Stability of in-phase quadruple and vortex solitons in the parity-time-symmetric periodic potentials Xiaoping Ren,1,2 Hong Wang,1,* Hongcheng Wang,3 and Yingji He4 1
Engineering Research Center for Optoelectronics of Guangdong Province, School of Science, South China University of Technology, Guangzhou, 510640, China 2 School of Electronics and Information, South China University of Technology, Guangzhou 510640, China 3 School of Electronic Engineering, Dongguan University of Technology, Dongguan, 523808, China 4 School of Electronics and Information, Guangdong Polytechnic Normal University, Guangzhou 510665, China * [email protected]
Abstract: We report the stability of in-phase quadruple and off-site vortex solitons in the parity-time-symmetric periodic potentials with defocusing Kerr nonlinearity. All solitons can exist in the first gap and can be stable in a certain range. It is shown that the power of vortex solitons decreases and the stable region shrinks with increase of the topological charge. Especially the stable region is very small for double charge vortex solitons. The power evolutions of vortex solitons along the propagation distance are also analysed. Increasing the lattice depth or decreasing the gain-loss component can stabilize vortex solitons. For both lattice depth and gain-loss component there exists a critical value, below or above which all vortex solitons will become unstable. ©2014 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (190.5530) Pulse propagation and temporal solitons.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
L. M. Pismen, Vortices in Nonlinear Fields (Clarendon, 1999). See M. S. Soskin and M. V. Vasnetsov, Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003). Experimental results on vortex solitons in 2D photonic lattices were presented by M. Segev at CLEO Europe, Munich, Germany (2003), EE3-1. B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(2), 026601 (2001). J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28(21), 2094–2096 (2003). D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004). J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004). J. Yang, “Stability of vortex solitons in a photorefractive optical lattice,” New J. Phys. 6, 47 (2004). C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear Lattices,” Phys. Rev. A 83(4), 041805 (2011). Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011). S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012). H. Wang, D. Ling, S. Zhang, X. Zhu, and Y. He, “Gap solitons in parity-time complex superlattice with dual periods,” Chin. Phys. B 23(6), 064208 (2014). X. Zhu, H. Wang, H. Li, W. He, and Y. J. He, “Two-dimensional multipeak gap solitons supported by paritytime-symmetric periodic potentials,” Opt. Lett. 38(15), 2723–2725 (2013). H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012). Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
#213467 - $15.00 USD Received 4 Jun 2014; revised 20 Jul 2014; accepted 25 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119774 | OPTICS EXPRESS 19774
18. C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT symmetry,” Opt. Express 21(3), 3917–3925 (2013). 19. H. Wang, W. He, S. Shi, X. Zhu, and Y. J. He, “Defect gap solitons in self-focusing Kerr media with paritytime-symmetric linear superlattices and modulated nonlinear lattices,” Phys. Scr. 89(2), 025502 (2014). 20. V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86(1), 013808 (2012). 21. H. G. Li, X. Zhu, Z. W. Shi, B. A. Malomed, T. S. Lai, and C. H. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89(5), 053811 (2014). 22. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008). 23. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). 24. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153 (2007). 25. J. Yang, Nonlinear Waves in Integrable Systems, (SLAM, 2010).
1. Introduction Vortices are fundamental objects in many branches of physics [1]. In optics, vortices are associated with the screw phase dislocations carried by diffracting optical beams [2]. When such optical vortices propagate in nonlinear media, the vortex cores become self-trapped and optical vortex solitons form [3]. In recent years, the investigations of optical waves carrying angular momentum and vortex soliton generating in optical lattices have been carried to reality because of their ability to create nonlinear waveguide arrays in 2D. The experimental observation of vortex solitons in nonlinear lattices has been reported [4]. In a periodic potentials optical lattice, the self-focusing or defocusing nonlinearity may balance the diffraction of the lattice, which can result in the stable vortex solitons. In particular, both onsite vortex solitons (vortices whose screw phase dislocation is located on a site) [5] and offsite vortex solitons (vortices whose screw phase dislocation is located between sites) [6] in an optical lattice