American Institute of Mathematical Sciences

Advanced
March  2012, 32(3): 703-715. doi: 10.3934/dcds.2012.32.703

On the Cauchy problem for nonlinear Schrödinger equations with rotation

Paolo Antonelli 1, , Daniel Marahrens 1, and  Christof Sparber 2,

1. 

Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge CB3 0WA, United Kingdom, United Kingdom
2. 

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 South Morgan Street, Chicago, Illinois 60607, United States

Received  September 2010 Revised  March 2011 Published  October 2011

We consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in [11, 12]. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case.
Keywords: Nonlinear Schrödinger equation, finite time blow-up., rotation, Bose-Einstein condensation, angular momentum operator.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35A0.
Citation: Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703
References:
[1]

A. Aftalion, "Vortices in Bose-Einstein Condensates," Progress in Nonlinear Differential Equations and their Applications, 67, Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

[2]

P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys., 287 (2009), 657-686.  Google Scholar

[3]

A. V. Babin, A. A. Ilyin and E. S. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Applied Math., 64 (2011), 591-648. doi: 10.1002/cpa.20356.  Google Scholar

[4]

W. Bao, Q. Du and Y. Z. Zhang, Dynamics of rotating Bose-Einstein condensates and its efficient and accurate numerical computation, SIAM J. Appl. Math., 66 (2006), 758-786. doi: 10.1137/050629392.  Google Scholar

[5]

R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843. doi: 10.1137/S0036141002416936.  Google Scholar

[6]

R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials, Discrete Contin. Dyn. Syst., 13 (2005), 385-398. doi: 10.3934/dcds.2005.13.385.  Google Scholar

[7]

R. Carles, Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., 9 (2011), no. 4, 937-964. Google Scholar

[8]

T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003.  Google Scholar

[9]

D. Choi and Q. Niu, Bose-Einstein condensates in an optical lattice, Phys. Rev. Lett., 82 (1999), 2022-2025. doi: 10.1103/PhysRevLett.82.2022.  Google Scholar

[10]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797. doi: 10.1063/1.523491.  Google Scholar

[11]

C. Hao, L. Hsiao and H.-L. Li, Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term, Math. Methods Appl. Sci., 31 (2008), 655-664. doi: 10.1002/mma.931.  Google Scholar

[12]

C. Hao, L. Hsiao and H.-L. Li, Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions, J. Math. Phys., 48 (2007), 102105, 11 pp.  Google Scholar

[13]

N. A. Jamaludin, N. G. Parker and A. M. Martin, Bright solitary waves of atomic Bose-Einstein condensates under rotation, Phys. Rev. A, 77 (2008), 051603. doi: 10.1103/PhysRevA.77.051603.  Google Scholar

[14]

O. Kavian and F. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173.  Google Scholar

[15]

H. Kitada, On a construction of the fundamental solution for Schrödinger equations, J. Fac. Sci. Univ. Tokyo Sec. IA Math., 27 (1980), 193-226.  Google Scholar

[16]

E. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Comm. Math. Phys., 264 (2006), 505-537. doi: 10.1007/s00220-006-1524-9.  Google Scholar

[17]

H. Liu, Critical thresholds in the semiclassical limit of 2-D rotational Schrödinger equations, Z. Angew. Math. Phys., 57 (2006), 42-58. doi: 10.1007/s00033-005-0004-y.  Google Scholar

[18]

H. Liu and C. Sparber, Rigorous derivation of the hydrodynamical equations for rotating superfluids, Math. Models Methods Appl. Sci., 18 (2008), 689-706. doi: 10.1142/S0218202508002826.  Google Scholar

[19]

H. Liu and E. Tadmor, Rotation prevents finite-time breakdown, Phys. D, 188 (2004), 262-276. doi: 10.1016/j.physd.2003.07.006.  Google Scholar

[20]

K. W. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 84 (2000), 806-809. doi: 10.1103/PhysRevLett.84.806.  Google Scholar

[21]

K. W. Madison, F. Chevy, V. Bretin and J. Dalibard, Stationary states of a rotating Bose-Einstein condensate: Routes to vortex nucleation, Phys. Rev. Lett., 86 (2001), 4443-4446. doi: 10.1103/PhysRevLett.86.4443.  Google Scholar

[22]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman and E. A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett., 83 (1999), 2498-2501. doi: 10.1103/PhysRevLett.83.2498.  Google Scholar

[23]

P. Raphaël, On the blow up phenomenon for the $L^2$ critical non linear Schrödinger equation, in "Lectures on Nonlinear Dispersive Equations," GAKUTO Internat. Ser. Math. Sci. Appl., 27, Gakkōtosho, Tokyo, (2006), 9-61.  Google Scholar

[24]

R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Comm. Math. Phys., 229 (2002), 491-509. doi: 10.1007/s00220-002-0695-2.  Google Scholar

[25]

S. Stock, B. Battelier, V. Bretin, Z. Hadzibabic and J. Dalibard, Bose-Einstein condensates in fast rotation, Laser Phy. Lett., 2 (2005), 275-284. doi: 10.1002/lapl.200410177.  Google Scholar

[26]

M. C. Tsatsos, Attractive Bose-Einstein Condensates in three dimensions under rotation: Revisiting the problem of stability of the ground state in harmonic traps, Phys. Rev. A, 83 (2011), 063615. doi: 10.1103/PhysRevA.83.063615.  Google Scholar

show all references

References:
[1]

