October  2019, 39(10): 5923-5944. doi: 10.3934/dcds.2019259

The nonlinear Schrödinger equations with harmonic potential in modulation spaces

TIFR Centre for Applicable Mathematics, Bangalore 560 065, India

Received  November 2018 Revised  January 2019 Published  July 2019

We study nonlinear Schrödinger $ i\partial_tu-Hu = F(u) $ (NLSH) equation associated to harmonic oscillator $ H = -\Delta +|x|^2 $ in modulation spaces $ M^{p,q}. $ When $ F(u) = (|x|^{-\gamma}\ast |u|^2)u, $ we prove global well-posedness for (NLSH) in modulation spaces $ M^{p,p}(\mathbb R^d) $ $ (1\leq p < 2d/(d+\gamma), 0<\gamma< \min \{ 2, d/2\}). $ When $ F(u) = (K\ast |u|^{2k})u $ with $ K\in \mathcal{F}L^q $ (Fourier-Lebesgue spaces) or $ M^{\infty,1} $ (Sjöstrand's class) or $ M^{1, \infty}, $ some local and global well-posedness for (NLSH) are obtained in some modulation spaces. As a consequence, we can get local and global well-posedness for (NLSH) in a function spaces$ - $which are larger than usual $ L^p_s- $Sobolev spaces.

Citation: Divyang G. Bhimani. The nonlinear Schrödinger equations with harmonic potential in modulation spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5923-5944. doi: 10.3934/dcds.2019259
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlineare Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar

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Á. BényiK. GröchenigK. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal., 246 (2007), 366-384. doi: 10.1016/j.jfa.2006.12.019. Google Scholar

[3]

Á. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558. doi: 10.1112/blms/bdp027. Google Scholar

[4]

D. G. Bhimani, The Cauchy problem for the Hartree type equation in modulation spaces, Nonlinear Anal., 130 (2016), 190-201. doi: 10.1016/j.na.2015.10.002. Google Scholar

[5]

D. G. Bhimani, Global well-posedness for fractional Hartree equation on modulation spaces and Fourier algebra, available at arXiv: 1810.04076.Google Scholar

[6]

D. G. Bhimani and P. K. Ratnakumar, Functions operating on modulation spaces and nonlinear dispersive equations, J. Funct. Anal., 270 (2016), 621-648. doi: 10.1016/j.jfa.2015.10.017. Google Scholar

[7]

D. G. Bhimani, R. Balhara and S. Thangavelu, Hermite Multipliers on Modulation Spaces, In: Delgado J., Ruzhansky M. (eds) Analysis and Partial Differential Equations: Perspectives from Developing Countries, Springer Proceedings in Mathematics & Statistics, Springer, Cham, 275 (2019), 42–64. Google Scholar

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C. C. BradleyC. A. Sackett and R. G. Hulet, Bose-Einstein condensation of lithium: Observation of limited condensate number, Phys. Rev. Lett., 78 (1997), 985-989. doi: 10.1103/PhysRevLett.78.985. Google Scholar

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P. Cao and R. Carles, Semi-classical wave packet dynamics for Hartree equations, Rev. Math. Phys., 23 (2011), 933-967. doi: 10.1142/S0129055X11004485. Google Scholar

[10]

R. Carles, Remarks on nonlinear Schrödinger equations with harmonic potential, Ann. Henri Poincaré, 3 (2002), 757-772. doi: 10.1007/s00023-002-8635-4. Google Scholar

[11]

R. Carles, Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., 9 (2011), 937-964. doi: 10.4310/CMS.2011.v9.n4.a1. Google Scholar

[12]

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

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R. CarlesN. Mauser and H. P. Stimming, (Semi)classical limit of the Hartree equation with harmonic potential, SIAM J. Appl. Math., 66 (2005), 29-56. doi: 10.1137/040609732. Google Scholar

[14]

E. Cordero and F. Nicola, On the Schrödinger equation with potential in modulation spaces, J. Pseudo-Differ. Oper. Appl., 5 (2014), 319-341. doi: 10.1007/s11868-014-0096-2. Google Scholar

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E. CorderoF. Nicola and L. Rodino, Schrödinger equations with rough Hamiltonians, Discrete Contin. Dyn. Syst., 35 (2015), 4805-4821. doi: 10.3934/dcds.2015.35.4805. Google Scholar

[16]

E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class, J. Math. Phys., 55 (2014), 081506, 17 pp. doi: 10.1063/1.4892459. Google Scholar

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J. CunananM. Kobayashi and M. Sugimoto, Mitsuru, Inclusion relations between $L^p-$Sobolev and Wiener amalgam spaces, J. Funct. Anal., 268 (2015), 239-254. doi: 10.1016/j.jfa.2014.10.017. Google Scholar

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F. DalfovoS. GiorginiP. L. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys., 71 (1999), 463-512. Google Scholar

