2015, 2015(special): 505-514. doi: 10.3934/proc.2015.0505

Manakov solitons and effects of external potential wells

1. 

Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee Blvd., So a 1784, Bulgaria, Bulgaria

2. 

Dept. of Applied Mathematics and Informatics, Technical University of Sofia, 1000 Sofia

Received  September 2014 Revised  April 2015 Published  November 2015

The effects of the external potential wells on the Manakov soliton interactions using the perturbed complex Toda chain (PCTC) model are analyzed. The superposition of a large number of wells/humps influences stronger the motion of the soliton envelopes and can cause a transition from asymptotically free and mixed asymptotic regime to a bound state regime and vice versa. Such external potentials are easier to implement in experiments and can be used to control the soliton motion in a given direction and to achieve a predicted motion of the optical pulse. A general feature of the conducted numerical experiments is that the long-time evolution of both CTC and PCTC match very well with the Manakov model numerics, often much longer than expected even for 9-soliton train configurations. This means that PCTC is reliable dynamical model for predicting the evolution of the multisoliton solutions of Manakov model in adiabatic approximation.
Citation: V. S. Gerdjikov, A. V. Kyuldjiev, M. D. Todorov. Manakov solitons and effects of external potential wells. Conference Publications, 2015, 2015 (special) : 505-514. doi: 10.3934/proc.2015.0505
References:
[1]

D. Anderson, and M. Lisak, Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,, Phys. Rev. A, 27 (1983), 1393.   Google Scholar

[2]

C. I. Christov, S. Dost, and G. A. Maugin, Inelasticity of soliton collisions in systems of coupled NLS equations,, Physica Scripta, 50 (1994), 449.   Google Scholar

[3]

V. S. Gerdjikov, B. B. Baizakov, and M. Salerno, Modelling adiabatic $N$-soliton interactions and perturbations,, Theor. Math. Phys., 144 (2005), 1138.   Google Scholar

[4]

V. S. Gerdjikov, "On soliton interactions of vector nonlinear Schrödinger equations,'', in AMiTaNS'11, (2011), 57.   Google Scholar

[5]

V. S. Gerdjikov, Modeling soliton interactions of the perturbed vector nonlinear Schrödinger equation,, Bulgarian J. Phys., 38 (2011), 274.   Google Scholar

[6]

V. S. Gerdjikov, B. B. Baizakov, M. Salerno, and N. A. Kostov, Adiabatic $N$-soliton interactions of Bose-Einstein condensates in external potentials,, Phys. Rev. E., 73 (2006).   Google Scholar

[7]

V. S. Gerdjikov, E. V. Doktorov, and N. P. Matsuka, $N$-soliton train and generalized complex Toda chain for Manakov system,, Theor. Math. Phys., 151 (2007), 762.   Google Scholar

[8]

V. S. Gerdjikov, E. G. Evstatiev, D. J. Kaup, G. L. Diankov, and I. M. Uzunov, Stability and quasi-equidistant propagation of NLS soliton trains,, Phys. Lett. A, 241 (1998), 323.   Google Scholar

[9]

V. S. Gerdjikov, G. G. Grahovski, "Two soliton interactions of BD.I multicomponent NLS equations and their gauge equivalent,'', in AMiTaNS'10, (2010), 561.   Google Scholar

[10]

V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, and E. G. Evstatiev, Asymptotic behavior of $N$-soliton trains of the nonlinear Schrödinger equation,, Phys. Rev. Lett., 77 (1996), 3943.   Google Scholar

[11]

V. S. Gerdjikov, N. A. Kostov, E. V. Doktorov, and N. P. Matsuka, Generalized perturbed complex Toda chain for Manakov system and exact solutions of the Bose-Einstein mixtures,, Mathematics and Computers in Simulation, 80 (2009), 112.   Google Scholar

[12]

V. S. Gerdjikov and M. D. Todorov, $N$-soliton interactions for the Manakov system. Effects of external potentials,, in Localized Excitations in Nonlinear Complex Systems, 7 (2014), 147.   Google Scholar

