Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations

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October 2011, 4(5): 1181-1197. doi: 10.3934/dcdss.2011.4.1181

Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations

Vladimir S. Gerdjikov ^1,

1.	Institute for Nuclear Research and Nuclear Energy, Bulgarian academy of sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria

Received October 2009 Revised December 2009 Published December 2010

Abstract
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We analyze the properties of the soliton solutions of a class of models describing one-dimensional BEC with spin $F$. We describe the minimal sets of scattering data which determine uniquely both the corresponding potential of the Lax operator and its scattering matrix. Next we give several reductions of these MNLS, derive their $N$-soliton solutions and analyze the soliton interactions. Finally we prove an important theorem proving that if the initial conditions satisfy the reduction then one gets a solution of the reduced MNLS.

Keywords: Soliton solutions, Soliton interactions, Reductions of MNLS., Multicomponent nonlinear Schrödinger equations, Bose-Einstein condensates.

Mathematics Subject Classification: Primary: 35Q51, 37K40; Secondary: 34K1.

Citation: Vladimir S. Gerdjikov. Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1181-1197. doi: 10.3934/dcdss.2011.4.1181

References:

[1]	M. J. Ablowitz, B. Prinari and A. D. Trubatch, "Discrete and Continuous Nonlinear Schrödinger Systems," London Mathematical Society Lecture Note Series, 302. Cambridge University Press, Cambridge, 2004.
[2]	E. V. Doktorov, J. Wang and J. Yang, Perturbation theory for bright spinor Bose-Einstein condensate solitons, Phys. Rev. A, 77 (2008), 043617. doi: 10.1103/PhysRevA.77.043617.
[3]	L. D. Faddeev and L. A. Takhtadjan, "Hamiltonian Approach in the Theory of Solitons," Springer Verlag, Berlin, 1987.
[4]	A. P. Fordy and P. P. Kulish, Nonlinear Schrodinger equations and simple Lie algebras, Commun. Math. Phys., 89 (1983), 427-443. doi: 10.1007/BF01214664.
[5]	V. S. Gerdjikov, Generalised Fourier transforms for the soliton equations. Gauge covariant formulation, Inverse Problems, 2 (1986), 51-74. doi: 10.1088/0266-5611/2/1/005.
[6]	V. S. Gerdjikov, The Zakharov-Shabat dressing method and the representation theory of the semisimple Lie algebras, Phys. Lett. A, 126 (1987), 184-188. doi: 10.1016/0375-9601(87)90457-9.
[7]	V. S. Gerdjikov, The generalized Zakharov-Shabat system and the soliton perturbations, Theor. Math. Phys., 99 (1994), 292-299. doi: 10.1007/BF01016144.
[8]	V. S. Gerdjikov, On reductions of soliton solutions of multi-component NLS models and spinor Bose-Einstein condensates, Application of mathematics in technical and natural sciences, 15-27, AIP Conf. Proc., 1186, Amer. Inst. Phys., Melville, NY, 2009.
[9]	V. S. Gerdjikov, G. G. Grahovski and N. A. Kostov, Multicomponent equations of the nonlinear Schrödinger type on symmetric spaces and their reductions, (Russian) Teoret. Mat. Fiz., 144 (2005), 313-323; translation in Theoret. and Math. Phys., 144 (2005), 1147-1156
[10]	V. S. Gerdjikov, D. J. Kaup, N. A. Kostov and T. I. Valchev, On classification of soliton solutions of multicomponent nonlinear evolution equations, J. Phys. A: Math. Theor., 41 (2008), 315213. doi: 10.1088/1751-8113/41/31/315213.
[11]	V. S. Gerdjikov, N. A. Kostov and T. I. Valchev, Bose-Einstein condensates with $F=1$ and $F=2$. Reductions and soliton interactions of multi-component NLS models, in "International Conference on Ultrafast and Nonlinear Optics 2009" (eds: S. M. Saltiel, A. A. Dreischuh and I. P. Christov), Proceedings of SPIE, Volume 7501, (2009).
