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 Zm of effective propagation distance and the peak locations Zi of AB and KM soliton trains, we can control the restraint, maintenance and postpone excitations of AB and KM soliton trains.

© 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)

iuz+12[β1(z)uxx+β2(z)uyy]+[χ1(z)|u|2+χ(z)|v|2]u=η(z)(v+iγu),ivz+12[β1(z)uxx+β2(z)uyy]+[χ(z)|u|2+χ1(z)|v|2]v=η(z)(u+iγv),
where 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

v(z,x,y)=±u(z,x,y)exp(±iθ),u(z,x,y)=A0α1(z)α2(z)U[Z(z),X(z,x,y)]exp{i[cos(θ)Ω(z)+ϕ(z,x,y)]},
with U obeying the standard NLSE
iUZ+12UXX+|U|2U=0,
where the effective propagation distance Z=14[k2D1(z)α1(z)+l2D2(z)α2(z)], the similarity variable X=12[kα1(z)x+lα2(z)y]12[kcD1(z)α1(z)+ldD2(z)α2(z)], the phase ϕ(z,x,y)=12[aα1(z)x2+bα2(z)y2]+cα1(z)x+dα2(z)y12[c2D1(z)α1(z)+d2D2(z)α2(z)], the chirp factors α1(z) = 1/[1 − aD1(z)] and α2(z) = 1/[1 − bD2(z)], the accumulated diffractions D1(z)=0zβ1(s)ds and D2(z)=0zβ2(s)ds and Ω(z)=0zη(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

χ(z)+χ1(z)=14A02[k2β1(z)α1(z)α2(z)+l2β2(z)α2(z)α1(z)],
which means that if parameters β1(z) and β2(z) are chosen to be free parameters, then χ(z) and χ1(z) will be decided from Eq. (4). Therefore solutions can transmit stably in an exponential nonlinear medium when the diffraction coefficients β1(z) and β2(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

u=A0α1(z)α2(z)[1+P+iQR]exp{i[(1v02)(ZZ0)+v0X+cos(θ)Ω(z)+ϕ]},
where P = −2κ{cos(δZ′)[(δ2 + κ2) cos(κX′) + δ2κX′sin(κX′) − 2δκ cosh(δZ′)] + cos(κX′) sinh(δZ′)(2δ2κ2)δZ′}/δ, Q = −{8δZ′(2δ2κ2)[δ cos(κX′) cosh(δZ′) − κ] + sinh(δZ′) 8δ3[cos(κX′)+ κX′sin(κX′)] + (κ4 − 4δ2)κ sinh(2δZ′)}/(2δκ), R = −{κ4(δ2 + κ2) + 8δ2κ2 (δ2X′2 + κ2Z′2) + 32δ4Z′2(δ2κ2) + 4[κ4 cosh(2δZ′) − δ4 cos(2κX′)] − 16δ2κZ′(2δ2κ2) cos(κX′) sinh(δZ′) −4δκ2[4δ2X′ sin(κX′) + κ3 cos(κX′)] cosh(δZ′)}/(4δ2κ2) with Z′ = ZZ0, X′ = Xv0(ZZ0), δ=κ4κ2/2. Here Z, X and ϕ are expressed below Eq. (3), Z0 and v0 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 ZX 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 Z1 = 5, and the first, second and third PS pairs appear at Z2 = 17, Z3 = 28, Z4 = 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 Z11 = 2, Z12 = 2.5 and Z13 = 3, respectively. The second-order PS appears at Z2 = 8. Three PS pairs in back wing are excited at distances Z31 = 13, Z32 = 13.5 and Z33 = 14, respectively.

 figure: Fig. 1

Fig. 1 (a) and (b) KM soliton and AB trains for Eq. (3) in the ZX coordinates, respectively. Parameters are chosen as v0 = 0.1 with (a) Z0 = 5, κ = 0.6i and (b) Z0 = 8, κ = 0.6.

