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), 1393-1398; D. Anderson, M. Lisak, and T. Reichel, Approximate analytical approaches to nonlinear pulse propagation in optical fibers: A comparison, Phys. Rev. A, 38 (1988), 1618-1620.

[2]

C. I. Christov, S. Dost, and G. A. Maugin, Inelasticity of soliton collisions in systems of coupled NLS equations, Physica Scripta, 50 (1994), 449-454.

[3]

V. S. Gerdjikov, B. B. Baizakov, and M. Salerno, Modelling adiabatic $N$-soliton interactions and perturbations, Theor. Math. Phys., 144(2) (2005), 1138-1146.

[4]

V. S. Gerdjikov, "On soliton interactions of vector nonlinear Schrödinger equations,'' in AMiTaNS'11, AIP CP1404 (eds. M. D. Todorov and C. I. Christov), AIP, Melville, NY (2011), 57-67.

[5]

V. S. Gerdjikov, Modeling soliton interactions of the perturbed vector nonlinear Schrödinger equation, Bulgarian J. Phys., 38 (2011), 274-283.

[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), 046606.

[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(3) (2007), 762-773.

[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), 323-328

[9]

V. S. Gerdjikov, G. G. Grahovski, "Two soliton interactions of BD.I multicomponent NLS equations and their gauge equivalent,'' in AMiTaNS'10, AIP CP1301 (eds. M. D. Todorov and C. I. Christov), AIP, Melville, NY (2010), 561-572.

[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), 3943-3946.

[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), 112-119.

[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, Nonlinear Systems and Complexity (ed. R. Carretero-Gonzalez et al.), Springer International Publishing Switzerland, 7 (2014), 147-169.

[13]

V. S. Gerdjikov and M. D. Todorov, "On the effects of sech-like potentials on Manakov solitons,'' in AMiTaNS'13, AIP CP1561 (ed. M. D. Todorov), AIP, Melville, NY (2013), 75-83.

[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(5) (1997), 6039-6060.

[15]

V. S. Gerdjikov, G. Vilasi, and A. B. Yanovski, Integrable Hamiltonian hierarchies. Spectral and geometric methods, Lecture Notes in Physics 748, Springer Verlag, Berlin-Heidelberg-New York, 2008.

[16]

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

[17]

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

[18]

T.-L. Ho, Spinor Bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), 742.

[19]

V. I. Karpman and V. V. Solov'ev, A perturbational approach to the two-solition systems, Physica D, 3 (1981), 487-502.

[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). 

[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), 056617.

[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), paper 071, .arXiv:nlin.SI/0703057.

[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), Cambridge Scholar Publishing (2014), 410-426.

[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), 6147-6165.

[25]

S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Zh. Eksp. Teor. Fiz., 65 (1973), 1392. English translation: Sov. Phys. JETP 38 (1974), 248.

[26]

S. P. Novikov, S. V. Manakov, L. P. Pitaevski and V. E. Zakharov, Theory of Solitons, the Inverse Scattering Method, Consultant Bureau, New York, 1984.

[27]

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, Oxford, UK, 2003.

[28]

T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases, J. Phys. Soc. Japan, 67 (1998), 1822.

[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), 063607.

[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), 1424-1432.

[31]

M. Toda, Theory of Nonlinear Lattices, Springer Verlag, Berlin, 1989.

[32]

M. D. Todorov and C. I. Christov, Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations, Discrete and Continuous Dynamical Systems, Supplement 2007, 982-992.

[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), 46-55.

[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), 1221-1232.

[35]

M. Uchiyama, J. Ieda and M. Wadati, Multicomponent bright solitons in $F=2$ spinor Bose-Einstein condensates, J. Phys. Soc. Japan, 76(7) (2007), 74005.

[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), 62-69.

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), 1393-1398; D. Anderson, M. Lisak, and T. Reichel, Approximate analytical approaches to nonlinear pulse propagation in optical fibers: A comparison, Phys. Rev. A, 38 (1988), 1618-1620.

[2]

C. I. Christov, S. Dost, and G. A. Maugin, Inelasticity of soliton collisions in systems of coupled NLS equations, Physica Scripta, 50 (1994), 449-454.

[3]

V. S. Gerdjikov, B. B. Baizakov, and M. Salerno, Modelling adiabatic $N$-soliton interactions and perturbations, Theor. Math. Phys., 144(2) (2005), 1138-1146.

[4]

V. S. Gerdjikov, "On soliton interactions of vector nonlinear Schrödinger equations,'' in AMiTaNS'11, AIP CP1404 (eds. M. D. Todorov and C. I. Christov), AIP, Melville, NY (2011), 57-67.

[5]

V. S. Gerdjikov, Modeling soliton interactions of the perturbed vector nonlinear Schrödinger equation, Bulgarian J. Phys., 38 (2011), 274-283.

[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), 046606.

[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(3) (2007), 762-773.

[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), 323-328

[9]

V. S. Gerdjikov, G. G. Grahovski, "Two soliton interactions of BD.I multicomponent NLS equations and their gauge equivalent,'' in AMiTaNS'10, AIP CP1301 (eds. M. D. Todorov and C. I. Christov), AIP, Melville, NY (2010), 561-572.

[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), 3943-3946.

[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), 112-119.

[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, Nonlinear Systems and Complexity (ed. R. Carretero-Gonzalez et al.), Springer International Publishing Switzerland, 7 (2014), 147-169.

[13]

V. S. Gerdjikov and M. D. Todorov, "On the effects of sech-like potentials on Manakov solitons,'' in AMiTaNS'13, AIP CP1561 (ed. M. D. Todorov), AIP, Melville, NY (2013), 75-83.

[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(5) (1997), 6039-6060.

[15]

V. S. Gerdjikov, G. Vilasi, and A. B. Yanovski, Integrable Hamiltonian hierarchies. Spectral and geometric methods, Lecture Notes in Physics 748, Springer Verlag, Berlin-Heidelberg-New York, 2008.

[16]

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

[17]

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

[18]

T.-L. Ho, Spinor Bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), 742.

[19]

V. I. Karpman and V. V. Solov'ev, A perturbational approach to the two-solition systems, Physica D, 3 (1981), 487-502.

[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). 

[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), 056617.

[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), paper 071, .arXiv:nlin.SI/0703057.

[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), Cambridge Scholar Publishing (2014), 410-426.

[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), 6147-6165.

[25]

S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Zh. Eksp. Teor. Fiz., 65 (1973), 1392. English translation: Sov. Phys. JETP 38 (1974), 248.

[26]

S. P. Novikov, S. V. Manakov, L. P. Pitaevski and V. E. Zakharov, Theory of Solitons, the Inverse Scattering Method, Consultant Bureau, New York, 1984.

[27]

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, Oxford, UK, 2003.

[28]

T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases, J. Phys. Soc. Japan, 67 (1998), 1822.

[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), 063607.

[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), 1424-1432.

[31]

M. Toda, Theory of Nonlinear Lattices, Springer Verlag, Berlin, 1989.

[32]

M. D. Todorov and C. I. Christov, Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations, Discrete and Continuous Dynamical Systems, Supplement 2007, 982-992.

[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), 46-55.

[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), 1221-1232.

[35]

M. Uchiyama, J. Ieda and M. Wadati, Multicomponent bright solitons in $F=2$ spinor Bose-Einstein condensates, J. Phys. Soc. Japan, 76(7) (2007), 74005.

[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), 62-69.

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