[12]	V. S. Gerdjikov, N. A. Kostov and T. I. Valchev, Solutions of multi-component NLS models and spinor Bose-Einstein condensates, Physica D, 238 (2009), 1306-1310. doi: 10.1016/j.physd.2008.06.007.
[13]	S. Helgasson, "Differential Geometry, Lie groups and Symmetric Spaces," Academic Press, 1978.
[14]	J. Ieda, T. Miyakawa and M. Wadati, Exact analysis of soliton dynamics in spinor Bose-Einstein condensates, Phys. Rev Lett., 93 (2004), 194102. doi: 10.1103/PhysRevLett.93.194102.
[15]	J. Ieda, T. Miyakawa and M. Wadati, Matter-wave solitons in an $F=1$ spinor Bose-Einstein condensate, J. Phys. Soc. Jpn., 73 (2004), 2996. doi: 10.1143/JPSJ.73.2996.
[16]	T. Kanna and M. Lakshmanan, Exact soliton solutions of coupled nonlinear Schrödinger equations: Shape-changing collisions, logic gates, and partially coherent solitons, Phys. Rev. E, 67 (2003), 046617. doi: 10.1103/PhysRevE.67.046617.
[17]	N. A. Kostov, V. A. Atanasov, V. S. Gerdjikov and G. G. Grahovski, On the soliton solutions of the spinor Bose-Einstein condensate, Proceedings of SPIE, 6604 (2007), 66041T. doi: 10.1117/12.727191.
[18]	N. A. Kostov and V. S. Gerdjikov, Reductions of multicomponent mKdV equations on symmetric spaces of DIII-type, SIGMA, 4 (2008), 30 pages, arXiv:0803.1651.
[19]	L. Li, Z. Li, B. A. Malomed, D. Mihalache and W. M. Liu, Exact soliton solutions and nonlinear modulation instability in spinor Bose-Einstein condensates, Phys. Rev. A, 72 (2005), 033611. doi: 10.1103/PhysRevA.72.033611.
[20]	S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Zh. Eksp. Teor. Fiz, 65 (1973), 505-516; Sov. Phys. JETP, 38 (1974), 248-253.
[21]	A. V. Mikhailov, The reduction problem and the inverse scattering method, Physica D: Nonlinear Phenomena, 3 (1981), 73-117. doi: 10.1016/0167-2789(81)90120-2.
[22]	H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed and R. Carretero-Gonzalez, Bright-dark soliton complexes in spinor Bose-Einstein condensates, Phys. Rev. A, 77 (2008), 033612. doi: 10.1103/PhysRevA.77.033612.
[23]	Q.-H. Park and H. J. Shin, Painlevé analysis of the coupled nonlinear Schrödinger equation for polarized optical waves in an isotropic medium, Phys. Rev. E, 59 (1999), 2373-2379. doi: 10.1103/PhysRevE.59.2373.
[24]	M. Salerno, Matter-wave quantum dots and antidots in ultracold atomic Bose-Fermi mixtures, Phys. Rev. A, 72 (2005), 063602 (7 pages), arXiv:cond-mat/0503097.
[25]	T. Tsuchida, N-soliton collision in the Manakov model, Prog. Theor. Phys., 111 (2004), 151-182. doi: 10.1143/PTP.111.151.
[26]	T. Tsuchida and M. Wadati, The coupled modified Korteweg-de Vries equations, J. Phys. Soc. Jpn., 67 (1998), 1175. doi: 10.1143/JPSJ.67.1175.
[27]	M. Uchiyama, J. Ieda and M. Wadati, Dark solitons in $F=1$ spinor Bose-Einstein condensate, J. Phys. Soc. Jpn., 75 (2006), 064002. doi: 10.1143/JPSJ.75.064002.
[28]	M. Uchiyama, J. Ieda and M. Wadati, Multicomponent bright solitons in $F=2$ spinor Bose-Einstein condensates, J. Phys. Soc. Japan, 76 (2007), 74005. doi: 10.1143/JPSJ.76.074005.
[29]	V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. I. Pitaevskii, "Theory of Solitons. The Inverse Scattering Method," Plenum, N. Y., 1984.
[30]	V. E. Zakharov and A. V. Mikhailov, On the integrability of classical spinor models in two-dimensional space-time, Comm. Math. Phys., 74 (1980), 21-40. doi: 10.1007/BF01197576.
[31]	V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics-JETP, 34 (1972), 62-69.
[32]	V. E. Zakharov and A. B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I., Functional Analysis and Its Applications, 8 (1974), 226-235. doi: 10.1007/BF01075696.
[33]	V. E. Zakharov and A. B. Shabat, Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II., Functional Analysis and Its Applications, 13 (1974), 166-174.

