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

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

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

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.
Citation: Vladimir S. Gerdjikov. Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations. Discrete & 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, (2004).   Google Scholar

[2]

E. V. Doktorov, J. Wang and J. Yang, Perturbation theory for bright spinor Bose-Einstein condensate solitons,, Phys. Rev. A, 77 (2008).  doi: 10.1103/PhysRevA.77.043617.  Google Scholar

[3]

L. D. Faddeev and L. A. Takhtadjan, "Hamiltonian Approach in the Theory of Solitons,", Springer Verlag, (1987).   Google Scholar

[4]

A. P. Fordy and P. P. Kulish, Nonlinear Schrodinger equations and simple Lie algebras,, Commun. Math. Phys., 89 (1983), 427.  doi: 10.1007/BF01214664.  Google Scholar

[5]

V. S. Gerdjikov, Generalised Fourier transforms for the soliton equations. Gauge covariant formulation,, Inverse Problems, 2 (1986), 51.  doi: 10.1088/0266-5611/2/1/005.  Google Scholar

[6]

V. S. Gerdjikov, The Zakharov-Shabat dressing method and the representation theory of the semisimple Lie algebras,, Phys. Lett. A, 126 (1987), 184.  doi: 10.1016/0375-9601(87)90457-9.  Google Scholar

[7]

V. S. Gerdjikov, The generalized Zakharov-Shabat system and the soliton perturbations,, Theor. Math. Phys., 99 (1994), 292.  doi: 10.1007/BF01016144.  Google Scholar

[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, (1186), 15.   Google Scholar

[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.   Google Scholar

[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).  doi: 10.1088/1751-8113/41/31/315213.  Google Scholar

[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, 7501 (2009).   Google Scholar

[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.  doi: 10.1016/j.physd.2008.06.007.  Google Scholar

[13]

S. Helgasson, "Differential Geometry, Lie groups and Symmetric Spaces,", Academic Press, (1978).   Google Scholar

[14]

J. Ieda, T. Miyakawa and M. Wadati, Exact analysis of soliton dynamics in spinor Bose-Einstein condensates,, Phys. Rev Lett., 93 (2004).  doi: 10.1103/PhysRevLett.93.194102.  Google Scholar

[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).  doi: 10.1143/JPSJ.73.2996.  Google Scholar

[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).  doi: 10.1103/PhysRevE.67.046617.  Google Scholar

[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).  doi: 10.1117/12.727191.  Google Scholar

[18]

N. A. Kostov and V. S. Gerdjikov, Reductions of multicomponent mKdV equations on symmetric spaces of DIII-type,, SIGMA, 4 (2008).   Google Scholar

[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).  doi: 10.1103/PhysRevA.72.033611.  Google Scholar

[20]

S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves,, Zh. Eksp. Teor. Fiz, 65 (1973), 505.   Google Scholar

[21]

A. V. Mikhailov, The reduction problem and the inverse scattering method,, Physica D: Nonlinear Phenomena, 3 (1981), 73.  doi: 10.1016/0167-2789(81)90120-2.  Google Scholar

[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).  doi: 10.1103/PhysRevA.77.033612.  Google Scholar

[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.  doi: 10.1103/PhysRevE.59.2373.  Google Scholar

[24]

M. Salerno, Matter-wave quantum dots and antidots in ultracold atomic Bose-Fermi mixtures,, Phys. Rev. A, 72 (2005).   Google Scholar

[25]

T. Tsuchida, N-soliton collision in the Manakov model,, Prog. Theor. Phys., 111 (2004), 151.  doi: 10.1143/PTP.111.151.  Google Scholar

[26]

T. Tsuchida and M. Wadati, The coupled modified Korteweg-de Vries equations,, J. Phys. Soc. Jpn., 67 (1998).  doi: 10.1143/JPSJ.67.1175.  Google Scholar

[27]