with Kerr nonlinearity have been shown theoretically. The vortex solitons in photorefractive crystal are quickly observed in experiments [7, 8]. Recently, a theoretical work demonstrated that both on- and off-site vortex solitons can be stable with certain range of parameters in a photorefractive crystal [9]. On the other hand, light propagation in PT-symmetric potentials has attracted intensive investigations in recent years. In 1998, Bender and Boettcher showed that if the system is parity-time (PT)-symmetry, non-Hermitian Hamiltonians can have entirely real spectra [10]. In recent, the existence of stable one-dimensional (1D) and two-dimensional (2D) optical solitons in a PT-symmetric nonlinear [11] lattice and linear lattice [12, 13] have been investigated. In the context of PT-symmetry, the gap solitons in optical lattices have been investigated extensively [14–20]. Very recently, the composite vortices and surface modes in PT-symmetric photonic lattices have been reported [21]. However, the properties of vortex solitons in the PT-symmetric separable periodic optical potentials have not been studied. Stability of vortex solitons in PT-symmetric periodic potentials is obviously an important issue. With the PT-symmetry, there are some characteristics which cannot be observed in the conservative media for vortex solitons in lattices. The study of vortex solitons in PTsymmetric periodic potentials lays the foundation for the future exploration in the interplay of PT-symmetry, nonlinearity and angular momentum. The evolution of the power along the propagation distance which is in accordance with the stability of solitons can serve as another condition to verify whether the solitons in PT-symmetric lattices are stable or not. In this paper, we numerically investigate the stability of in-phase quadruple solitons and off-site vortex solitons with unit and double charge in the PT-symmetric periodic potentials based on defocusing Kerr nonlinearity. The interplay of angular momentum, diffraction of beams, nonlinearity, and PT-symmetry can lower the power of vortex solitons. The stable region of vortex solitons shrinks by increasing the topological charge. Furthermore, the influence of lattice depth and gain-loss component on the stability of vortex solitons has been analysed. For fixed lattice depth or fixed gain-loss component, both exists a critical value, below or above which all vortex solitons are unstable. The power evolution of solitons along
#213467 - $15.00 USD Received 4 Jun 2014; revised 20 Jul 2014; accepted 25 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119774 | OPTICS EXPRESS 19775
the propagation distance has been studied and it is in accordance with the stability of solitons, and this is very different from the case in conservative media. 2. Theoretical model The mathematical model for light propagation in the PT-symmetric periodic potentials with defocusing Kerr nonlinearity is the normalized nonlinear Schrödinger equation, which can be written as [22] iU + U + U + V ( x, y )U − U 2 U = 0 z xx yy
(1)
where U is the slowly varying amplitude of the beam, z is the normalized longitudinal coordinate, ( x, y ) are the normalized distances along the transverse directions, and V ( x, y ) is the 2D PT-symmetric periodic potential. In particular, we take the 2D PT-symmetric periodic potential V ( x, y ) as [22] V ( x, y ) = V0 {[sin 2 ( x) + sin 2 ( y )] + iW0 [sin(2 x) + sin(2 y )]} ,
Here V0 is the parameter which controls the depth of the PT-symmetric optical lattice, and W0 is the parameter corresponding to the amplitude of the imaginary part compared with the real part. The band structure of the PT-symmetric optical lattice can be calculated by the plane wave expansion method. The band structures corresponding to this separable PTsymmetric periodic potential for V0 = 9 with W0 = 0.1 and W0 = 0.5 are displayed in Figs. 1(a) and 1(b), respectively. Numerical analysis shows that the critical threshold for this potential is W0th = 0.5, which is identical to the case
V ( x, y ) = V0 {[cos 2 ( x) + cos 2 ( y )] + iW0 [sin(2 x) + sin(2 y )]} [22,23]. If W0 is above 0.5, the PT-
symmetric will be destroyed.
Fig. 1. The band structure of the PT-symmetric optical lattices associate with (a) W0 = 0.1 , (b)
W0 = 0.5 , for all cases V0 = 9 .
In this paper, we take V0 = 9,W0 = 0.1 . As shown in Fig. 1(a), the first gap is 8.331 ≤ μ ≤ 12.337 , and the semi-infinite gap is 12.539 ≤ μ < +∞ . The lattice solitons in Eq. (1) are sought in the form of U = u ( x, y ) exp(i μ z ) , Where μ is the real propagation constant, u ( x, y ) is a complex-value localized function. Thus, the function u ( x, y ) satisfies the following equation 2
(2) u xx + u yy + V ( x, y )u − u u − μ u = 0 These soliton solutions are resolved numerically by using the modified squared-operator iteration method [24]. Define P =
+∞
−∞
+∞
−∞
2
u dxdy as the power of a soliton [22].