show all references

References:

[1]	M. J. Ablowitz, B. Prinari and A. D. Trubatch, "Discrete and Continuous Nonlinear Schrödinger Systems," London Mathematical Society Lecture Note Series, 302. Cambridge University Press, Cambridge, 2004.
[2]	E. V. Doktorov, J. Wang and J. Yang, Perturbation theory for bright spinor Bose-Einstein condensate solitons, Phys. Rev. A, 77 (2008), 043617. doi: 10.1103/PhysRevA.77.043617.
[3]	L. D. Faddeev and L. A. Takhtadjan, "Hamiltonian Approach in the Theory of Solitons," Springer Verlag, Berlin, 1987.
[4]	A. P. Fordy and P. P. Kulish, Nonlinear Schrodinger equations and simple Lie algebras, Commun. Math. Phys., 89 (1983), 427-443. doi: 10.1007/BF01214664.
[5]	V. S. Gerdjikov, Generalised Fourier transforms for the soliton equations. Gauge covariant formulation, Inverse Problems, 2 (1986), 51-74. doi: 10.1088/0266-5611/2/1/005.
[6]	V. S. Gerdjikov, The Zakharov-Shabat dressing method and the representation theory of the semisimple Lie algebras, Phys. Lett. A, 126 (1987), 184-188. doi: 10.1016/0375-9601(87)90457-9.
[7]	V. S. Gerdjikov, The generalized Zakharov-Shabat system and the soliton perturbations, Theor. Math. Phys., 99 (1994), 292-299. doi: 10.1007/BF01016144.
[8]	V. S. Gerdjikov, On reductions of soliton solutions of multi-component NLS models and spinor Bose-Einstein condensates, Application of mathematics in technical and natural sciences, 15-27, AIP Conf. Proc., 1186, Amer. Inst. Phys., Melville, NY, 2009.
[9]	V. S. Gerdjikov, G. G. Grahovski and N. A. Kostov, Multicomponent equations of the nonlinear Schrödinger type on symmetric spaces and their reductions, (Russian) Teoret. Mat. Fiz., 144 (2005), 313-323; translation in Theoret. and Math. Phys., 144 (2005), 1147-1156
[10]	V. S. Gerdjikov, D. J. Kaup, N. A. Kostov and T. I. Valchev, On classification of soliton solutions of multicomponent nonlinear evolution equations, J. Phys. A: Math. Theor., 41 (2008), 315213. doi: 10.1088/1751-8113/41/31/315213.
[11]	V. S. Gerdjikov, N. A. Kostov and T. I. Valchev, Bose-Einstein condensates with $F=1$ and $F=2$. Reductions and soliton interactions of multi-component NLS models, in "International Conference on Ultrafast and Nonlinear Optics 2009" (eds: S. M. Saltiel, A. A. Dreischuh and I. P. Christov), Proceedings of SPIE, Volume 7501, (2009).
[12]	V. S. Gerdjikov, N. A. Kostov and T. I. Valchev, Solutions of multi-component NLS models and spinor Bose-Einstein condensates, Physica D, 238 (2009), 1306-1310. doi: 10.1016/j.physd.2008.06.007.
[13]	S. Helgasson, "Differential Geometry, Lie groups and Symmetric Spaces," Academic Press, 1978.
[14]	J. Ieda, T. Miyakawa and M. Wadati, Exact analysis of soliton dynamics in spinor Bose-Einstein condensates, Phys. Rev Lett., 93 (2004), 194102. doi: 10.1103/PhysRevLett.93.194102.
[15]	J. Ieda, T. Miyakawa and M. Wadati, Matter-wave solitons in an $F=1$ spinor Bose-Einstein condensate, J. Phys. Soc. Jpn., 73 (2004), 2996. doi: 10.1143/JPSJ.73.2996.
