[TABLE]

which maintains equilibrium between the gain and loss.

The stability region for the exact symmetric solitons in the static model ($\gamma_{1}=\delta=0$), given by Eqs. (4) and (5), can also be found in an exact form Driben ; Sukho : they are stable if the squared amplitude takes values

[TABLE]

while the antisymmetric solitons are completely unstable (although their instability may be very weak, depending on the parameters). For this reason, antisymmetric solitons are not considered in detail below below (they may be made stable in a discrete version of Eqs. (LABEL:uv) Barash-discr ).

As mentioned above, an essential difference of the $\mathcal{PT}$-symmetric system (LABEL:uv) from its conservative counterpart, with $\gamma_{0}=0$, is that, at $A^{2}>A_{\mathrm{crit}}^{2}$, unstable $\mathcal{PT}$-symmetric solitons are not replaced by stable asymmetric ones (cf. works Wabnitz -Peng , where asymmetric solitons are considered in the conservative system), because asymmetric states, that do not obey condition (6), cannot maintain the LG balance (i.e., asymmetric solitons cannot exist in the case of $\gamma_{0}>0$). As a result, unstable $\mathcal{PT}$-symmetric soliton suffer blowup, similar to what is shown below in Fig. 2(c).

3.  Stabilization and dynamical control of solitons by the management

3.1.  Numerical results

We focus on the case of zero phase shift between the variations of the LG and ICC coefficients, i.e., $\varphi=0$ in Eq. (LABEL:uv), as the management format with $\varphi\neq 0$ (in particular, $\varphi=\pi$, which, as mentioned above, is tantamount to taking $\delta<0$) leads to strong instability. The stability of the solitons under the action of the management was identified from sufficiently long direct simulations, with the input taken in the form of Eqs. (4) and (5) at $z=0$ and a given value of $\gamma_{0},$ while $A$ was varied, to collect systematic results for the stability of the solitons with different amplitudes, cf. Eq. (7). The simulations were carried out by means of the split-step numerical algorithm, similar to those employed in Refs. Driben and Sukho . A rigorous study of the stability against small perturbations, which makes it necessary to solve linearized equations around the periodically varying solution, is a challenging task, which we do not tackle here.

A combination of panels displayed in Fig. 1 show stability areas for the solitons with different values of the constant part of the LG coefficient, $\gamma_{0}$, for a fixed management period, $L=\pi/3$, and two different values of amplitude $A$ in the input expression (5). Naturally, the stability areas are larger for smaller $A$ (similar to what is predicted by Eq. (7) in the absence of the management) , and they shrink with the increase of the modulation amplitudes, $\gamma_{1}$ and $\delta$. It is clearly seen that the strongest stability is provided by the management scheme in which the ratio of $\delta$ and $\gamma_{1}$ is the same as the ratio of their counterparts in the constant parts of the ICC and LG coefficients (in other words, the management is applied* coherently *with the static part of the system):

[TABLE]

This finding is explained by the fact that, when relation (8) holds, Eqs. (LABEL:uv) admit exact solutions for $\mathcal{PT}$-symmetric and antisymmetric solitons:

[TABLE]

cf. Eq. (5), with $u\left(z,t\right)$ given by exactly the same equation (4) as above. Of course, stability conditions for these exact solutions cannot be found in the same simple form (7) which is valid in the absence of the management.

A principally different case is one shown in Fig. 1(a), which corresponds to $\gamma_{0}=0$ (i.e., the static system is the conservative one). In this case, relation (8) does not exist, and, accordingly, the shape of the stability area is completely different from those displayed in panels (b)-(f).

The evolution of a stable soliton which satisfies constraint (8) is displayed in Fig. 2(a), showing that it keeps a constant shape, in exact agreement with Eqs. (9) and (4). On the other hand, in the case when the management parameters deviate from condition (8), a typical example of the evolution of stable solitons shows small but visible fluctuations in Fig. 2(b). On the contrary, unstable solitons blow up due to the failure of the LG balance, see an example of the quick onset of the blowup in Fig. 2(c), for parameters chosen deep in the instability area. Moderately unstable solitons develop into breathers, which evolve as quasi-stable states (cf. similar dynamical modes reported in Ref. Sukho ), but eventually they are destroyed by the collapse, as shown in Fig. 3.

Similar results, pertaining to a larger management period, $L=\pi$, are presented in Fig. 4. It is seen that they are qualitatively similar to those in Fig. 1, but the large period supports smaller stability areas, both in the case of $\gamma_{0}=0$ and $\gamma_{0}>0$. Note that the same condition (8) determines the condition of the optimum stability in this case too, when exact $\mathcal{PT}$-symmetric solitons are given by Eqs. (4) and (9).

