doi: 10.3934/mcrf.2021036
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Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

* Corresponding author: Jian Zhang

Received  December 2020 Revised  April 2021 Early access July 2021

Fund Project: The project is supported by the National Natural Science Foundation of China (Grant No. 11871138)

In this paper, we study the nonlinear Schrödinger equation with a partial confinement. By applying the cross-constrained variational arguments and invariant manifolds of the evolution flow, the sharp condition for global existence and blowup of the solution is derived.

Citation: Chenglin Wang, Jian Zhang. Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021036
References:
[1]

P. AntonelliR. Carles and J. D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Comm. Math. Phys., 334 (2015), 367-396.  doi: 10.1007/s00220-014-2166-y.

[2]

P. AntonelliA. AthanassoulisZ. Y. Huang and P. A. Markowich, Numerical simulations of x-ray free electron lasers (XFEL), Multiscale Model. Simul., 12) (2014), 1607-1621.  doi: 10.1137/130927838.

[3]

W. Z. Bao and Y. Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.

[4]

J. BellazziniN. BoussaïdL. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Commun. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1.

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Institute of Mathematical Sciences, American Mathematical Society, New York, 2003.

[6]

R. CarlesP. A. Markowich and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), 2569-2590.  doi: 10.1088/0951-7715/21/11/006.

[7]

T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.

[8]

Z. H. Gan and J. Zhang, Sharp threshold of global existence and insatbility of standing wave for a Davey-Stewartson system, Comm. Math. Phys., 283 (2008), 93-125.  doi: 10.1007/s00220-008-0456-y.

[9]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.

[10]

C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates, Nonlinearity, 14 (2001), R25–R62. doi: 10.1088/0951-7715/14/5/201.

[11]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.

[12]

M. Lu, N. Q. Burdick, S. H. Youn and B. L. Lev, Strongly dipolar Bose-Einstein condensate of dysprosium, Phys. Rev. Lett., 107 (2011), 190401. doi: 10.1103/PhysRevLett.107.190401.

[13]

M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement, Commun. Pure. Appl. Anal., 17 (2018), 1671-1680.  doi: 10.3934/cpaa.2018080.

[14]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel. J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[15]

J. J. Pan and J. Zhang, Mass concentration for nonlinear Schrödinger equation with partial confinement, J. Math. Anal. Appl., 481 (2020), 123484, 14 pp. doi: 10.1016/j.jmaa.2019.123484.

[16]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation. International Series of Monographs on Physics, The Clarendon Press, Oxford University Press, Oxford, 2003.

[17]

J. Stubbe, Global solutions and stable ground states of nonlinear Schrödinger equations, Phys. D., 48 (1991), 259-272.  doi: 10.1016/0167-2789(91)90087-P.

[18]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. 

[19]

R. Z. Xu and C. Xu, Cross-constrained problems for nonlinear Schrödiner equation with harmonic potential, Electron. J. Differential Equations, 211 (2012), 12 pp.

[20]

J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statist. Phys., 101 (2000), 731-746.  doi: 10.1023/A:1026437923987.

[21]

J. Zhang, Cross-constrained variational problem and nonlinear Schrödinger equation, Foundations of Computational Mathematics, World Scientific Publishing, River Edge, NJ, 2002,457–469.

[22]

J. Zhang, Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, 30 (2005), 1429-1443.  doi: 10.1080/03605300500299539.

[23]

J. Zhang and S. H. Zhu, Sharp energy criteria and singularity of blow-up solutions for the Davey-Stewartson system, Commun. Math. Sci., 17 (2019), 653-667.  doi: 10.4310/CMS.2019.v17.n3.a4.

[24]

J. Zhang, Sharp threshold of global existence for nonlinear Schrödinger equation with partial confinement, Nonlinear Anal., 196 (2020), 111832, 7 pp. doi: 10.1016/j.na.2020.111832.

show all references

References:
[1]

P. AntonelliR. Carles and J. D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Comm. Math. Phys., 334 (2015), 367-396.  doi: 10.1007/s00220-014-2166-y.

[2]

P. AntonelliA. AthanassoulisZ. Y. Huang and P. A. Markowich, Numerical simulations of x-ray free electron lasers (XFEL), Multiscale Model. Simul., 12) (2014), 1607-1621.  doi: 10.1137/130927838.

[3]

W. Z. Bao and Y. Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.

[4]

J. BellazziniN. BoussaïdL. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Commun. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1.

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Institute of Mathematical Sciences, American Mathematical Society, New York, 2003.

[6]

R. CarlesP. A. Markowich and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), 2569-2590.  doi: 10.1088/0951-7715/21/11/006.

[7]

T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.

[8]

Z. H. Gan and J. Zhang, Sharp threshold of global existence and insatbility of standing wave for a Davey-Stewartson system, Comm. Math. Phys., 283 (2008), 93-125.  doi: 10.1007/s00220-008-0456-y.

[9]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.

[10]

C. Josserand and Y. Pomeau, Nonlinear aspects of the theory of Bose-Einstein condensates, Nonlinearity, 14 (2001), R25–R62. doi: 10.1088/0951-7715/14/5/201.

[11]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.

[12]

M. Lu, N. Q. Burdick, S. H. Youn and B. L. Lev, Strongly dipolar Bose-Einstein condensate of dysprosium, Phys. Rev. Lett., 107 (2011), 190401. doi: 10.1103/PhysRevLett.107.190401.

[13]

M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement, Commun. Pure. Appl. Anal., 17 (2018), 1671-1680.  doi: 10.3934/cpaa.2018080.

[14]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel. J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[15]

J. J. Pan and J. Zhang, Mass concentration for nonlinear Schrödinger equation with partial confinement, J. Math. Anal. Appl., 481 (2020), 123484, 14 pp. doi: 10.1016/j.jmaa.2019.123484.

[16]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation. International Series of Monographs on Physics, The Clarendon Press, Oxford University Press, Oxford, 2003.

[17]

J. Stubbe, Global solutions and stable ground states of nonlinear Schrödinger equations, Phys. D., 48 (1991), 259-272.  doi: 10.1016/0167-2789(91)90087-P.

[18]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. 

[19]

R. Z. Xu and C. Xu, Cross-constrained problems for nonlinear Schrödiner equation with harmonic potential, Electron. J. Differential Equations, 211 (2012), 12 pp.

[20]

J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statist. Phys., 101 (2000), 731-746.  doi: 10.1023/A:1026437923987.

[21]

J. Zhang, Cross-constrained variational problem and nonlinear Schrödinger equation, Foundations of Computational Mathematics, World Scientific Publishing, River Edge, NJ, 2002,457–469.

[22]

J. Zhang, Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, 30 (2005), 1429-1443.  doi: 10.1080/03605300500299539.

[23]

J. Zhang and S. H. Zhu, Sharp energy criteria and singularity of blow-up solutions for the Davey-Stewartson system, Commun. Math. Sci., 17 (2019), 653-667.  doi: 10.4310/CMS.2019.v17.n3.a4.

[24]

J. Zhang, Sharp threshold of global existence for nonlinear Schrödinger equation with partial confinement, Nonlinear Anal., 196 (2020), 111832, 7 pp. doi: 10.1016/j.na.2020.111832.

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