[13]

V. S. Gerdjikov and M. D. Todorov, "On the effects of sech-like potentials on Manakov solitons,'', in AMiTaNS'13, (2013), 75.   Google Scholar

[14]

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, Nonlinear Schrödinger equation and $N$-soliton interactions: Generalized Karpman-Soloviev approach and the complex Toda chain,, Phys. Rev. E, 55 (1997), 6039.   Google Scholar

[15]

V. S. Gerdjikov, G. Vilasi, and A. B. Yanovski, Integrable Hamiltonian hierarchies. Spectral and geometric methods,, Lecture Notes in Physics 748, 748 (2008).   Google Scholar

[16]

V. S. Gerdjikov, M. D. Todorov, and A. V. Kyuldjiev, Asymptotic behavior of Manakov solitons: Effects of potential wells and humps,, preprint, ().   Google Scholar

[17]

A. Griffin, T. Nikuni, and E. Zaremba, Bose-Condensed Gases at Finite Temperatures,, Cambridge University Press, (2009).   Google Scholar

[18]

T.-L. Ho, Spinor Bose condensates in optical traps,, Phys. Rev. Lett., 81 (1998).   Google Scholar

[19]

V. I. Karpman and V. V. Solov'ev, A perturbational approach to the two-solition systems,, Physica D, 3 (1981), 487.   Google Scholar

[20]

, Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment (eds. P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonzalez), Springer, 45 (2008).   Google Scholar

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N. A. Kostov, V. Z. Enol'skii, V. S. Gerdjikov, V. V. Konotop and M. Salerno, On two-component Bose-Einstein condensates in periodic potential,, Phys. Rev. E, 70 (2004).   Google Scholar

[22]

N. A. Kostov, V. S. Gerdjikov and T. I. Valchev, Exact solutions for equations of Bose-Fermi mixtures in one-dimensional optical lattice,, SIGMA 3 (2007), 3 (2007).   Google Scholar

[23]

A. V. Kyuldjiev, V. S. Gerdjikov, M. D. Todorov, Asymptotic Behavior of Manakov Solitons: Effects of of shallow and wide potential wells and humps,, in Mathematics in Industry (ed. A. Slavova), (2014), 410.   Google Scholar

[24]

T. I. Lakoba and D. J. Kaup, Perturbation theory for the Manakov soliton and its applications to pulse propagation in randomly birefringent fibers,, Phys. Rev. E, 56 (1997), 6147.   Google Scholar

[25]

S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves,, Zh. Eksp. Teor. Fiz., 65 (1973).   Google Scholar

[26]

S. P. Novikov, S. V. Manakov, L. P. Pitaevski and V. E. Zakharov, Theory of Solitons, the Inverse Scattering Method,, Consultant Bureau, (1984).   Google Scholar

[27]

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation,, Oxford University Press, (2003).   Google Scholar

[28]

T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases,, J. Phys. Soc. Japan, 67 (1998).   Google Scholar

[29]

M. Modugno, F. Dalfovo, C. Fort, P. Maddaloni and F. Minardi, Dynamics of two colliding Bose-Einstein condensates in an elongated magnetostatic trap,, Phys. Rev. A, 62 (2000).   Google Scholar

[30]

V. M. Perez-Garcia, H. Michinel, J. I. Cirac, M. Lewenstein and P. Zoller, Dynamics of Bose-Einstein condensates: Variational solutions of the Gross-Pitaevskii equations,, Phys. Rev. A, 56 (1997), 1424.   Google Scholar

[31]

M. Toda, Theory of Nonlinear Lattices,, Springer Verlag, (1989).   Google Scholar

[32]

M. D. Todorov and C. I. Christov, Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations,, Discrete and Continuous Dynamical Systems, (2007), 982.   Google Scholar

[33]

M. D. Todorov and C. I. Christov, Impact of the large cross-modulation parameter on the collision dynamics of quasi-particles governed by vector NLSE,, Mathematics and Computers in Simulation, 80 (2008), 46.   Google Scholar