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

β1(z)=β10exp(σz),β2(z)=β20exp(σz),
where βi0(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 Zm in DDS. Inserting this expression (6) into the expression of Z in (2) yields Z = {σ(k2β10 + l2β20) − β10β20(al2 + bk2)[1 − exp(−σz)]}/{4(σ10[1 − exp(−σz)])(σ20[1 − exp(−σz)])}. If σ > 0, the maximum value is Zm = [σ(k2β10 + l2β20) − β10β20(al2 + bk2)]/[4(σ10)(σ20)] as z → ∞. The maximum Zm can be modulated by changing the value of σ when a, b, k, l, β10 and β20 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 Zm and peak locations Zi to discuss the degree of excitation of KM soliton train in Fig. 1(a).

When Zm < Z1, Zm = Z1 and Zm > Z1, 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 Zm, the restraint, maintenance and postpone of the first, second and third PS pairs will happen. When the value of Zm is smaller than Z2, 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 Zm = Z2, the first PS pair in the KM soliton train maintains its peak a long distance with self-similar propagation behaviors [Fig. 2(c)]. When Zm is slightly bigger than Z2, the complete excitation of the first PS is also postponed, and it has a long tail [Fig. 2(d)]. When Zm continues to add to Zm < Z3, Zm = Z3, Zm > Z3, the restraint, maintenance and postpone of the second PS pairs will appear respectively. For the length of limit, we neglect these related plots.

 figure: Fig. 2

Fig. 2 We take y = 2. The physical quantity I = |u| of controllable behaviors for KM soliton train: (a), (c) and (h) maintenance of the second-order PS, the first and third PS pairs, (b),(g) and (d),(i) restraint and postpone of the first and third PS pairs. Parameters are chosen as A0 = 0.5, a = 0.05, b = 0.06, β10 = 0.1, β20 = 0.12, k = 2.5, l = 3, γ = 0.2, c = 1, d = 1.2 with (a)–(d), (g)–(i) σ = 0.07, 0.032, 0.0315, 0.0295, 0.018, 0.0178, 0.0174, respectively. Results are similar for other values of y. (e) and (f) Magnitude and phase of 𝒫𝒯 -symmetric and -antisymmetric solution corresponding to (c). (j)–(l) Evolutional plots corresponding to (a), (c) and (g) at z = 60, 180, 190, respectively.

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In solution (2), γ = sin(θ) defines two different angles, that is, θ = arcsinγ ∈ [0, π/2] and θ = π − arcsinγ ∈ [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 cos(θ)=1γ2 and another has cos(θ)=1γ2, 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 Zm increases to be equal to the value smaller than Z4, the excitation of the third PS in the KM soliton train is restrained, and only initial part is produced [Fig. 2(g)]. When Zm = Z4, the third PS pair maintains its peak in a self-similar form [Fig. 2(h)]. When Zm > Z4, the postpone of the third PS pair in the KM soliton train also happens. If Zm 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.

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 Zm < Z11, the excitations of all PS pairs in the AB train are restrained, and only initial shapes are excited [Fig. 3(a)]. When Zm = Z13, the third PS pair in the front wing maintains its peak a long distance [Fig. 3(b)]. When Zm is slightly bigger than Z13, the full excitations of all PS pairs in front wing are postponed [Fig. 3(c)]. If the value Zm adds to Zm < Z2, Zm = Z2 and Zm > Z2, 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.

 figure: Fig. 3

Fig. 3 We take y = 2. The physical quantity I = |u| of controllable behaviors for AB train: (a), (d) and (g) restraint of the front wing, the second-order PS and back wing, (b), (e) and (h) maintenance of the front wing, the second-order PS and back wing, (c), (f) and (i) postpone of the front wing, the second-order PS and back wing. Parameters are chosen as the same as those in Fig. 2 except for (a)–(i) σ = 0.5, 0.148, 0.11, 0.075, 0.0597, 0.054, 0.043, 0.0372, 0.035, respectively. Results are similar for other values of y. (j)–(l) Evolutional plots corresponding to (b), (d) and (i) at z = 60, 80, 150, respectively.

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When Zm adds to the value slightly smaller than Z31, 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 Zm = Z33, 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 Zm adds to the value slightly bigger than Z33, 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 Zm, and the peaks of the AB and KM soliton trains exist the periodic locations Zi. By modulating the relation between values of Zm and Zi, we realize the control of the restraint, maintenance and postpone excitations of the AB and KM soliton trains.

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).

