## Abstract

The *𝒫𝒯* -symmetric and *𝒫𝒯* -antisymmetric Akhmediev breather (AB) and Kuznetsov-Ma (KM) soliton train solutions of a (2+1)-dimensional variable-coefficient coupled nonlinear Schrödinger equation in *𝒫𝒯* -symmetric coupled waveguides with gain and loss are derived via the Darboux transformation method. From these analytical solutions, we investigate the controllable behaviors of AB and KM soliton trains in a diffraction decreasing system with exponential profile. By adjusting the relation between the maximum *Z _{m}* of effective propagation distance and the peak locations

*Z*of AB and KM soliton trains, we can control the restraint, maintenance and postpone excitations of AB and KM soliton trains.

_{i}© 2014 Optical Society of America

## 1. Introduction

The parity-time(*𝒫𝒯*) symmetry, which originates from quantum mechanics [1], is presently attracting a great interest both from the theoretical and from the applicative points of view in nonlinear optics. *𝒫𝒯* -symmetric systems have been realized in optics by combining a spatially symmetric profile of the refractive index with symmetrically placed mutually balanced gain and loss [2]. Since the pioneer study of solitonic dynamics in *𝒫𝒯* -symmetric potentials [3], stable spatiotemporal solitons [4] and soliton interactions [5] in nonlinear media with *𝒫𝒯* -symmetric potentials have been extensively discussed.

The Peregrine solution (PS) [6], space-periodic Akhmediev breather (AB) [7] and time-periodic Kuznetsov-Ma (KM) soliton [8] are considered as theoretical prototypes to describe rogue waves, which are spontaneous waves several times higher than the average wave crests [9]. Controllable behaviors of the single PS, AB and KM soliton have been studied [10,11]. Although analytical PS in *𝒫𝒯* -coupled nonlinear waveguides has been reported [12], the controllable behaviors of AB and KM soliton trains in *𝒫𝒯* -symmetric systems are less studied.

The nonlinear coupled waveguide with gain and loss was presented as an improvement of the conventional twin core coupler [13], whose structure is made up of two optical waveguides in close proximity to one another with an equal amount of optical gain and loss. In a real waveguide, the variation of the geometry leads to its inhomogeneity [14]. Therefore, we discuss the dynamics of controllable behaviors of AB and KM soliton trains in the following variable-coefficient (vc) coupled nonlinear Schrödinger equations (CNLSE)

*u*(

*z*,

*x*,

*y*) and

*v*(

*z*,

*x*,

*y*) represent two normalized complex mode fields in two parallel planar waveguides with dimensionless propagation coordinate

*z*and transverse coordinates

*x*,

*y*. In Eqs. (1), the second and third terms in the left-hand sides both denote diffractions in different directions, the last two terms in the left-hand sides describe the nonlinearly coupled terms of the self-phase-modulation (SPM) and cross-phase-modulation (XPM). The first terms in the right-hand sides account for the coupling between the modes propagating in two waveguides. The opposite signs of the

*γ*term are the

*𝒫𝒯*-balanced gain and loss in the first and second equations of Eqs. (1), respectively. When all coefficients are constant, Eqs. (1) are two-dimensional (2D) case of CNLSE in [12]. When the coefficients of coupling terms are variable, Eqs. (1) are 2D case of CNLSE in [5].

## 2. AB and KM soliton train solutions

We consider the gain/loss term is small enough, such as *γ* ≤ 1 [12], thus the energy through linear coupling is transferred from the core with gain to the lossy one, and modes can be excited in the system by input beams but do not arise spontaneously. Without loss of generality, considering *γ* = sin(*θ*), we construct the *𝒫𝒯* -symmetric (+) and -antisymmetric (−) relation

*U*obeying the standard NLSE where the effective propagation distance $Z=\frac{1}{4}\left[{k}^{2}{D}_{1}(z){\alpha}_{1}(z)+{l}^{2}{D}_{2}(z){\alpha}_{2}(z)\right]$, the similarity variable $X=\frac{1}{2}\left[k{\alpha}_{1}(z)x+l{\alpha}_{2}(z)y\right]-\frac{1}{2}\left[kc{D}_{1}(z){\alpha}_{1}(z)+ld{D}_{2}(z){\alpha}_{2}(z)\right]$, the phase $\varphi (z,x,y)=-\frac{1}{2}\left[a{\alpha}_{1}(z){x}^{2}+b{\alpha}_{2}(z){y}^{2}\right]+c{\alpha}_{1}(z)x+d{\alpha}_{2}(z)y-\frac{1}{2}\left[{c}^{2}{D}_{1}(z){\alpha}_{1}(z)+{d}^{2}{D}_{2}(z){\alpha}_{2}(z)\right]$, the chirp factors

