On the Cauchy problem for nonlinear Schrödinger equations with rotation

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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

Abstract
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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 - A, 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 (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.
[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. 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. doi: 10.1137/050629392.
[5]	R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications,, SIAM J. Math. Anal., 35 (2003), 823. doi: 10.1137/S0036141002416936.
[6]	R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials,, Discrete Contin. Dyn. Syst., 13 (2005), 385. doi: 10.3934/dcds.2005.13.385.
[7]	R. Carles, Nonlinear Schrödinger equation with time dependent potential,, Commun. Math. Sci., 9 (2011), 937.
[8]	T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003).
[9]	D. Choi and Q. Niu, Bose-Einstein condensates in an optical lattice,, Phys. Rev. Lett., 82 (1999), 2022. 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. 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. 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).
[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). 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.
[15]	H. Kitada, On a construction of the fundamental solution for Schrödinger equations,, J. Fac. Sci. Univ. Tokyo Sec. IA Math., 27 (1980), 193.
[16]	E. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases,, Comm. Math. Phys., 264 (2006), 505. 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. 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. doi: 10.1142/S0218202508002826.
[19]	H. Liu and E. Tadmor, Rotation prevents finite-time breakdown,, Phys. D, 188 (2004), 262. 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. 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. 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. doi: 10.1103/PhysRevLett.83.2498.
[23]	P. Raphaël, On the blow up phenomenon for the $L^2$ critical non linear Schrödinger equation,, in, 27 (2006), 9.
[24]	R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas,, Comm. Math. Phys., 229 (2002), 491. 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. 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). doi: 10.1103/PhysRevA.83.063615.

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References:

[1]	A. Aftalion, "Vortices in Bose-Einstein Condensates,", Progress in Nonlinear Differential Equations and their Applications, 67 (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.
[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. 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. doi: 10.1137/050629392.
[5]	R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications,, SIAM J. Math. Anal., 35 (2003), 823. doi: 10.1137/S0036141002416936.
[6]	R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials,, Discrete Contin. Dyn. Syst., 13 (2005), 385. doi: 10.3934/dcds.2005.13.385.
[7]	R. Carles, Nonlinear Schrödinger equation with time dependent potential,, Commun. Math. Sci., 9 (2011), 937.
[8]	T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003).
[9]	D. Choi and Q. Niu, Bose-Einstein condensates in an optical lattice,, Phys. Rev. Lett., 82 (1999), 2022. 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. 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. 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).
[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). 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.
[15]	H. Kitada, On a construction of the fundamental solution for Schrödinger equations,, J. Fac. Sci. Univ. Tokyo Sec. IA Math., 27 (1980), 193.
[16]	E. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases,, Comm. Math. Phys., 264 (2006), 505. 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. 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. doi: 10.1142/S0218202508002826.
[19]	H. Liu and E. Tadmor, Rotation prevents finite-time breakdown,, Phys. D, 188 (2004), 262. 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. 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. 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. doi: 10.1103/PhysRevLett.83.2498.
[23]	P. Raphaël, On the blow up phenomenon for the $L^2$ critical non linear Schrödinger equation,, in, 27 (2006), 9.
[24]	R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas,, Comm. Math. Phys., 229 (2002), 491. 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. 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). doi: 10.1103/PhysRevA.83.063615.

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