#213467 - $15.00 USD Received 4 Jun 2014; revised 20 Jul 2014; accepted 25 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119774 | OPTICS EXPRESS 19776
In order to investigate the linear stability of these solitons, we add the perturbations p( x, y ) and q ( x, y ) to the solution, which is written as: U ( x, y, z ) = exp(i μ z ) {u ( x, y ) + [ p ( x, y ) − q ( x, y )]exp(λ z ) + [ p( x, y ) + q ( x, y )]* exp(λ * z )}
where p
Engineering Research Center for Optoelectronics of Guangdong Province, School of Science, South China University of Technology, Guangzhou, 510640, China 2 School of Electronics and Information, South China University of Technology, Guangzhou 510640, China 3 School of Electronic Engineering, Dongguan University of Technology, Dongguan, 523808, China 4 School of Electronics and Information, Guangdong Polytechnic Normal University, Guangzhou 510665, China * [email protected]
Abstract: We report the stability of in-phase quadruple and off-site vortex solitons in the parity-time-symmetric periodic potentials with defocusing Kerr nonlinearity. All solitons can exist in the first gap and can be stable in a certain range. It is shown that the power of vortex solitons decreases and the stable region shrinks with increase of the topological charge. Especially the stable region is very small for double charge vortex solitons. The power evolutions of vortex solitons along the propagation distance are also analysed. Increasing the lattice depth or decreasing the gain-loss component can stabilize vortex solitons. For both lattice depth and gain-loss component there exists a critical value, below or above which all vortex solitons will become unstable. ©2014 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (190.5530) Pulse propagation and temporal solitons.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
L. M. Pismen, Vortices in Nonlinear Fields (Clarendon, 1999). See M. S. Soskin and M. V. Vasnetsov, Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003). Experimental results on vortex solitons in 2D photonic lattices were presented by M. Segev at CLEO Europe, Munich, Germany (2003), EE3-1. B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(2), 026601 (2001). J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28(21), 2094–2096 (2003). D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004). J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004). J. Yang, “Stability of vortex solitons in a photorefractive optical lattice,” New J. Phys. 6, 47 (2004). C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear Lattices,” Phys. Rev. A 83(4), 041805 (2011). Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011). S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012). H. Wang, D. Ling, S. Zhang, X. Zhu, and Y. He, “Gap solitons in parity-time complex superlattice with dual periods,” Chin. Phys. B 23(6), 064208 (2014). X. Zhu, H. Wang, H. Li, W. He, and Y. J. He, “Two-dimensional multipeak gap solitons supported by paritytime-symmetric periodic potentials,” Opt. Lett. 38(15), 2723–2725 (2013). H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012). Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
#213467 - $15.00 USD Received 4 Jun 2014; revised 20 Jul 2014; accepted 25 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119774 | OPTICS EXPRESS 19774
18. C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT symmetry,” Opt. Express 21(3), 3917–3925 (2013). 19. H. Wang, W. He, S. Shi, X. Zhu, and Y. J. He, “Defect gap solitons in self-focusing Kerr media with paritytime-symmetric linear superlattices and modulated nonlinear lattices,” Phys. Scr. 89(2), 025502 (2014). 20. V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86(1), 013808 (2012). 21. H. G. Li, X. Zhu, Z. W. Shi, B. A. Malomed, T. S. Lai, and C. H. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89(5), 053811 (2014). 22. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008). 23. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). 24. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153 (2007). 25. J. Yang, Nonlinear Waves in Integrable Systems, (SLAM, 2010).
1. Introduction Vortices are fundamental objects in many branches of physics [1]. In optics, vortices are associated with the screw phase dislocations carried by diffracting optical beams [2]. When such optical vortices propagate in nonlinear media, the vortex cores become self-trapped and optical vortex solitons form [3]. In recent years, the investigations of optical waves carrying angular momentum and vortex soliton generating in optical lattices have been carried to reality because of their ability to create nonlinear waveguide arrays in 2D. The experimental observation of vortex solitons in nonlinear lattices has been reported [4]. In a periodic potentials optical lattice, the self-focusing or defocusing nonlinearity may balance the diffraction of the lattice, which can result in the stable vortex solitons. In particular, both onsite vortex solitons (vortices whose screw phase dislocation is located on a site) [5] and offsite vortex solitons (vortices whose screw phase dislocation is located between sites) [6] in an optical lattice with Kerr nonlinearity have been shown theoretically. The vortex solitons in photorefractive