*α*

_{1}(

*z*) = 1/[1 −

*aD*

_{1}(

*z*)] and

*α*

_{2}(

*z*) = 1/[1 −

*bD*

_{2}(

*z*)], the accumulated diffractions ${D}_{1}(z)={\int}_{0}^{z}{\beta}_{1}(s)ds$ and ${D}_{2}(z)={\int}_{0}^{z}{\beta}_{2}(s)ds$ and $\mathrm{\Omega}(z)={\int}_{0}^{z}\eta (s)ds$. Moreover, free and adjustable parameters

*k*and

*l*are related to beam widths,

*a*and

*b*are the chirp factors,

*c*and

*d*are related to the positions of the wavefront in different directions.

Moreover, the parameters of systems satisfy the following relation

*β*

_{1}(

*z*) and

*β*

_{2}(

*z*) are chosen to be free parameters, then

*χ*(

*z*) and

*χ*

_{1}(

*z*) will be decided from Eq. (4). Therefore solutions can transmit stably in an exponential nonlinear medium when the diffraction coefficients

*β*

_{1}(

*z*) and

*β*

_{2}(

*z*) change exponentially.

According to the modified Darboux transformation (DT) technique [15], based on solution of NLSE (3), the AB and KM soliton train solutions of Eqs. (1) read

*P*= −2

*κ*{cos(

*δZ′*)[(

*δ*

^{2}+

*κ*

^{2}) cos(

*κX′*) +

*δ*

^{2}

*κX′*sin(

*κX′*) − 2

*δκ*cosh(

*δZ′*)] + cos(

*κX′*) sinh(

*δZ′*)(2

*δ*

^{2}−

*κ*

^{2})

*δZ′*}/

*δ*,

*Q*= −{8

*δZ′*(2

*δ*

^{2}−

*κ*

^{2})[

*δ*cos(

*κX′*) cosh(

*δZ′*) −

*κ*] + sinh(

*δZ′*) 8

*δ*

^{3}[cos(

*κX′*)+

*κX′*sin(

*κX′*)] + (

*κ*

^{4}− 4

*δ*

^{2})

*κ*sinh(2

*δZ′*)}/(2

*δκ*),

*R*= −{

*κ*

^{4}(

*δ*

^{2}+

*κ*

^{2}) + 8

*δ*

^{2}

*κ*

^{2}(

*δ*

^{2}

*X′*

^{2}+

*κ*

^{2}

*Z′*

^{2}) + 32

*δ*

^{4}

*Z′*

^{2}(

*δ*

^{2}−

*κ*

^{2}) + 4[

*κ*

^{4}cosh(2

*δZ′*) −

*δ*

^{4}cos(2

*κX′*)] − 16

*δ*

^{2}

*κZ′*(2

*δ*

^{2}−

*κ*

^{2}) cos(

*κX′*) sinh(

*δZ′*) −4

*δκ*

^{2}[4

*δ*

^{2}

*X′*sin(

*κX′*) +

*κ*

^{3}cos(

*κX′*)] cosh(

*δZ′*)}/(4

*δ*

^{2}

*κ*

^{2}) with

*Z′*=

*Z*−

*Z*

_{0},

*X′*=

*X*−

*v*

_{0}(

*Z*−

*Z*

_{0}), $\delta =\kappa \sqrt{4-{\kappa}^{2}}/2$. Here

*Z*,

*X*and

*ϕ*are expressed below Eq. (3),

*Z*

_{0}and

*v*

_{0}are two arbitrary constants,

*κ*is the modulation frequency. When

*κ*is a real or an imaginary value, solution (5) describes the AB train or KM soliton train, respectively.