Another essential summary of the results is presented in Fig. 5, in the form of stability maps in the plane of $\left(\gamma_{1},A\right)$, while $\delta$ is linked to $\gamma_{1}$ by the optimum-stability condition (8). At $\gamma_{1}=0$ (hence $\delta=0$ too), i.e., in the absence the management, the largest values of $A$ admitting the stability in Figs. 5(a) and (b) are precisely the same as predicted analytically by Eq. (7) for the static $\mathcal{PT}$-symmetric coupler.

A natural trend evidenced by Figs. 5(a) and (b) is that the stability limit, given by the largest value of $A$ up to which the solitons persist, decreases with the increase of the modulation strength, $\gamma_{1}$. A noteworthy feature observed by Fig. 5(b) is that, for a relatively large management period, $L=\pi$, the stability boundary for larger $\gamma_{0}$ may be located higher, in terms of $A$, than its counterpart for smaller $\gamma_{0}$, see, for instance, the boundaries for $\gamma_{0}=0.15$ and $\gamma_{0}=0.5$. This feature is counter-intuitive, as the exact result (7) for the solitons in the static model demonstrates monotonic decay of $A_{\mathrm{crit}}$ with the increase of $\gamma_{0}$. An explanation for this point is that, at smaller $\gamma_{0}$, the stability is more sensitive to changes of period $L$. The analysis of the situation for the long-period modulations with large $L$ is presented in the next section, with the help of the adiabatic approximation.

Lastly, Fig. 5(d) demonstrates that, in the case of $\gamma_{0}=0$, when the constant term is absent in the LG coefficient (hence Eq. (8) is irrelevant), the increase of the modulation amplitudes of both the LG and ICC terms, i.e., $\gamma_{1}$ and $\delta$, naturally causes shrinkage of the stability area.

3.2.  The adiabatic approximation

The long-period management may be considered as adiabatic under the condition that the period is much larger than the intrinsic period of phase oscillations of soliton (5):

[TABLE]

This condition makes it possible to separate the scales of the slow evolution of stable solitons, driven by the management, and their rapid phase oscillations, which correspond to the nearly-constant values of the parameters

The adiabatic limit allows one to approximately transform Eq. (LABEL:uv) into equations with constant coefficients, by defining

[TABLE]

where it is assumed that $\gamma_{1}$ and $\delta$ are related by Eq. (8), to address the case which is most relevant for the stability analysis. The substitution of variables (11)-(13) leads, in the first approximation, to the following equations replacing Eq. (LABEL:uv):

[TABLE]

This approximate transformation is relevant under condition

[TABLE]

which is necessary to secure condition $1+\delta\sin\left(2\pi z/L\right)>0$; otherwise, the transformation given  by (11)-(13) becomes singular. Taking into regard the currently imposed relation (8), Eq. (15) may also be written as $\gamma_{1}<\gamma_{0}$.

Being tantamount in their form to Eq. (LABEL:uv), equations (LABEL:tilde) produce solutions in the form of (4) and (5), which, in turn, are stable under the accordingly transformed criterion (7). The critical condition corresponds to the largest amplitude, in terms of the transformed fields (13), which is $\tilde{A}_{\mathrm{crit}}^{2}=A^{2}/\left(1-\delta\right)$, attained at $\sin\left(2\pi z/L\right)=-1$. In terms of the original notation, the respective approximate stability criterion for the slowly varying solitons takes the form of

[TABLE]

(Eq. (8) is used to write the result in Eq. (16) in two equivalent forms).

The global picture of the transformation of the stability boundary in the plane of $\left(\gamma_{1},A\right)$ is illustrated in Fig. (6), by showing it for six different management periods, which cover the range of four order of magnitude (from $L=\pi/20$ to $L=1000\pi$), and two values $\gamma_{0}=0.25,0.5$. It is observed that, in each case, there is a critical value, $L_{\mathrm{c}}$, such that dependence $A\left(\gamma_{1}\right)$ along the stability boundary is monotonous at $L<L_{\mathrm{c}}$, and non-monotonous at $L>L_{\mathrm{c}}$.

Further, the analytical prediction given by Eq. (16) is compared to the numerically found stability boundaries, for a very large period, $L\equiv 1000\pi$, in Fig. 7(a). It is seen that the prediction is close to the numerical counterparts in the region of $\gamma_{1}<\gamma_{0}$. At $\gamma_{1}=\gamma_{0}$ the analytically predicted critical amplitude vanishes in Eq. (16), and it ceases to exist at $\gamma_{1}>\gamma_{0}$, i.e., in the case when the sign of the total LG periodically changes, according to Eq. (LABEL:uv). In fact, the analytical approximation breaks down close to $\gamma_{1}=\gamma_{0}$ (i.e., close to $\delta=1$, see Eq. (15)), as mentioned above. In fact, the numerically found critical amplitude does not vanish at point $\gamma_{1}=\gamma_{0}$, but, instead, it attains a finite minimum value. At $\gamma_{1}>\gamma_{0}$, the stability area still exists, slowly expanding with the increase of $\gamma_{1}/\gamma_{0}\equiv\delta$. This trend is a natural one, as the increase of the absolute value of the ICC suppresses the symmetry-breaking instability driven by the self-focusing nonlinearity Wabnitz -Peng .