[34]

M. D. Todorov and C. I. Christov, Collision dynamics of elliptically polarized solitons in coupled nonlinear Schrödinger equations,, Mathematics and Computers in Simulation, 82 (2012), 1221.   Google Scholar

[35]

M. Uchiyama, J. Ieda and M. Wadati, Multicomponent bright solitons in $F=2$ spinor Bose-Einstein condensates,, J. Phys. Soc. Japan, 76 (2007).   Google Scholar

[36]

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. English translation:, Soviet Physics-JETP, 34 (1972), 62.   Google Scholar

show all references

References:
[1]

D. Anderson, and M. Lisak, Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,, Phys. Rev. A, 27 (1983), 1393.   Google Scholar

[2]

C. I. Christov, S. Dost, and G. A. Maugin, Inelasticity of soliton collisions in systems of coupled NLS equations,, Physica Scripta, 50 (1994), 449.   Google Scholar

[3]

V. S. Gerdjikov, B. B. Baizakov, and M. Salerno, Modelling adiabatic $N$-soliton interactions and perturbations,, Theor. Math. Phys., 144 (2005), 1138.   Google Scholar

[4]

V. S. Gerdjikov, "On soliton interactions of vector nonlinear Schrödinger equations,'', in AMiTaNS'11, (2011), 57.   Google Scholar

[5]

V. S. Gerdjikov, Modeling soliton interactions of the perturbed vector nonlinear Schrödinger equation,, Bulgarian J. Phys., 38 (2011), 274.   Google Scholar

[6]

V. S. Gerdjikov, B. B. Baizakov, M. Salerno, and N. A. Kostov, Adiabatic $N$-soliton interactions of Bose-Einstein condensates in external potentials,, Phys. Rev. E., 73 (2006).   Google Scholar

[7]

V. S. Gerdjikov, E. V. Doktorov, and N. P. Matsuka, $N$-soliton train and generalized complex Toda chain for Manakov system,, Theor. Math. Phys., 151 (2007), 762.   Google Scholar

[8]

V. S. Gerdjikov, E. G. Evstatiev, D. J. Kaup, G. L. Diankov, and I. M. Uzunov, Stability and quasi-equidistant propagation of NLS soliton trains,, Phys. Lett. A, 241 (1998), 323.   Google Scholar

[9]

V. S. Gerdjikov, G. G. Grahovski, "Two soliton interactions of BD.I multicomponent NLS equations and their gauge equivalent,'', in AMiTaNS'10, (2010), 561.   Google Scholar

[10]

V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, and E. G. Evstatiev, Asymptotic behavior of $N$-soliton trains of the nonlinear Schrödinger equation,, Phys. Rev. Lett., 77 (1996), 3943.   Google Scholar

[11]

V. S. Gerdjikov, N. A. Kostov, E. V. Doktorov, and N. P. Matsuka, Generalized perturbed complex Toda chain for Manakov system and exact solutions of the Bose-Einstein mixtures,, Mathematics and Computers in Simulation, 80 (2009), 112.   Google Scholar

[12]

V. S. Gerdjikov and M. D. Todorov, $N$-soliton interactions for the Manakov system. Effects of external potentials,, in Localized Excitations in Nonlinear Complex Systems, 7 (2014), 147.   Google Scholar

[13]

V. S. Gerdjikov and M. D. Todorov, "On the effects of sech-like potentials on Manakov solitons,'', in AMiTaNS'13, (2013), 75.   Google Scholar

[14]

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, Nonlinear Schrödinger equation and $N$-soliton interactions: Generalized Karpman-Soloviev approach and the complex Toda chain,, Phys. Rev. E, 55 (1997), 6039.   Google Scholar

[15]

V. S. Gerdjikov, G. Vilasi, and A. B. Yanovski, Integrable Hamiltonian hierarchies. Spectral and geometric methods,, Lecture Notes in Physics 748, 748 (2008).   Google Scholar