crystal are quickly observed in experiments [7, 8]. Recently, a theoretical work demonstrated that both on- and off-site vortex solitons can be stable with certain range of parameters in a photorefractive crystal [9]. On the other hand, light propagation in PT-symmetric potentials has attracted intensive investigations in recent years. In 1998, Bender and Boettcher showed that if the system is parity-time (PT)-symmetry, non-Hermitian Hamiltonians can have entirely real spectra [10]. In recent, the existence of stable one-dimensional (1D) and two-dimensional (2D) optical solitons in a PT-symmetric nonlinear [11] lattice and linear lattice [12, 13] have been investigated. In the context of PT-symmetry, the gap solitons in optical lattices have been investigated extensively [14–20]. Very recently, the composite vortices and surface modes in PT-symmetric photonic lattices have been reported [21]. However, the properties of vortex solitons in the PT-symmetric separable periodic optical potentials have not been studied. Stability of vortex solitons in PT-symmetric periodic potentials is obviously an important issue. With the PT-symmetry, there are some characteristics which cannot be observed in the conservative media for vortex solitons in lattices. The study of vortex solitons in PTsymmetric periodic potentials lays the foundation for the future exploration in the interplay of PT-symmetry, nonlinearity and angular momentum. The evolution of the power along the propagation distance which is in accordance with the stability of solitons can serve as another condition to verify whether the solitons in PT-symmetric lattices are stable or not. In this paper, we numerically investigate the stability of in-phase quadruple solitons and off-site vortex solitons with unit and double charge in the PT-symmetric periodic potentials based on defocusing Kerr nonlinearity. The interplay of angular momentum, diffraction of beams, nonlinearity, and PT-symmetry can lower the power of vortex solitons. The stable region of vortex solitons shrinks by increasing the topological charge. Furthermore, the influence of lattice depth and gain-loss component on the stability of vortex solitons has been analysed. For fixed lattice depth or fixed gain-loss component, both exists a critical value, below or above which all vortex solitons are unstable. The power evolution of solitons along
#213467 - $15.00 USD Received 4 Jun 2014; revised 20 Jul 2014; accepted 25 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119774 | OPTICS EXPRESS 19775
the propagation distance has been studied and it is in accordance with the stability of solitons, and this is very different from the case in conservative media. 2. Theoretical model The mathematical model for light propagation in the PT-symmetric periodic potentials with defocusing Kerr nonlinearity is the normalized nonlinear Schrödinger equation, which can be written as [22] iU + U + U + V ( x, y )U − U 2 U = 0 z xx yy
(1)
where U is the slowly varying amplitude of the beam, z is the normalized longitudinal coordinate, ( x, y ) are the normalized distances along the transverse directions, and V ( x, y ) is the 2D PT-symmetric periodic potential. In particular, we take the 2D PT-symmetric periodic potential V ( x, y ) as [22] V ( x, y ) = V0 {[sin 2 ( x) + sin 2 ( y )] + iW0 [sin(2 x) + sin(2 y )]} ,
Here V0 is the parameter which controls the depth of the PT-symmetric optical lattice, and W0 is the parameter corresponding to the amplitude of the imaginary part compared with the real part. The band structure of the PT-symmetric optical lattice can be calculated by the plane wave expansion method. The band structures corresponding to this separable PTsymmetric periodic potential for V0 = 9 with W0 = 0.1 and W0 = 0.5 are displayed in Figs. 1(a) and 1(b), respectively. Numerical analysis shows that the critical threshold for this potential is W0th = 0.5, which is identical to the case
V ( x, y ) = V0 {[cos 2 ( x) + cos 2 ( y )] + iW0 [sin(2 x) + sin(2 y )]} [22,23]. If W0 is above 0.5, the PT-
symmetric will be destroyed.
Fig. 1. The band structure of the PT-symmetric optical lattices associate with (a) W0 = 0.1 , (b)
W0 = 0.5 , for all cases V0 = 9 .
In this paper, we take V0 = 9,W0 = 0.1 . As shown in Fig. 1(a), the first gap is 8.331 ≤ μ ≤ 12.337 , and the semi-infinite gap is 12.539 ≤ μ < +∞ . The lattice solitons in Eq. (1) are sought in the form of U = u ( x, y ) exp(i μ z ) , Where μ is the real propagation constant, u ( x, y ) is a complex-value localized function. Thus, the function u ( x, y ) satisfies the following equation 2
(2) u xx + u yy + V ( x, y )u − u u − μ u = 0 These soliton solutions are resolved numerically by using the modified squared-operator iteration method [24]. Define P =
+∞
−∞
+∞
−∞
2
u dxdy as the power of a soliton [22].
#213467 - $15.00 USD Received 4 Jun 2014; revised 20 Jul 2014; accepted 25 Jul 2014; published 8 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.01914119774 | OPTICS EXPRESS 19776
In order to investigate the linear stability of these solitons, we add the perturbations p( x, y ) and q ( x, y ) to the solution, which is written as: U ( x, y, z ) = exp(i μ z ) {u ( x, y ) + [ p ( x, y ) − q ( x, y )]exp(λ z ) + [ p( x, y ) + q ( x, y )]* exp(λ * z )}
where p