At all values of $\gamma_{0}$, the minimum of $A_{\mathrm{crit}}$ is attained exactly at $\gamma_{1}=\gamma_{0}$, in full agreement with the analytical prediction. This fact is confirmed by the numerically generated value of $\gamma_{0}/\gamma_{1}$ at the minimum point, which is shown, versus $L$, by the dashed line in Fig. 7(b). As concerns value $A_{\min}$ of the critical amplitude at the minimum point, it slowly decreasing with the increase of $L$, as is also shown in Fig. 7(b).

4.  Collisions of solitons

Because Eq. (LABEL:uv) keeps the Galilean invariance, a stable soliton can be set in motion by applying kick $\eta$ to it, i.e., multiplying the quiescent solution $\left\{u_{0}\left(z,t\right),v_{0}\left(z,t\right)\right\}$ by $\exp\left(i\eta t\right)$. As a result, it will be transformed into a moving one,

[TABLE]

This fact suggests to simulate collisions between initially separated solitons, boosted in opposite directions by kicks $\pm\eta$, cf. Refs. Driben and Sukho .

A typical set of collisions, simulated for a set of different values of the kicks, is displayed in Fig.(8). Panels(a) to (f) show different outcomes of the collisions, produced by increasing values of $\eta$. In all the cases, the colliding solitons eventually separate. However, in panels (a) and (b) slowly moving soliton pairs form quasi-bound states, in which they perform several oscillations before re-emerging with larger values of opposite velocities $\pm c$ (see Eq. (17)) than they had prior to the collision. The formation of the intermediate bound state resembles the effect which was previously found in simulations of soliton-soliton collisions in other nonintegrable models Campbell1 ; Campbell2 ; RMP . Fast moving solitons, boosted by stronger kicks, pass through each other elastically (panels (d,e,f)), which is typical for collisions between solitons in conservative systems RMP , and remains true in the present $\mathcal{PT}$-symmetric one.

Lastly, Fig. 9 displays simulations of the collisions with the same values of $L,\gamma_{0}$, and $A$, under the action of the same kicks as in the top row in Fig. 8, but in the absence of the management, i.e., for $\gamma_{1}=\delta=0$. It is clearly seen that the collisions are completely elastic (similar to those simulated in Driben ), and the intermediate quasi-bound states do not emerge. Thus, the presence of the management accounts for the creation of the those states. For larger kicks, corresponding to the bottom row in Fig. 8, the collisions remain the same (elastic) as displayed in Fig. 8.

As said above, in this work we do not address $\mathcal{PT}$-antisymmetric solitons, which correspond to the bottom signs in Eqs. (4), (5) and (9), as they are unstable even in the absence of the management. The instability of some antisymmetric solitons being weak, it is possible to consider their collisions too, with each other or with $\mathcal{PT}$-symmetric counterparts. As shown in Ref. Sukho , in the latter case the collision may excite intrinsic oscillations in the solitons. The consideration of this case is beyond the scope of the present work.

5.  Conclusion

The objective of this work is to generalize the known model of the $\mathcal{PT}$-symmetric coupler, based on linearly coupled waveguides with the intrinsic cubic nonlinearity and equal gain and loss coefficients carried by the guiding cores. The generalization introduces “management”, which makes the LG (loss-gain) and ICC (inter-core-coupling) coefficients periodically varying functions of the evolutional variable. The model may be realized in optics, in the temporal and spatial domains alike. Stability of $\mathcal{PT}$-symmetric solitons and possibilities of applying the dynamical control to them by means of the management are explored by means of systematic simulations, and also analytically in the adiabatic approximation, which corresponds to the long-period limit. The stability is strongest when the ratio of the amplitudes of the modulation of the LG and ICC coefficients is equal to its counterpart for the constant parts of the same coefficients. In the latter case, an exact solution is found for $\mathcal{PT}$-symmetric solitons. Collisions between moving solitons were briefly considered too.

A challenging possibility for the development of the present analysis is to develop analysis of the management for solitons in two-dimensional $\mathcal{PT}$-symmetric systems.

Acknowledgements.

This work was supported, in part, by the Israel Science Foundation through grant No. 1287/17. We appreciate a helpful discussion with Zhaopin Chen (Tel Aviv University) and technical assistance provided by Ms. Shiyue Liu (Chinese University of Hong Kong).