References and links

1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998). [CrossRef]  

2. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007). [CrossRef]   [PubMed]  

3. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008). [CrossRef]   [PubMed]  

4. C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014). [CrossRef]  

5. Y. V. Bludov, C. Hang, G. X. Huang, and V. V. Konotop, “PT-symmetric coupler with a coupling defect: soliton interaction with exceptional point,” Opt. Lett. 39, 3382–3385, 2014. [CrossRef]   [PubMed]  

6. D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. Ser. B 25, 16–43 (1983). [CrossRef]  

7. N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986). [CrossRef]  

8. E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).

9. N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009). [CrossRef]  

10. C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012). [CrossRef]  

11. H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014). [CrossRef]  

12. Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013). [CrossRef]  

13. Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992). [CrossRef]  

14. F. Abdullaeev, Theory of Solitons in Inhomogeneous Media, (Wiley, 1994).

15. D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrodinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012). [CrossRef]  

16. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005). [CrossRef]  

References

  • View by:

  1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  2. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
    [Crossref] [PubMed]
  3. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [Crossref] [PubMed]
  4. C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
    [Crossref]
  5. Y. V. Bludov, C. Hang, G. X. Huang, and V. V. Konotop, “PT-symmetric coupler with a coupling defect: soliton interaction with exceptional point,” Opt. Lett. 39, 3382–3385, 2014.
    [Crossref] [PubMed]
  6. D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. Ser. B 25, 16–43 (1983).
    [Crossref]
  7. N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
    [Crossref]
  8. E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).
  9. N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
    [Crossref]
  10. C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
    [Crossref]
  11. H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
    [Crossref]
  12. Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
    [Crossref]
  13. Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
    [Crossref]
  14. F. Abdullaeev, Theory of Solitons in Inhomogeneous Media, (Wiley, 1994).
  15. D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrodinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
    [Crossref]
  16. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
    [Crossref]

2014 (3)

C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

Y. V. Bludov, C. Hang, G. X. Huang, and V. V. Konotop, “PT-symmetric coupler with a coupling defect: soliton interaction with exceptional point,” Opt. Lett. 39, 3382–3385, 2014.
[Crossref] [PubMed]

H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
[Crossref]

2013 (1)

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

2012 (2)

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrodinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[Crossref]

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
[Crossref]

2009 (1)

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[Crossref]

2008 (1)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1992 (1)

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

1986 (1)

N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
[Crossref]

1983 (1)

D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. Ser. B 25, 16–43 (1983).
[Crossref]

1977 (1)

E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).

Abdullaeev, F.

F. Abdullaeev, Theory of Solitons in Inhomogeneous Media, (Wiley, 1994).

Akhmediev, N.

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrodinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[Crossref]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[Crossref]

N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
[Crossref]

Ankiewicz, A.

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrodinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[Crossref]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[Crossref]

Bender, C. M.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Bludov, Y. V.

Bludov, Yu V.

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Chen, Y.

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

Christodoulides, D. N.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Dai, C. Q.

C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
[Crossref]

Driben, R.

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

El-Ganainy, R.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Fang, J. P.

H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
[Crossref]

Hang, C.

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[Crossref]

Huang, G. X.

Kedziora, D. J.

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrodinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[Crossref]

Konotop, V. V.

Y. V. Bludov, C. Hang, G. X. Huang, and V. V. Konotop, “PT-symmetric coupler with a coupling defect: soliton interaction with exceptional point,” Opt. Lett. 39, 3382–3385, 2014.
[Crossref] [PubMed]

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

Korneev, V.I.

N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
[Crossref]

Kruglov, V. I.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[Crossref]

Kuznetsov, E. A.

E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).

Makris, K. G.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Malomed, B. A.

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

Musslimani, Z. H.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Pan, Z. H.

H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
[Crossref]

Payne, D. N.

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[Crossref]

Peregrine, D. H.

D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. Ser. B 25, 16–43 (1983).
[Crossref]

Snyder, A. W.

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

Taki, M.

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[Crossref]

Wang, X. G.

C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

Zhang, J. F.

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
[Crossref]

Zhou, G. Q.

C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
[Crossref]

Zhu, H. P.

H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
[Crossref]

Dokl. Akad. Nauk SSSR (1)

E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).