[19]

H. G. Feichtinger, Modulation Spaces on Locally Compact Abelian Groups, Technical report, University of Vienna, 1983.Google Scholar

[20]

K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.Google Scholar

[21]

K. KatoM. Kobayashi and S. Ito, Remark on wave front sets of solutions to Schrödinger equation of a free particle and a harmonic oscillator, SUT J. Math., 47 (2011), 175-183. Google Scholar

[22]

K. KatoM. Kobayashi and S. Ito, Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials, J. Funct. Anal., 266 (2014), 733-753. doi: 10.1016/j.jfa.2013.08.017. Google Scholar

[23]

M. Kobayashi and M. Sugimoto, The inclusion relation between Sobolev and modulation spaces, J. Funct. Anal., 260 (2011), 3189-3208. doi: 10.1016/j.jfa.2011.02.015. Google Scholar

[24]

F. Nicola, Phase space analysis of semilinear parabolic equations, J. Funct. Anal., 267 (2014), 727-743. doi: 10.1016/j.jfa.2014.05.007. Google Scholar

[25]

K. Okoudjou, Embedding of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc., 132 (2004), 1639-1647. doi: 10.1090/S0002-9939-04-07401-5. Google Scholar

[26]

M. RuzhanskyM. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, Progr Math, 301 (2012), 267-283. doi: 10.1007/978-3-0348-0454-7_14. Google Scholar

[27]

T. Saanouni, Global well-posedness and instability of a nonlinear Schrödinger equation with harmonic potential, J. Aust. Math. Soc., 98 (2015), 78-103. doi: 10.1017/S1446788714000391. Google Scholar

[28]

M. Sugimoto and N. Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal., 248 (2007), 79-106. doi: 10.1016/j.jfa.2007.03.015. Google Scholar

[29]

T. Tao, Nonlinear Dispersive Equations, Local and global analysis, CBMS Regional Conference Series in Mathematics, 106, American Mathematical Society, 2006. doi: 10.1090/cbms/106. Google Scholar

[30]

S. Thangavelu, Lectures on Hermite and Laguerre Expansions, vol. 42 of Mathematical Notes. Princeton University Press, Princeton, 1993. Google Scholar

[31]

J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I., J. Funct. Anal., 207 (2004), 399-429. doi: 10.1016/j.jfa.2003.10.003. Google Scholar

[32]

T. Tsurumi and M. Wadati, Collapses of wave functions in multidimensional nonlinear Schrödinger equations under harmonic potential, J. Phys. Soc. Japan, 67 (1998), 93-95. doi: 10.1143/JPSJ.67.93. Google Scholar

[33]

B. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 232 (2007), 36-73. doi: 10.1016/j.jde.2006.09.004. Google Scholar

[34]

B. WangL. Zhao and B. Guo, Isometric decomposition operators, function spaces $E^{\lambda}_{p, q}$ and applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39. doi: 10.1016/j.jfa.2005.06.018. Google Scholar

[35]

B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I. World Scientific Publishing Co. Pte. Lt., 2011. doi: 10.1142/9789814360746. Google Scholar

[36]

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. Google Scholar

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlineare Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar

[2]

Á. BényiK. GröchenigK. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal., 246 (2007), 366-384. doi: 10.1016/j.jfa.2006.12.019. Google Scholar

[3]

Á. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558. doi: 10.1112/blms/bdp027. Google Scholar

[4]

D. G. Bhimani, The Cauchy problem for the Hartree type equation in modulation spaces, Nonlinear Anal., 130 (2016), 190-201. doi: 10.1016/j.na.2015.10.002. Google Scholar

[5]

D. G. Bhimani, Global well-posedness for fractional Hartree equation on modulation spaces and Fourier algebra, available at arXiv: 1810.04076.Google Scholar

[6]

D. G. Bhimani and P. K. Ratnakumar, Functions operating on modulation spaces and nonlinear dispersive equations, J. Funct. Anal., 270 (2016), 621-648. doi: 10.1016/j.jfa.2015.10.017. Google Scholar

[7]

D. G. Bhimani, R. Balhara and S. Thangavelu, Hermite Multipliers on Modulation Spaces, In: Delgado J., Ruzhansky M. (eds) Analysis and Partial Differential Equations: Perspectives from Developing Countries, Springer Proceedings in Mathematics & Statistics, Springer, Cham, 275 (2019), 42–64. Google Scholar

[8]

C. C. BradleyC. A. Sackett and R. G. Hulet, Bose-Einstein condensation of lithium: Observation of limited condensate number, Phys. Rev. Lett., 78 (1997), 985-989. doi: 10.1103/PhysRevLett.78.985. Google Scholar

[9]

P. Cao and R. Carles, Semi-classical wave packet dynamics for Hartree equations, Rev. Math. Phys., 23 (2011), 933-967. doi: 10.1142/S0129055X11004485. Google Scholar