M. Uchiyama, J. Ieda and M. Wadati, Dark solitons in $F=1$ spinor Bose-Einstein condensate,, J. Phys. Soc. Jpn., 75 (2006).  doi: 10.1143/JPSJ.75.064002.  Google Scholar

[28]

M. Uchiyama, J. Ieda and M. Wadati, Multicomponent bright solitons in $F=2$ spinor Bose-Einstein condensates,, J. Phys. Soc. Japan, 76 (2007).  doi: 10.1143/JPSJ.76.074005.  Google Scholar

[29]

V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. I. Pitaevskii, "Theory of Solitons. The Inverse Scattering Method,", Plenum, (1984).   Google Scholar

[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.  doi: 10.1007/BF01197576.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/BF01075696.  Google Scholar

[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.   Google Scholar

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, (2004).   Google Scholar

[2]

E. V. Doktorov, J. Wang and J. Yang, Perturbation theory for bright spinor Bose-Einstein condensate solitons,, Phys. Rev. A, 77 (2008).  doi: 10.1103/PhysRevA.77.043617.  Google Scholar

[3]

L. D. Faddeev and L. A. Takhtadjan, "Hamiltonian Approach in the Theory of Solitons,", Springer Verlag, (1987).   Google Scholar

[4]

A. P. Fordy and P. P. Kulish, Nonlinear Schrodinger equations and simple Lie algebras,, Commun. Math. Phys., 89 (1983), 427.  doi: 10.1007/BF01214664.  Google Scholar

[5]

V. S. Gerdjikov, Generalised Fourier transforms for the soliton equations. Gauge covariant formulation,, Inverse Problems, 2 (1986), 51.  doi: 10.1088/0266-5611/2/1/005.  Google Scholar

[6]

V. S. Gerdjikov, The Zakharov-Shabat dressing method and the representation theory of the semisimple Lie algebras,, Phys. Lett. A, 126 (1987), 184.  doi: 10.1016/0375-9601(87)90457-9.  Google Scholar

[7]

V. S. Gerdjikov, The generalized Zakharov-Shabat system and the soliton perturbations,, Theor. Math. Phys., 99 (1994), 292.  doi: 10.1007/BF01016144.  Google Scholar

[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, (1186), 15.   Google Scholar

[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.   Google Scholar

[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).  doi: 10.1088/1751-8113/41/31/315213.  Google Scholar

[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, 7501 (2009).   Google Scholar

[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.  doi: 10.1016/j.physd.2008.06.007.  Google Scholar

[13]

S. Helgasson, "Differential Geometry, Lie groups and Symmetric Spaces,", Academic Press, (1978).   Google Scholar

[14]

J. Ieda, T. Miyakawa and M. Wadati, Exact analysis of soliton dynamics in spinor Bose-Einstein condensates,, Phys. Rev Lett., 93 (2004).  doi: 10.1103/PhysRevLett.93.194102.  Google Scholar

[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).  doi: 10.1143/JPSJ.73.2996.  Google Scholar

[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).  doi: 10.1103/PhysRevE.67.046617.  Google Scholar

[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).  doi: 10.1117/12.727191.  Google Scholar

[18]

N. A. Kostov and V. S. Gerdjikov, Reductions of multicomponent mKdV equations on symmetric spaces of DIII-type,, SIGMA, 4 (2008).   Google Scholar

[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).  doi: 10.1103/PhysRevA.72.033611.  Google Scholar

[20]

S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves,, Zh. Eksp. Teor. Fiz, 65 (1973), 505.   Google Scholar

[21]

A. V. Mikhailov, The reduction problem and the inverse scattering method,, Physica D: Nonlinear Phenomena, 3 (1981), 73.  doi: 10.1016/0167-2789(81)90120-2.  Google Scholar

[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).  doi: 10.1103/PhysRevA.77.033612.  Google Scholar

[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.  doi: 10.1103/PhysRevE.59.2373.  Google Scholar

[24]

