International Journal of Nonlinear Analysis and Applications
Numerical investigation of the solitary and periodic waves in the nonlocal discrete Manakov system
Document Type : Research Paper
Authors
Department of Mathematics, College of Education for Pure Sciences, University of Mosul, Mosul, Iraq
Abstract
Solitary waves are interesting phenomena arising in various fields of physics, chemistry, and biology. Nonlinear continuous and discrete models supporting wave solutions of solitary behaviour have received increasing attention in recent years. Some examples of such integrable systems include Korteweg de-Vries (KdV) equation, the nonlinear Schrödinger (NLS) equation, and the Manakov system (MS). In this paper, we propose a discrete nonlocal version of the nonlinear Manakov system which admits spatial and temporal PT-symmetry. PT-symmetry property gives relevance to various fields in physics and has received a lot of attention in the studies of integrable nonlinear equations. In this work, the time evolution of solitary and periodic wave solutions in the proposed system has been numerically investigated. Suitable initial conditions have been considered to construct bright and dark solitons. The variational iteration method (VIM) was used to simulate the solution of the system. The error measurement of the simulation demonstrates the efficiency of the numerical method in constructing the different types of wave solutions.
Keywords
[1] A.T. Abed and A.S.Y. Aladool, Applying particle swarm optimization based on Pad´e approximant to solve ordinary
differential equation, Numer. Algebr. Control Optim. 12 (2022), 321—337.
[2] M.J. Ablowitz, X.D. Luo and Z.H. Musslimani, Discrete nonlocal nonlinear Schr¨odinger systems: Integrability,
inverse scattering and solitons, Nonlinearity 33 (2020), 3653—3707.
[3] M.J. Ablowitz and Z.H. Musslimani, Integrable nonlocal nonlinear Schr¨odinger equation, Phys. Rev. Lett. 110
(2013), 64105.
[4] M.J. Ablowitz and Z.H. Musslimani, Integrable discrete PT symmetric model, Phys. Rev. E - Stat. Nonlinear,
Soft Matter Phys. 90 (2014), 32912.
[5] M. J. Ablowitz and Musslimani, Z. H. Musslimani, Integrable nonlocal nonlinear equations, Stud. Appl. Math.
139 (2017), 7-–59.
[6] M. Akbarzade and J. Langari, Application of variational iteration method to partial differential equation systems,
Int. J. Math. Anal. 5 (2011), 863-–870.
[7] T. Aktosun, F. Demontis and C.V.D. Mee, Exact solutions to the focusing nonlinear Schr¨odinger equation, Inverse
Probl. 23 (2007), 2171—2195.
[8] C.M. Bender, Introduction to PT-symmetric quantum theory, Contemp. Phys. 46 (2005), 277—292.
[9] C.M. Bender, Making sense of non-Hermitian Hamiltonians, Reports Prog. Phys. 70 (2007), 947.
[10] C.M. Bender and S. Boettcher, Real spectra in non-hermitian hamiltonians having PT symmetry, Phys. Rev.
Lett. 80 (1998), 5243-–5246.
[11] A. Beygi, S.P. Klevansky and C.M. Bender, Coupled oscillator systems having partial PT symmetry, Phys. Rev.
A - At. Mol. Opt. Phys. 91 (2015), 62101.
[12] N. Bildik and A. Konuralp, The use of variational iteration method, differential transform method and adomian
decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci.
Numer. Simul. 7 (2006), 65—70.
[13] A. Bratsos, M. Ehrhardt and I.T. Famelis, A discrete Adomian decomposition method for discrete nonlinear
Schr¨odinger equations, Appl. Math. Comput. 197 (2008), 190—205.
[14] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter and D.N. Christodoulides, NonHermitian physics and PT symmetry, Nat. Phys. 14 (2017), 11-–19.
[15] T.A. Gadzhimuradov and A.M. Agalarov, Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear
Schr¨odinger equation, Phys. Rev. A 93 (2016), 62124.