[16]

V. S. Gerdjikov, M. D. Todorov, and A. V. Kyuldjiev, Asymptotic behavior of Manakov solitons: Effects of potential wells and humps,, preprint, ().   Google Scholar

[17]

A. Griffin, T. Nikuni, and E. Zaremba, Bose-Condensed Gases at Finite Temperatures,, Cambridge University Press, (2009).   Google Scholar

[18]

T.-L. Ho, Spinor Bose condensates in optical traps,, Phys. Rev. Lett., 81 (1998).   Google Scholar

[19]

V. I. Karpman and V. V. Solov'ev, A perturbational approach to the two-solition systems,, Physica D, 3 (1981), 487.   Google Scholar

[20]

, Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment (eds. P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonzalez), Springer, 45 (2008).   Google Scholar

[21]

N. A. Kostov, V. Z. Enol'skii, V. S. Gerdjikov, V. V. Konotop and M. Salerno, On two-component Bose-Einstein condensates in periodic potential,, Phys. Rev. E, 70 (2004).   Google Scholar

[22]

N. A. Kostov, V. S. Gerdjikov and T. I. Valchev, Exact solutions for equations of Bose-Fermi mixtures in one-dimensional optical lattice,, SIGMA 3 (2007), 3 (2007).   Google Scholar

[23]

A. V. Kyuldjiev, V. S. Gerdjikov, M. D. Todorov, Asymptotic Behavior of Manakov Solitons: Effects of of shallow and wide potential wells and humps,, in Mathematics in Industry (ed. A. Slavova), (2014), 410.   Google Scholar

[24]

T. I. Lakoba and D. J. Kaup, Perturbation theory for the Manakov soliton and its applications to pulse propagation in randomly birefringent fibers,, Phys. Rev. E, 56 (1997), 6147.   Google Scholar

[25]

S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves,, Zh. Eksp. Teor. Fiz., 65 (1973).   Google Scholar

[26]

S. P. Novikov, S. V. Manakov, L. P. Pitaevski and V. E. Zakharov, Theory of Solitons, the Inverse Scattering Method,, Consultant Bureau, (1984).   Google Scholar

[27]

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation,, Oxford University Press, (2003).   Google Scholar

[28]

T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases,, J. Phys. Soc. Japan, 67 (1998).   Google Scholar

[29]

M. Modugno, F. Dalfovo, C. Fort, P. Maddaloni and F. Minardi, Dynamics of two colliding Bose-Einstein condensates in an elongated magnetostatic trap,, Phys. Rev. A, 62 (2000).   Google Scholar

[30]

V. M. Perez-Garcia, H. Michinel, J. I. Cirac, M. Lewenstein and P. Zoller, Dynamics of Bose-Einstein condensates: Variational solutions of the Gross-Pitaevskii equations,, Phys. Rev. A, 56 (1997), 1424.   Google Scholar

[31]

M. Toda, Theory of Nonlinear Lattices,, Springer Verlag, (1989).   Google Scholar

[32]

M. D. Todorov and C. I. Christov, Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations,, Discrete and Continuous Dynamical Systems, (2007), 982.   Google Scholar

[33]

M. D. Todorov and C. I. Christov, Impact of the large cross-modulation parameter on the collision dynamics of quasi-particles governed by vector NLSE,, Mathematics and Computers in Simulation, 80 (2008), 46.   Google Scholar

[34]

M. D. Todorov and C. I. Christov, Collision dynamics of elliptically polarized solitons in coupled nonlinear Schrödinger equations,, Mathematics and Computers in Simulation, 82 (2012), 1221.   Google Scholar

[35]

M. Uchiyama, J. Ieda and M. Wadati, Multicomponent bright solitons in $F=2$ spinor Bose-Einstein condensates,, J. Phys. Soc. Japan, 76 (2007).   Google Scholar

[36]

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. English translation:, Soviet Physics-JETP, 34 (1972), 62.   Google Scholar

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