[16]	T. Kanna and M. Lakshmanan, Exact soliton solutions of coupled nonlinear Schrödinger equations: Shape-changing collisions, logic gates, and partially coherent solitons, Phys. Rev. E, 67 (2003), 046617. doi: 10.1103/PhysRevE.67.046617.
[17]	N. A. Kostov, V. A. Atanasov, V. S. Gerdjikov and G. G. Grahovski, On the soliton solutions of the spinor Bose-Einstein condensate, Proceedings of SPIE, 6604 (2007), 66041T. doi: 10.1117/12.727191.
[18]	N. A. Kostov and V. S. Gerdjikov, Reductions of multicomponent mKdV equations on symmetric spaces of DIII-type, SIGMA, 4 (2008), 30 pages, arXiv:0803.1651.
[19]	L. Li, Z. Li, B. A. Malomed, D. Mihalache and W. M. Liu, Exact soliton solutions and nonlinear modulation instability in spinor Bose-Einstein condensates, Phys. Rev. A, 72 (2005), 033611. doi: 10.1103/PhysRevA.72.033611.
[20]	S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Zh. Eksp. Teor. Fiz, 65 (1973), 505-516; Sov. Phys. JETP, 38 (1974), 248-253.
[21]	A. V. Mikhailov, The reduction problem and the inverse scattering method, Physica D: Nonlinear Phenomena, 3 (1981), 73-117. doi: 10.1016/0167-2789(81)90120-2.
[22]	H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed and R. Carretero-Gonzalez, Bright-dark soliton complexes in spinor Bose-Einstein condensates, Phys. Rev. A, 77 (2008), 033612. doi: 10.1103/PhysRevA.77.033612.
[23]	Q.-H. Park and H. J. Shin, Painlevé analysis of the coupled nonlinear Schrödinger equation for polarized optical waves in an isotropic medium, Phys. Rev. E, 59 (1999), 2373-2379. doi: 10.1103/PhysRevE.59.2373.
[24]	M. Salerno, Matter-wave quantum dots and antidots in ultracold atomic Bose-Fermi mixtures, Phys. Rev. A, 72 (2005), 063602 (7 pages), arXiv:cond-mat/0503097.
[25]	T. Tsuchida, N-soliton collision in the Manakov model, Prog. Theor. Phys., 111 (2004), 151-182. doi: 10.1143/PTP.111.151.
[26]	T. Tsuchida and M. Wadati, The coupled modified Korteweg-de Vries equations, J. Phys. Soc. Jpn., 67 (1998), 1175. doi: 10.1143/JPSJ.67.1175.
[27]	M. Uchiyama, J. Ieda and M. Wadati, Dark solitons in $F=1$ spinor Bose-Einstein condensate, J. Phys. Soc. Jpn., 75 (2006), 064002. doi: 10.1143/JPSJ.75.064002.
[28]	M. Uchiyama, J. Ieda and M. Wadati, Multicomponent bright solitons in $F=2$ spinor Bose-Einstein condensates, J. Phys. Soc. Japan, 76 (2007), 74005. doi: 10.1143/JPSJ.76.074005.
[29]	V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. I. Pitaevskii, "Theory of Solitons. The Inverse Scattering Method," Plenum, N. Y., 1984.
[30]	V. E. Zakharov and A. V. Mikhailov, On the integrability of classical spinor models in two-dimensional space-time, Comm. Math. Phys., 74 (1980), 21-40. doi: 10.1007/BF01197576.
[31]	V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics-JETP, 34 (1972), 62-69.
[32]	V. E. Zakharov and A. B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I., Functional Analysis and Its Applications, 8 (1974), 226-235. doi: 10.1007/BF01075696.
[33]	V. E. Zakharov and A. B. Shabat, Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II., Functional Analysis and Its Applications, 13 (1974), 166-174.

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