[16] G.G. Grahovski, A.J. Mohammed and H. Susanto, Nonlocal Reductions of the Ablowitz–Ladik Equation, Theor.
Math. Phys. Russian Fed. 197 (2018), 1412—1429.
[17] G.G. Grahovski, J.I. Mustafa and H. Susanto, Nonlocal reductions of the multicomponent nonlinear Schr¨odinger
equation on symmetric spaces, Theor. Math. Phys. 197 (2018), 1430—1450.[18] A. Guo and G.J. Salamo, Observation of PT-symmetry breaking in complex optical potentials, Phys. Rev. Lett.
103 (2009), 93902.
[19] J.H. He, Variational iteration method - A kind of non-linear analytical technique: Some examples, Int. J. Nonlnear.
Mech. 34 (1999), 699—708.
[20] X. Huang and L. Ling, Soliton solutions for the nonlocal nonlinear Schr¨odinger equation, Eur. Phys. J. Plus 131
(2016), 1—11.
[21] J.L. Ji, Z.W. Xu and Z.N. Zhu, Nonintegrable spatial discrete nonlocal nonlinear schr¨odinger equation, Chaos 29
(2019), 103129.
[22] Y.V. Kartashov, V.V. Konotop and L. Torner, Topological States in Partially-PT -Symmetric Azimuthal Potentials, Phys. Rev. Lett. 115 (2015), 193902.
[23] K.B. Kazemi, Solving differential equations with least square and collocation methods, UNLV Theses, Diss. Prof.
Pap. Capstones 66 (2015).
[24] A. Khare and A. Saxena, Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations,
J. Math. Phys. 56 (2015), 32104.
[25] A. Khare, A. Saxena and A. Khare, Solutions of several coupled discrete models in terms of Lam´e polynomials of
arbitrary order, Pramana J. Phys. 79 (2012), 377-–392.
[26] V.V. Konotop, J. Yang and D.A. Zezyulin, Nonlinear waves in PT -symmetric systems, Rev. Mod. Phys. 88
(2016), 35002.
[27] T. Kottos, Broken symmetry makes light work, Nat. Phys. 6 (2010), 166-–167.
[28] J. Liu, Reductions of nonlocal nonlinear Schr¨odinger equations to Painlev’e type functions, arXiv Prepr. arXiv2104,
(2021), 10589.
[29] Y.C. Liu and C.S. Gurram, Solving nonlinear differential difference equations using He’s variational iteration
method, Appl. Math. Comput. Sci. 3 (2011).
[30] L.Y. Ma and Z.N. Zhu, N-soliton solution for an integrable nonlocal discrete focusing nonlinear Schr¨odinger
equation, Appl. Math. Lett. 59 (2016), 115–121.
[31] L.Y. Ma and Z.N. Zhu, Nonlocal nonlinear Schr¨odinger equation and its discrete version: Soliton solutions and
gauge equivalence, J. Math. Phys. 57 (2016), 83507.
[32] K.G. Makris, R. El-Ganainy, D.N. Christodoulides and Z.H. Musslimani, Beam dynamics in PT symmetric optical
lattices, Phys. Rev. Lett. 100 (2008), 103904.
[33] S.V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. PhysicsJETP 38 (1974), 248-–253.
[34] H.K. Mishra, A Comparative Study of Variational Iteration Method and He-Laplace Method, Appl. Math. 03
(2012), 1193—1201.
[35] M. Mitchell, M. Segev, T.H. Coskun and D.N. Christodoulides, Theory of self-trapped spatially incoherent light
beams, Phys. Rev. Lett. 79 (1997), 4990.
[36] S.T. Mohyud-Din and M.A. Noor, Variational iteration method for solving discrete KdV equation, Bull. Inst.
Acad. Sin. 5 (2010), no. 1, 69–73.
[37] Z.H. Musslimani, K.G. Makris, R. El-Ganainy and D.N. Christodoulides, Optical solitons in PT periodic potentials,
Conf. Quantum Electron. Laser Sci. Tech. Dig. Ser. 100 (2008), 30402.