## 3. Controllable behaviors of KM soliton train

At first, we discuss analytical solution (5) in the framework of NLSE (3). Figure 1 displays KM soliton train and AB train in the *Z* − *X* coordinates. They are both made up of two lines of peaks hinged upon a second-order PS. In Fig. 1(a), peaks of the KM soliton train along *Z*-axis exist at different locations. Here we only list these locations in the positive *Z*-axis, that is, the second-order PS appears at *Z*_{1} = 5, and the first, second and third PS pairs appear at *Z*_{2} = 17, *Z*_{3} = 28, *Z*_{4} = 39, respectively. In Fig. 1(b), the AB train has two lines of PS hinged upon a second-order PS and looks like two wings. We call these two lines of PS as the front and back wings along the *Z*-axis. In each wing, there are three PS pairs. We name the first, second and third PS pairs after the sequence of appearance along the *Z*-axis. Peaks of PS in AB appear at different locations. Among them, three PS pairs in front wing are excited at distances *Z*_{11} = 2, *Z*_{12} = 2.5 and *Z*_{13} = 3, respectively. The second-order PS appears at *Z*_{2} = 8. Three PS pairs in back wing are excited at distances *Z*_{31} = 13, *Z*_{32} = 13.5 and *Z*_{33} = 14, respectively.

Next, we discuss controllable behaviors of the KM soliton train in a diffraction decreasing system (DDS) with the exponential profile [16]

*β*

_{i0(i=1,2)}are positive parameters related to diffractions, and

*σ*> 0 corresponds to DDS.

From the expression of *Z* in (2), *Z* is not free and exists a maximum *Z _{m}* in DDS. Inserting this expression (6) into the expression of

*Z*in (2) yields

*Z*= {

*σ*(

*k*

^{2}

*β*

_{10}+

*l*

^{2}

*β*

_{20}) −

*β*

_{10}

*β*

_{20}(

*al*

^{2}+

*bk*

^{2})[1 − exp(−

*σz*)]}/{4(

*σ*−

*aβ*

_{10}[1 − exp(−

*σz*)])(

*σ*−

*bβ*

_{20}[1 − exp(−

*σz*)])}. If

*σ*> 0, the maximum value is

*Z*= [

_{m}*σ*(

*k*

^{2}

*β*

_{10}+

*l*

^{2}

*β*

_{20}) −

*β*

_{10}

*β*

_{20}(

*al*

^{2}+

*bk*

^{2})]/[4(

*σ*−

*aβ*

_{10})(

*σ*−

*bβ*

_{20})] as

*z*→ ∞. The maximum

*Z*can be modulated by changing the value of

_{m}*σ*when

*a*,

*b*,

*k*,

*l*,

*β*

_{10}and

*β*

_{20}are chosen as certain fixed values. On the other side, maximum amplitudes of PS in the KM soliton train appear in different locations along the

*Z*-axis in the framework of NLSE (3) in Fig. 1(a). Therefore, we can adjust the relation between the maximum

*Z*and peak locations

_{m}*Z*to discuss the degree of excitation of KM soliton train in Fig. 1(a).

_{i}When *Z _{m}* <

*Z*

_{1},

*Z*=

_{m}*Z*

_{1}and

*Z*>

_{m}*Z*

_{1}, the restraint, maintenance and postpone of the second-order PS in the KM soliton train appear. Fig. 2(a) displays this kind of maintenance, that is, the second-order PS in the KM soliton train maintains its peak a long propagation distance with a self-similar form. With the add of

*Z*, the restraint, maintenance and postpone of the first, second and third PS pairs will happen. When the value of

_{m}*Z*is smaller than

_{m}*Z*

_{2}, the threshold of exciting the first PS pair in the KM soliton train is never reached, thus the complete excitation is restrained [Fig. 2(b)]. That is, the first PS pair is partially excited and maintains its initial shape. When

*Z*=

_{m}*Z*

_{2}, the first PS pair in the KM soliton train maintains its peak a long distance with self-similar propagation behaviors [Fig. 2(c)]. When

*Z*is slightly bigger than

_{m}*Z*

_{2}, the complete excitation of the first PS is also postponed, and it has a long tail [Fig. 2(d)]. When

*Z*continues to add to

_{m}*Z*<

_{m}*Z*

_{3},

*Z*=

_{m}*Z*

_{3},

*Z*>

_{m}*Z*

_{3}, the restraint, maintenance and postpone of the second PS pairs will appear respectively. For the length of limit, we neglect these related plots.