M. Salerno, Matter-wave quantum dots and antidots in ultracold atomic Bose-Fermi mixtures,, Phys. Rev. A, 72 (2005).   Google Scholar

[25]

T. Tsuchida, N-soliton collision in the Manakov model,, Prog. Theor. Phys., 111 (2004), 151.  doi: 10.1143/PTP.111.151.  Google Scholar

[26]

T. Tsuchida and M. Wadati, The coupled modified Korteweg-de Vries equations,, J. Phys. Soc. Jpn., 67 (1998).  doi: 10.1143/JPSJ.67.1175.  Google Scholar

[27]

M. Uchiyama, J. Ieda and M. Wadati, Dark solitons in $F=1$ spinor Bose-Einstein condensate,, J. Phys. Soc. Jpn., 75 (2006).  doi: 10.1143/JPSJ.75.064002.  Google Scholar

[28]

M. Uchiyama, J. Ieda and M. Wadati, Multicomponent bright solitons in $F=2$ spinor Bose-Einstein condensates,, J. Phys. Soc. Japan, 76 (2007).  doi: 10.1143/JPSJ.76.074005.  Google Scholar

[29]

V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. I. Pitaevskii, "Theory of Solitons. The Inverse Scattering Method,", Plenum, (1984).   Google Scholar

[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.  doi: 10.1007/BF01197576.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/BF01075696.  Google Scholar

[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.   Google Scholar

[1]

Kui Li, Zhitao Zhang. A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 851-860. doi: 10.3934/dcds.2016.36.851

[2]

Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306

[3]

Vadym Vekslerchik, Víctor M. Pérez-García. Exact solution of the two-mode model of multicomponent Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 179-192. doi: 10.3934/dcdsb.2003.3.179

[4]

Liren Lin, Tai-Peng Tsai. Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 295-336. doi: 10.3934/dcds.2017013

[5]

Florian Méhats, Christof Sparber. Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5097-5118. doi: 10.3934/dcds.2016021

[6]

P.G. Kevrekidis, Dimitri J. Frantzeskakis. Multiple dark solitons in Bose-Einstein condensates at finite temperatures. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1199-1212. doi: 10.3934/dcdss.2011.4.1199

[7]

Weizhu Bao, Loïc Le Treust, Florian Méhats. Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime. Kinetic & Related Models, 2017, 10 (3) : 553-571. doi: 10.3934/krm.2017022

[8]

Pedro J. Torres, R. Carretero-González, S. Middelkamp, P. Schmelcher, Dimitri J. Frantzeskakis, P.G. Kevrekidis. Vortex interaction dynamics in trapped Bose-Einstein condensates. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1589-1615. doi: 10.3934/cpaa.2011.10.1589

[9]

Yi He, Gongbao Li. Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 731-762. doi: 10.3934/dcds.2016.36.731

[10]

Abbas Moameni. Soliton solutions for quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$. Communications on Pure & Applied Analysis, 2008, 7 (1) : 89-105. doi: 10.3934/cpaa.2008.7.89

[11]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[12]

W. Josh Sonnier, C. I. Christov. Repelling soliton collisions in coupled Schrödinger equations with negative cross modulation. Conference Publications, 2009, 2009 (Special) : 708-718. doi: 10.3934/proc.2009.2009.708

[13]

Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119

[14]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903

[15]

Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138

[16]

Anne de Bouard, Reika Fukuizumi, Romain Poncet. Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2793-2817. doi: 10.3934/dcdsb.2015.20.2793

[17]

Xuguang Lu. Long time strong convergence to Bose-Einstein distribution for low temperature. Kinetic & Related Models, 2018, 11 (4) : 715-734. doi: 10.3934/krm.2018029

[18]

Weizhu Bao, Yongyong Cai. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic & Related Models, 2013, 6 (1) : 1-135. doi: 10.3934/krm.2013.6.1

[19]

Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651

[20]

Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor. Soliton solutions for the elastic metric on spaces of curves. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1161-1185. doi: 10.3934/dcds.2018049

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]