[10]

R. Carles, Remarks on nonlinear Schrödinger equations with harmonic potential, Ann. Henri Poincaré, 3 (2002), 757-772. doi: 10.1007/s00023-002-8635-4. Google Scholar

[11]

R. Carles, Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., 9 (2011), 937-964. doi: 10.4310/CMS.2011.v9.n4.a1. Google Scholar

[12]

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

[13]

R. CarlesN. Mauser and H. P. Stimming, (Semi)classical limit of the Hartree equation with harmonic potential, SIAM J. Appl. Math., 66 (2005), 29-56. doi: 10.1137/040609732. Google Scholar

[14]

E. Cordero and F. Nicola, On the Schrödinger equation with potential in modulation spaces, J. Pseudo-Differ. Oper. Appl., 5 (2014), 319-341. doi: 10.1007/s11868-014-0096-2. Google Scholar

[15]

E. CorderoF. Nicola and L. Rodino, Schrödinger equations with rough Hamiltonians, Discrete Contin. Dyn. Syst., 35 (2015), 4805-4821. doi: 10.3934/dcds.2015.35.4805. Google Scholar

[16]

E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class, J. Math. Phys., 55 (2014), 081506, 17 pp. doi: 10.1063/1.4892459. Google Scholar

[17]

J. CunananM. Kobayashi and M. Sugimoto, Mitsuru, Inclusion relations between $L^p-$Sobolev and Wiener amalgam spaces, J. Funct. Anal., 268 (2015), 239-254. doi: 10.1016/j.jfa.2014.10.017. Google Scholar

[18]

F. DalfovoS. GiorginiP. L. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys., 71 (1999), 463-512. Google Scholar

[19]

H. G. Feichtinger, Modulation Spaces on Locally Compact Abelian Groups, Technical report, University of Vienna, 1983.Google Scholar

[20]

K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.Google Scholar

[21]

K. KatoM. Kobayashi and S. Ito, Remark on wave front sets of solutions to Schrödinger equation of a free particle and a harmonic oscillator, SUT J. Math., 47 (2011), 175-183. Google Scholar

[22]

K. KatoM. Kobayashi and S. Ito, Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials, J. Funct. Anal., 266 (2014), 733-753. doi: 10.1016/j.jfa.2013.08.017. Google Scholar

[23]

M. Kobayashi and M. Sugimoto, The inclusion relation between Sobolev and modulation spaces, J. Funct. Anal., 260 (2011), 3189-3208. doi: 10.1016/j.jfa.2011.02.015. Google Scholar

[24]

F. Nicola, Phase space analysis of semilinear parabolic equations, J. Funct. Anal., 267 (2014), 727-743. doi: 10.1016/j.jfa.2014.05.007. Google Scholar

[25]

K. Okoudjou, Embedding of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc., 132 (2004), 1639-1647. doi: 10.1090/S0002-9939-04-07401-5. Google Scholar

[26]

M. RuzhanskyM. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, Progr Math, 301 (2012), 267-283. doi: 10.1007/978-3-0348-0454-7_14. Google Scholar

[27]

T. Saanouni, Global well-posedness and instability of a nonlinear Schrödinger equation with harmonic potential, J. Aust. Math. Soc., 98 (2015), 78-103. doi: 10.1017/S1446788714000391. Google Scholar

[28]

M. Sugimoto and N. Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal., 248 (2007), 79-106. doi: 10.1016/j.jfa.2007.03.015. Google Scholar

[29]

T. Tao, Nonlinear Dispersive Equations, Local and global analysis, CBMS Regional Conference Series in Mathematics, 106, American Mathematical Society, 2006. doi: 10.1090/cbms/106. Google Scholar

[30]

S. Thangavelu, Lectures on Hermite and Laguerre Expansions, vol. 42 of Mathematical Notes. Princeton University Press, Princeton, 1993. Google Scholar

[31]

J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I., J. Funct. Anal., 207 (2004), 399-429. doi: 10.1016/j.jfa.2003.10.003. Google Scholar

[32]

T. Tsurumi and M. Wadati, Collapses of wave functions in multidimensional nonlinear Schrödinger equations under harmonic potential, J. Phys. Soc. Japan, 67 (1998), 93-95. doi: 10.1143/JPSJ.67.93. Google Scholar

[33]

B. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 232 (2007), 36-73. doi: 10.1016/j.jde.2006.09.004. Google Scholar

[34]

B. WangL. Zhao and B. Guo, Isometric decomposition operators, function spaces $E^{\lambda}_{p, q}$ and applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39. doi: 10.1016/j.jfa.2005.06.018. Google Scholar

[35]

B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I. World Scientific Publishing Co. Pte. Lt., 2011. doi: 10.1142/9789814360746. Google Scholar

[36]

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. Google Scholar

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