In solution (2), *γ* = sin(*θ*) defines two different angles, that is, *θ* = arcsin*γ* ∈ [0, *π*/2] and *θ* = *π* − arcsin*γ* ∈ [*π*/2, *π*] correspond to *𝒫𝒯* -symmetric and -antisymmetric solutions [12], respectively. Moreover, two families of solutions are also distinguished by the sign of cos(*θ*). One has
$\text{cos}(\theta )=\sqrt{1-{\gamma}^{2}}$ and another has
$\text{cos}(\theta )=-\sqrt{1-{\gamma}^{2}}$, thus we denote solution (2) as *𝒫𝒯* -symmetric solution {*u*_{+}, *v*_{+}} and *𝒫𝒯* -antisymmetric solution {*u*_{−}, *v*_{−}}. As an example, Fig. 2(e) exhibits the magnitude of *𝒫𝒯* -antisymmetric solution {*u*_{−}, *v*_{−}} for the corresponding case in Fig. 2(c), and the difference of phase of *v*_{+} and *v*_{−} is shown in Fig. 2(f).

When *Z _{m}* increases to be equal to the value smaller than

*Z*

_{4}, the excitation of the third PS in the KM soliton train is restrained, and only initial part is produced [Fig. 2(g)]. When

*Z*=

_{m}*Z*

_{4}, the third PS pair maintains its peak in a self-similar form [Fig. 2(h)]. When

*Z*>

_{m}*Z*

_{4}, the postpone of the third PS pair in the KM soliton train also happens. If

*Z*continues to add, the restraint, maintenance and postpone of the next fourth and fifth PS pairs, etc. will happen again and again. For the length of limit, we omit these detailed discussions.

_{m}The controllable behaviors including postpone, maintenance and restraint can be further exhibited in the (*x*, *y*)-plane for different *z*. As some examples, in Figs. 2(j)–2(l), we display evolutional plots corresponding to postpone of the second-order PS in Fig. 2(a), maintenance of the first PS pair in Fig. 2(c), and restraint of the third PS pair in Fig. 2(g), respectively.

## 4. Controllable behaviors of AB train

The controllable behaviors of AB train can been also studied by modulating the relation between the maximum value and peak location values in the DDS (6).

When *Z _{m}* <

*Z*

_{11}, the excitations of all PS pairs in the AB train are restrained, and only initial shapes are excited [Fig. 3(a)]. When

*Z*=

_{m}*Z*

_{13}, the third PS pair in the front wing maintains its peak a long distance [Fig. 3(b)]. When

*Z*is slightly bigger than

_{m}*Z*

_{13}, the full excitations of all PS pairs in front wing are postponed [Fig. 3(c)]. If the value

*Z*adds to

_{m}*Z*<

_{m}*Z*

_{2},

*Z*=

_{m}*Z*

_{2}and

*Z*>

_{m}*Z*

_{2}, the full excitations of all PS pairs in front wing are still postponed, and the restraint, maintenance and postpone of the second PS are produced in Figs. 3(d)–3(f) respectively.

When *Z _{m}* adds to the value slightly smaller than

*Z*

_{31}, all PS pairs in the front wing are completely excited, and the excitation of the second-order PS is still postponed, however, all PS pairs in the back wing are only excited to initial parts in Fig. 3(g). When

*Z*=

_{m}*Z*

_{33}, the third PS pair in the back wing is excited to the peak and maintains this value a long distance, and the excitations of the first and second PS pairs are both postponed in Fig. 3(h). When

*Z*adds to the value slightly bigger than

_{m}*Z*

_{33}, all PS pairs in the front wing and the second-order PS are completely excited, however, the excitations of all PS pairs in the back wing are postponed and have long tails along the propagation distance

*z*.

The controllable behaviors of AB train on (*x*, *y*)-plane for different *z* are also shown in Figs. 3(j)–3(l). From them, we can further understand the maintenance of the front wing in Fig. 3(b), restraint of the second-order PS in Fig. 3(d), and postpone of the back wing in Fig. 3(i).

## 5. Conclusions

In summary, we study a 2D vcCNLSE in *𝒫𝒯* -symmetric nonlinear couplers with gain and loss, and analytically derive the AB and KM soliton train solutions via the DT method. When the modulation frequency *κ* is a real or imaginary value, the AB or KM soliton train solution can be derived, respectively. Moreover, we study controllable behaviors of AB and KM soliton trains in a DDS with the exponential profile. In this system, the effective propagation distance *Z* exists a maximal value *Z _{m}*, and the peaks of the AB and KM soliton trains exist the periodic locations

*Z*. By modulating the relation between values of

_{i}*Z*and

_{m}*Z*, we realize the control of the restraint, maintenance and postpone excitations of the AB and KM soliton trains.

_{i}## Acknowledgments

This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY13F050006) and the NSFC (Grant Nos. 11375007 and 11404289).

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