Eur. Phys. J. D (1)

H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
[Crossref]

IEEE J. Quantum Electron. (1)

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

J. Aust. Math. Soc. Ser. B (1)

D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. Ser. B 25, 16–43 (1983).
[Crossref]

J. Opt. (1)

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

Opt. Lett. (2)

Phys. Lett. A (1)

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[Crossref]

Phys. Rev. A (1)

C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

Phys. Rev. E (3)

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
[Crossref]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrodinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[Crossref]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[Crossref]

Phys. Rev. Lett. (2)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

Theor. Math. Phys. (1)

N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
[Crossref]

Other (1)

F. Abdullaeev, Theory of Solitons in Inhomogeneous Media, (Wiley, 1994).

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Figures (3)

Fig. 1
Fig. 1 (a) and (b) KM soliton and AB trains for Eq. (3) in the ZX coordinates, respectively. Parameters are chosen as v0 = 0.1 with (a) Z0 = 5, κ = 0.6i and (b) Z0 = 8, κ = 0.6.
Fig. 2
Fig. 2 We take y = 2. The physical quantity I = |u| of controllable behaviors for KM soliton train: (a), (c) and (h) maintenance of the second-order PS, the first and third PS pairs, (b),(g) and (d),(i) restraint and postpone of the first and third PS pairs. Parameters are chosen as A0 = 0.5, a = 0.05, b = 0.06, β10 = 0.1, β20 = 0.12, k = 2.5, l = 3, γ = 0.2, c = 1, d = 1.2 with (a)–(d), (g)–(i) σ = 0.07, 0.032, 0.0315, 0.0295, 0.018, 0.0178, 0.0174, respectively. Results are similar for other values of y. (e) and (f) Magnitude and phase of 𝒫𝒯 -symmetric and -antisymmetric solution corresponding to (c). (j)–(l) Evolutional plots corresponding to (a), (c) and (g) at z = 60, 180, 190, respectively.
Fig. 3
Fig. 3 We take y = 2. The physical quantity I = |u| of controllable behaviors for AB train: (a), (d) and (g) restraint of the front wing, the second-order PS and back wing, (b), (e) and (h) maintenance of the front wing, the second-order PS and back wing, (c), (f) and (i) postpone of the front wing, the second-order PS and back wing. Parameters are chosen as the same as those in Fig. 2 except for (a)–(i) σ = 0.5, 0.148, 0.11, 0.075, 0.0597, 0.054, 0.043, 0.0372, 0.035, respectively. Results are similar for other values of y. (j)–(l) Evolutional plots corresponding to (b), (d) and (i) at z = 60, 80, 150, respectively.

Equations (6)

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i u z + 1 2 [ β 1 ( z ) u x x + β 2 ( z ) u y y ] + [ χ 1 ( z ) | u | 2 + χ ( z ) | v | 2 ] u = η ( z ) ( v + i γ u ) , i v z + 1 2 [ β 1 ( z ) u x x + β 2 ( z ) u y y ] + [ χ ( z ) | u | 2 + χ 1 ( z ) | v | 2 ] v = η ( z ) ( u + i γ v ) ,
v ( z , x , y ) = ± u ( z , x , y ) exp ( ± i θ ) , u ( z , x , y ) = A 0 α 1 ( z ) α 2 ( z ) U [ Z ( z ) , X ( z , x , y ) ] exp { i [ cos ( θ ) Ω ( z ) + ϕ ( z , x , y ) ] } ,
i U Z + 1 2 U X X + | U | 2 U = 0 ,
χ ( z ) + χ 1 ( z ) = 1 4 A 0 2 [ k 2 β 1 ( z ) α 1 ( z ) α 2 ( z ) + l 2 β 2 ( z ) α 2 ( z ) α 1 ( z ) ] ,
u = A 0 α 1 ( z ) α 2 ( z ) [ 1 + P + i Q R ] exp { i [ ( 1 v 0 2 ) ( Z Z 0 ) + v 0 X + cos ( θ ) Ω ( z ) + ϕ ] } ,
β 1 ( z ) = β 10 exp ( σ z ) , β 2 ( z ) = β 20 exp ( σ z ) ,

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