[38] N. Okiotor, F. Ogunfiditimi and M.O. Durojaye, On the computation of the Lagrange multiplier for the variational
iteration ,ethod (VIM) for solving differential equations, J. Adv. Math. Comput. Sci. 35 (2020), no. 3, 74-–92.
[39] J. Patra, Some problems on variational iteration method, MSc thesis, Department of Mathematics, National
Institute of Technology Rourkela-769008, 2015.
[40] D.E. Pelinovsky and V.M. Rothos, Bifurcations of travelling wave solutions in the discrete NLS equations, Phys.
D Nonlinear Phenom. 202 (2005), 16-–36.[41] A. Regensburger, C. Bersch, M.A. Miri, G. Onishchukov, D.N. Christodoulides and U. Peschel, Parity-time
synthetic photonic lattices, Nature 488 (2012), 167–171.
[42] A. Ruschhaupt, F. Delgado and J.G. Muga, Physical realization of PT-symmetric potential scattering in a planar
slab waveguide, J. Phys. A. Math. Gen. 38 (2005), L171.
[43] C.E. R¨uter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev and D. Kip, Observation of parity-time
symmetry in optics, Nat. Phys. 6 (2010), 192-–195.
[44] A. Sadollah, H. Eskandar, D.G. Yoo and J.H. Kim, Approximate solving of nonlinear ordinary differential equations using least square weight function and metaheuristic algorithms, Eng. Appl. Artif. Intell. 40 (2015), 117—132.
[45] D. Sinha and P.K. Ghosh, Integrable nonlocal vector nonlinear Schr¨odinger equation with self-induced parity-timesymmetric potential, Phys. Lett. Sect. A Gen. At. Solid State Phys. 381 (2017), 124-–128.
[46] S. Stalin, M. Senthilvelan and M. Lakshmanan, Degenerate soliton solutions and their dynamics in the nonlocal
Manakov system: II Interactions between solitons, arXiv Prepr. arXiv1806. (2018), 06735.
[47] S. Stalin, M. Senthilvelan and M. Lakshmanan, Degenerate soliton solutions and their dynamics in the nonlocal
Manakov system: I symmetry preserving and symmetry breaking solutions, Nonlinear Dyn. 95 (2019), 343—360.
[48] S.V. Suchkov, A.A. Sukhorukov, J. Huang, S.V. Dmitriev, C. Lee and Y.S. Kivshar, Nonlinear switching and
solitons in PT-symmetric photonic systems, Laser Photonics Rev. 10 (2016), 177-–213.
[49] B. Sun, General soliton solutions to a nonlocal long-wave–short-wave resonance interaction equation with nonzero
boundary condition, Nonlinear Dyn. 92 (2018), 1369—1377.
[50] N.H. Sweilam, Variational iteration method for solving cubic nonlinear Schr¨odinger equation, J. Comput. Appl.
Math. 207 (2007), 155-–163.
[51] A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, 2009.
[52] A.M. Wazwaz, A variety of optical solitons for nonlinear Schr¨odinger equation with detuning term by the variational iteration method, Optik (Stuttg) 196 (2019), 163169.
[53] A.M. Wazwaz, Optical bright and dark soliton solutions for coupled nonlinear Schr¨odinger (CNLS) equations by
the variational iteration method, Optik (Stuttg) 207 (2020), 164457.
[54] J. Wu, Least squares methods for solving partial differential equations by using B´ezier control points, Appl. Math.
Comput. 219 (2012), 3655-–3663.
[55] J. Yang, Physically significant nonlocal nonlinear Schr¨odinger equation and its soliton solutions, Phys. Rev. E 98
(2018), 42202.
[56] B. Yang and Y. Chen, General rogue waves and their dynamics in several reverse time integrable nonlocal nonlinear
equations, arXiv Prepr. arXiv1712, (2017), 05974.
[57] A. Yildirim, Applying He’s variational iteration method for solving differential-difference equation, Math. Probl.
Eng. 2008 (2008).