A. Aftalion, "Vortices in Bose-Einstein Condensates," Progress in Nonlinear Differential Equations and their Applications, 67, Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

[2]

P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys., 287 (2009), 657-686.  Google Scholar

[3]

A. V. Babin, A. A. Ilyin and E. S. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Applied Math., 64 (2011), 591-648. doi: 10.1002/cpa.20356.  Google Scholar

[4]

W. Bao, Q. Du and Y. Z. Zhang, Dynamics of rotating Bose-Einstein condensates and its efficient and accurate numerical computation, SIAM J. Appl. Math., 66 (2006), 758-786. doi: 10.1137/050629392.  Google Scholar

[5]

R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843. doi: 10.1137/S0036141002416936.  Google Scholar

[6]

R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials, Discrete Contin. Dyn. Syst., 13 (2005), 385-398. doi: 10.3934/dcds.2005.13.385.  Google Scholar

[7]

R. Carles, Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., 9 (2011), no. 4, 937-964. Google Scholar

[8]

T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003.  Google Scholar

[9]

D. Choi and Q. Niu, Bose-Einstein condensates in an optical lattice, Phys. Rev. Lett., 82 (1999), 2022-2025. doi: 10.1103/PhysRevLett.82.2022.  Google Scholar

[10]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797. doi: 10.1063/1.523491.  Google Scholar

[11]

C. Hao, L. Hsiao and H.-L. Li, Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term, Math. Methods Appl. Sci., 31 (2008), 655-664. doi: 10.1002/mma.931.  Google Scholar

[12]

C. Hao, L. Hsiao and H.-L. Li, Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions, J. Math. Phys., 48 (2007), 102105, 11 pp.  Google Scholar

[13]

N. A. Jamaludin, N. G. Parker and A. M. Martin, Bright solitary waves of atomic Bose-Einstein condensates under rotation, Phys. Rev. A, 77 (2008), 051603. doi: 10.1103/PhysRevA.77.051603.  Google Scholar

[14]

O. Kavian and F. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173.  Google Scholar

[15]

H. Kitada, On a construction of the fundamental solution for Schrödinger equations, J. Fac. Sci. Univ. Tokyo Sec. IA Math., 27 (1980), 193-226.  Google Scholar

[16]

E. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Comm. Math. Phys., 264 (2006), 505-537. doi: 10.1007/s00220-006-1524-9.  Google Scholar

[17]

H. Liu, Critical thresholds in the semiclassical limit of 2-D rotational Schrödinger equations, Z. Angew. Math. Phys., 57 (2006), 42-58. doi: 10.1007/s00033-005-0004-y.  Google Scholar

[18]

H. Liu and C. Sparber, Rigorous derivation of the hydrodynamical equations for rotating superfluids, Math. Models Methods Appl. Sci., 18 (2008), 689-706. doi: 10.1142/S0218202508002826.  Google Scholar

[19]

H. Liu and E. Tadmor, Rotation prevents finite-time breakdown, Phys. D, 188 (2004), 262-276. doi: 10.1016/j.physd.2003.07.006.  Google Scholar

[20]

K. W. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 84 (2000), 806-809. doi: 10.1103/PhysRevLett.84.806.  Google Scholar

[21]

K. W. Madison, F. Chevy, V. Bretin and J. Dalibard, Stationary states of a rotating Bose-Einstein condensate: Routes to vortex nucleation, Phys. Rev. Lett., 86 (2001), 4443-4446. doi: 10.1103/PhysRevLett.86.4443.  Google Scholar

[22]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman and E. A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett., 83 (1999), 2498-2501. doi: 10.1103/PhysRevLett.83.2498.  Google Scholar

[23]

P. Raphaël, On the blow up phenomenon for the $L^2$ critical non linear Schrödinger equation, in "Lectures on Nonlinear Dispersive Equations," GAKUTO Internat. Ser. Math. Sci. Appl., 27, Gakkōtosho, Tokyo, (2006), 9-61.  Google Scholar

[24]

R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Comm. Math. Phys., 229 (2002), 491-509. doi: 10.1007/s00220-002-0695-2.  Google Scholar

[25]

S. Stock, B. Battelier, V. Bretin, Z. Hadzibabic and J. Dalibard, Bose-Einstein condensates in fast rotation, Laser Phy. Lett., 2 (2005), 275-284. doi: 10.1002/lapl.200410177.  Google Scholar

[26]

M. C. Tsatsos, Attractive Bose-Einstein Condensates in three dimensions under rotation: Revisiting the problem of stability of the ground state in harmonic traps, Phys. Rev. A, 83 (2011), 063615. doi: 10.1103/PhysRevA.83.063615.  Google Scholar

[1]

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
[2]

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
[3]

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
[4]

Kui Li, Zhitao Zhang. A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 851-860. doi: 10.3934/dcds.2016.36.851
[5]

Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781
[6]

Brahim Alouini. Long-time behavior of a Bose-Einstein equation in a two-dimensional thin domain. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1629-1643. doi: 10.3934/cpaa.2011.10.1629
[7]

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
[8]

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
[9]

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
[10]

Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639
[11]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
[12]

Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050
[13]

Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11
[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]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
[16]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034
[17]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[18]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085
[19]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169
[20]

José A. Carrillo, Katharina Hopf, Marie-Therese Wolfram. Numerical study of Bose–Einstein condensation in the Kaniadakis–Quarati model for bosons. Kinetic & Related Models, 2020, 13 (3) : 507-529. doi: 10.3934/krm.2020017

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (157)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences