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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

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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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

[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.

[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.

[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.

[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.

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