[58] G. Zhang and Z. Yan, Multi-rational and semi-rational solitons and interactions for the nonlocal coupled nonlinear
Schr¨odinger equations, Epl 118 (2017), 60004.
[59] Y. Zhang, Y. Liu and X. Tang, A general integrable three-component coupled nonlocal nonlinear Schr¨odinger
equation, Nonlinear Dyn. 89 (2017), 2729-–2738.
[60] H.Q. Zhang, M.Y. Zhang and R. Hu, Darboux transformation and soliton solutions in the parity-time-symmetric
nonlocal vector nonlinear Schr¨odinger equation, Appl. Math. Lett. 76 (2018), 170-–174.
differential equation, Numer. Algebr. Control Optim. 12 (2022), 321—337.
[2] M.J. Ablowitz, X.D. Luo and Z.H. Musslimani, Discrete nonlocal nonlinear Schr¨odinger systems: Integrability,
inverse scattering and solitons, Nonlinearity 33 (2020), 3653—3707.
[3] M.J. Ablowitz and Z.H. Musslimani, Integrable nonlocal nonlinear Schr¨odinger equation, Phys. Rev. Lett. 110
(2013), 64105.
[4] M.J. Ablowitz and Z.H. Musslimani, Integrable discrete PT symmetric model, Phys. Rev. E - Stat. Nonlinear,
Soft Matter Phys. 90 (2014), 32912.
[5] M. J. Ablowitz and Musslimani, Z. H. Musslimani, Integrable nonlocal nonlinear equations, Stud. Appl. Math.
139 (2017), 7-–59.
[6] M. Akbarzade and J. Langari, Application of variational iteration method to partial differential equation systems,
Int. J. Math. Anal. 5 (2011), 863-–870.
[7] T. Aktosun, F. Demontis and C.V.D. Mee, Exact solutions to the focusing nonlinear Schr¨odinger equation, Inverse
Probl. 23 (2007), 2171—2195.
[8] C.M. Bender, Introduction to PT-symmetric quantum theory, Contemp. Phys. 46 (2005), 277—292.
[9] C.M. Bender, Making sense of non-Hermitian Hamiltonians, Reports Prog. Phys. 70 (2007), 947.
[10] C.M. Bender and S. Boettcher, Real spectra in non-hermitian hamiltonians having PT symmetry, Phys. Rev.
Lett. 80 (1998), 5243-–5246.
[11] A. Beygi, S.P. Klevansky and C.M. Bender, Coupled oscillator systems having partial PT symmetry, Phys. Rev.
A - At. Mol. Opt. Phys. 91 (2015), 62101.
[12] N. Bildik and A. Konuralp, The use of variational iteration method, differential transform method and adomian
decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci.
Numer. Simul. 7 (2006), 65—70.
[13] A. Bratsos, M. Ehrhardt and I.T. Famelis, A discrete Adomian decomposition method for discrete nonlinear
Schr¨odinger equations, Appl. Math. Comput. 197 (2008), 190—205.
[14] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter and D.N. Christodoulides, NonHermitian physics and PT symmetry, Nat. Phys. 14 (2017), 11-–19.
[15] T.A. Gadzhimuradov and A.M. Agalarov, Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear
Schr¨odinger equation, Phys. Rev. A 93 (2016), 62124.
[16] G.G. Grahovski, A.J. Mohammed and H. Susanto, Nonlocal Reductions of the Ablowitz–Ladik Equation, Theor.
Math. Phys. Russian Fed. 197 (2018), 1412—1429.
[17] G.G. Grahovski, J.I. Mustafa and H. Susanto, Nonlocal reductions of the multicomponent nonlinear Schr¨odinger
equation on symmetric spaces, Theor. Math. Phys. 197 (2018), 1430—1450.[18] A. Guo and G.J. Salamo, Observation of PT-symmetry breaking in complex optical potentials, Phys. Rev. Lett.
103 (2009), 93902.
[19] J.H. He, Variational iteration method - A kind of non-linear analytical technique: Some examples, Int. J. Nonlnear.
Mech. 34 (1999), 699—708.
