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On the Cauchy problem for nonlinear Schrödinger equations with rotation
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 |
References:
[1] |
A. Aftalion, "Vortices in Bose-Einstein Condensates," Progress in Nonlinear Differential Equations and their Applications, 67, Birkhäuser Boston, Inc., Boston, MA, 2006. |
[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. |
[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. |
[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. |
[5] |
R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843.
doi: 10.1137/S0036141002416936. |
[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. |
[7] |
R. Carles, Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., 9 (2011), no. 4, 937-964. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
O. Kavian and F. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843.
doi: 10.1137/S0036141002416936. |
[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. |
[7] |
R. Carles, Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., 9 (2011), no. 4, 937-964. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
O. Kavian and F. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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