[20] X. Huang and L. Ling, Soliton solutions for the nonlocal nonlinear Schr¨odinger equation, Eur. Phys. J. Plus 131
(2016), 1—11.
[21] J.L. Ji, Z.W. Xu and Z.N. Zhu, Nonintegrable spatial discrete nonlocal nonlinear schr¨odinger equation, Chaos 29
(2019), 103129.
[22] Y.V. Kartashov, V.V. Konotop and L. Torner, Topological States in Partially-PT -Symmetric Azimuthal Potentials, Phys. Rev. Lett. 115 (2015), 193902.
[23] K.B. Kazemi, Solving differential equations with least square and collocation methods, UNLV Theses, Diss. Prof.
Pap. Capstones 66 (2015).
[24] A. Khare and A. Saxena, Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations,
J. Math. Phys. 56 (2015), 32104.
[25] A. Khare, A. Saxena and A. Khare, Solutions of several coupled discrete models in terms of Lam´e polynomials of
arbitrary order, Pramana J. Phys. 79 (2012), 377-–392.
[26] V.V. Konotop, J. Yang and D.A. Zezyulin, Nonlinear waves in PT -symmetric systems, Rev. Mod. Phys. 88
(2016), 35002.
[27] T. Kottos, Broken symmetry makes light work, Nat. Phys. 6 (2010), 166-–167.
[28] J. Liu, Reductions of nonlocal nonlinear Schr¨odinger equations to Painlev’e type functions, arXiv Prepr. arXiv2104,
(2021), 10589.
[29] Y.C. Liu and C.S. Gurram, Solving nonlinear differential difference equations using He’s variational iteration
method, Appl. Math. Comput. Sci. 3 (2011).
[30] L.Y. Ma and Z.N. Zhu, N-soliton solution for an integrable nonlocal discrete focusing nonlinear Schr¨odinger
equation, Appl. Math. Lett. 59 (2016), 115–121.
[31] L.Y. Ma and Z.N. Zhu, Nonlocal nonlinear Schr¨odinger equation and its discrete version: Soliton solutions and
gauge equivalence, J. Math. Phys. 57 (2016), 83507.
[32] K.G. Makris, R. El-Ganainy, D.N. Christodoulides and Z.H. Musslimani, Beam dynamics in PT symmetric optical
lattices, Phys. Rev. Lett. 100 (2008), 103904.
[33] S.V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. PhysicsJETP 38 (1974), 248-–253.
[34] H.K. Mishra, A Comparative Study of Variational Iteration Method and He-Laplace Method, Appl. Math. 03
(2012), 1193—1201.
[35] M. Mitchell, M. Segev, T.H. Coskun and D.N. Christodoulides, Theory of self-trapped spatially incoherent light
beams, Phys. Rev. Lett. 79 (1997), 4990.
[36] S.T. Mohyud-Din and M.A. Noor, Variational iteration method for solving discrete KdV equation, Bull. Inst.
Acad. Sin. 5 (2010), no. 1, 69–73.
[37] Z.H. Musslimani, K.G. Makris, R. El-Ganainy and D.N. Christodoulides, Optical solitons in PT periodic potentials,
Conf. Quantum Electron. Laser Sci. Tech. Dig. Ser. 100 (2008), 30402.
[38] N. Okiotor, F. Ogunfiditimi and M.O. Durojaye, On the computation of the Lagrange multiplier for the variational
iteration ,ethod (VIM) for solving differential equations, J. Adv. Math. Comput. Sci. 35 (2020), no. 3, 74-–92.
[39] J. Patra, Some problems on variational iteration method, MSc thesis, Department of Mathematics, National
Institute of Technology Rourkela-769008, 2015.
[40] D.E. Pelinovsky and V.M. Rothos, Bifurcations of travelling wave solutions in the discrete NLS equations, Phys.
D Nonlinear Phenom. 202 (2005), 16-–36.[41] A. Regensburger, C. Bersch, M.A. Miri, G. Onishchukov, D.N. Christodoulides and U. Peschel, Parity-time
synthetic photonic lattices, Nature 488 (2012), 167–171.
[42] A. Ruschhaupt, F. Delgado and J.G. Muga, Physical realization of PT-symmetric potential scattering in a planar
slab waveguide, J. Phys. A. Math. Gen. 38 (2005), L171.
[43] C.E. R¨uter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev and D. Kip, Observation of parity-time
symmetry in optics, Nat. Phys. 6 (2010), 192-–195.
[44] A. Sadollah, H. Eskandar, D.G. Yoo and J.H. Kim, Approximate solving of nonlinear ordinary differential equations using least square weight function and metaheuristic algorithms, Eng. Appl. Artif. Intell. 40 (2015), 117—132.
[45] D. Sinha and P.K. Ghosh, Integrable nonlocal vector nonlinear Schr¨odinger equation with self-induced parity-timesymmetric potential, Phys. Lett. Sect. A Gen. At. Solid State Phys. 381 (2017), 124-–128.
[46] S. Stalin, M. Senthilvelan and M. Lakshmanan, Degenerate soliton solutions and their dynamics in the nonlocal
Manakov system: II Interactions between solitons, arXiv Prepr. arXiv1806. (2018), 06735.
[47] S. Stalin, M. Senthilvelan and M. Lakshmanan, Degenerate soliton solutions and their dynamics in the nonlocal
Manakov system: I symmetry preserving and symmetry breaking solutions, Nonlinear Dyn. 95 (2019), 343—360.
[48] S.V. Suchkov, A.A. Sukhorukov, J. Huang, S.V. Dmitriev, C. Lee and Y.S. Kivshar, Nonlinear switching and
solitons in PT-symmetric photonic systems, Laser Photonics Rev. 10 (2016), 177-–213.
[49] B. Sun, General soliton solutions to a nonlocal long-wave–short-wave resonance interaction equation with nonzero
boundary condition, Nonlinear Dyn. 92 (2018), 1369—1377.
[50] N.H. Sweilam, Variational iteration method for solving cubic nonlinear Schr¨odinger equation, J. Comput. Appl.
Math. 207 (2007), 155-–163.
[51] A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, 2009.
[52] A.M. Wazwaz, A variety of optical solitons for nonlinear Schr¨odinger equation with detuning term by the variational iteration method, Optik (Stuttg) 196 (2019), 163169.
[53] A.M. Wazwaz, Optical bright and dark soliton solutions for coupled nonlinear Schr¨odinger (CNLS) equations by
the variational iteration method, Optik (Stuttg) 207 (2020), 164457.
[54] J. Wu, Least squares methods for solving partial differential equations by using B´ezier control points, Appl. Math.
Comput. 219 (2012), 3655-–3663.
[55] J. Yang, Physically significant nonlocal nonlinear Schr¨odinger equation and its soliton solutions, Phys. Rev. E 98
(2018), 42202.
[56] B. Yang and Y. Chen, General rogue waves and their dynamics in several reverse time integrable nonlocal nonlinear
equations, arXiv Prepr. arXiv1712, (2017), 05974.
[57] A. Yildirim, Applying He’s variational iteration method for solving differential-difference equation, Math. Probl.
Eng. 2008 (2008).
[58] G. Zhang and Z. Yan, Multi-rational and semi-rational solitons and interactions for the nonlocal coupled nonlinear
Schr¨odinger equations, Epl 118 (2017), 60004.
[59] Y. Zhang, Y. Liu and X. Tang, A general integrable three-component coupled nonlocal nonlinear Schr¨odinger
equation, Nonlinear Dyn. 89 (2017), 2729-–2738.
[60] H.Q. Zhang, M.Y. Zhang and R. Hu, Darboux transformation and soliton solutions in the parity-time-symmetric
nonlocal vector nonlinear Schr¨odinger equation, Appl. Math. Lett. 76 (2018), 170-–174.
)
Volume 13, Issue 2
July 2022Pages 2059-2070
- Receive Date: 06 February 2022
- Revise Date: 19 May 2022
- Accept Date: 11 June 2022