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November  2021, 29(5): 3449-3469. doi: 10.3934/era.2021047

On a fractional Schrödinger equation in the presence of harmonic potential

1. 

Mathematics Department, University of Wisconsin-Madison, Madison, WI 53706, USA

2. 

Department of Mathematics, California State University, Los Angeles, Los Angeles, CA 90032, USA

* Corresponding author: Hichem Hajaiej

Received  November 2020 Revised  March 2021 Published  November 2021 Early access  June 2021

In this paper, we establish the existence of ground state solutions for a fractional Schrödinger equation in the presence of a harmonic trapping potential. We also address the orbital stability of standing waves. Additionally, we provide interesting numerical results about the dynamics and compare them with other types of Schrödinger equations [11,18]. Our results explain the effect of each term of the Schrödinger equation: the fractional power, the power of the nonlinearity, and the harmonic potential.

Citation: Zhiyan Ding, Hichem Hajaiej. On a fractional Schrödinger equation in the presence of harmonic potential. Electronic Research Archive, 2021, 29 (5) : 3449-3469. doi: 10.3934/era.2021047
References:
[1]

B. Alouini, Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential, Commun. Pure Appl. Anal., 19 (2020), 4545-4573.  doi: 10.3934/cpaa.2020206.  Google Scholar

[2]

W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697.  doi: 10.1137/S1064827503422956.  Google Scholar

[3]

A. Burchard and H. Hajaiej, Rearrangement inequalities for functionals with monotone integrands, J. Funct. Anal., 233 (2006), 561-582.  doi: 10.1016/j.jfa.2005.08.010.  Google Scholar

[4]

R. Carles and H. Hajaiej, Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases, Bull. Lond. Math. Soc., 47 (2015), 509-518.  doi: 10.1112/blms/bdv024.  Google Scholar

[5]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.  Google Scholar

[6]

X. Chang, Ground state solutions of asymptotically linear fractional Schrödinger equation, J. Math. Phys., 54 (2013), 061504, 10 pp. doi: 10.1063/1.4809933.  Google Scholar

[7]

M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507, 7 pp. doi: 10.1063/1.3701574.  Google Scholar

[8]

S. Duo and Y. Zhang, Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl., 71 (2016), 2257-2271.  doi: 10.1016/j.camwa.2015.12.042.  Google Scholar

[9]

H. Emamirad and A. Rougirel, Feynman path formula for the time fractional Schrödinger equation, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3391-3400.  doi: 10.3934/dcdss.2020246.  Google Scholar

[10]

G. Fibich and X.-P. Wang, Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Phys. D, 175 (2003), 96-108.  doi: 10.1016/S0167-2789(02)00626-7.  Google Scholar

[11]

F. Hadj SelemH. HajaiejP. A. Markowich and S. Trabelsi, Variational approach to the orbital stability of standing waves of the Gross-Pitaevskii equation, Milan J. Math., 82 (2014), 273-295.  doi: 10.1007/s00032-014-0227-5.  Google Scholar

[12]

H. Hajaiej, On the optimality of the assumptions used to prove the existence and symmetry of minimizers of some fractional constrained variational problems, Annales Henri Poincaré, 14 (2013), 1425–1433. doi: 10.1007/s00023-012-0212-x.  Google Scholar

[13]

H. HajaiejP. A. Markowich and S. Trabelsi, Minimizers of a class of constrained vectorial variational problems: Part I., Milan J. Math., 82 (2014), 81-98.  doi: 10.1007/s00032-014-0218-6.  Google Scholar

[14]

H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Sufficient and necessary conditions for the fractional Gagliardo-Nirenberg inequalities and applications to {Navier-Stokes} and generalized Boson equations, RIMS Kôkyȗroku Bessatsu, (2011), 159–175.  Google Scholar

[15]

H. Hajaiej and C. A. Stuart, Symmetrization inequalities for composition operators of Carathéodory type, Proc. London Math. Soc., 87 (2003), 396-418.  doi: 10.1112/S0024611503014473.  Google Scholar

[16]

H. Hajaiej and C. A. Stuart, On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation, Adv. Nonlinear Stud., 4 (2004), 469-501.  doi: 10.1515/ans-2004-0407.  Google Scholar

[17]

K. Kirkpatrick and Y. Zhang, Fractional Schrödinger dynamics and decoherence, Phys. D, 332 (2016), 41-54.  doi: 10.1016/j.physd.2016.05.015.  Google Scholar

[18]

C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 470 (2014), 20140364, 26 pp. doi: 10.1098/rspa.2014.0364.  Google Scholar

[19]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[20]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[21]

H. A. Rose and M. I. Weinstein, On the bound states of the nonlinear Schrödinger equation with a linear potential, Phys. D, 30 (1988), 207-218.  doi: 10.1016/0167-2789(88)90107-8.  Google Scholar

[22]

Y. Su, H. Chen, S. Liu and X. Fang, Fractional Schrödinger- Poisson systems with weighted Hardy potential and critical exponent, Electron. J. Differential Equations, 16 (2020), Paper No. 1, 17 pp.  Google Scholar

[23]

J. Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51 (2000), 498-503.  doi: 10.1007/PL00001512.  Google Scholar

[24]

F. ZhaoL. Zhao and Y. Ding, Existence and multiplicity of solutions for a non-periodic Schrödinger equation, Nonlinear Anal., 69 (2008), 3671-3678.  doi: 10.1016/j.na.2007.10.024.  Google Scholar

show all references

References:
[1]

B. Alouini, Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential, Commun. Pure Appl. Anal., 19 (2020), 4545-4573.  doi: 10.3934/cpaa.2020206.  Google Scholar

[2]

W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697.  doi: 10.1137/S1064827503422956.  Google Scholar

[3]

A. Burchard and H. Hajaiej, Rearrangement inequalities for functionals with monotone integrands, J. Funct. Anal., 233 (2006), 561-582.  doi: 10.1016/j.jfa.2005.08.010.  Google Scholar

[4]

R. Carles and H. Hajaiej, Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases, Bull. Lond. Math. Soc., 47 (2015), 509-518.  doi: 10.1112/blms/bdv024.  Google Scholar

[5]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.  Google Scholar

[6]

X. Chang, Ground state solutions of asymptotically linear fractional Schrödinger equation, J. Math. Phys., 54 (2013), 061504, 10 pp. doi: 10.1063/1.4809933.  Google Scholar

[7]

M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507, 7 pp. doi: 10.1063/1.3701574.  Google Scholar

[8]

S. Duo and Y. Zhang, Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl., 71 (2016), 2257-2271.  doi: 10.1016/j.camwa.2015.12.042.  Google Scholar

[9]

H. Emamirad and A. Rougirel, Feynman path formula for the time fractional Schrödinger equation, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3391-3400.  doi: 10.3934/dcdss.2020246.  Google Scholar

[10]

G. Fibich and X.-P. Wang, Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Phys. D, 175 (2003), 96-108.  doi: 10.1016/S0167-2789(02)00626-7.  Google Scholar

[11]

F. Hadj SelemH. HajaiejP. A. Markowich and S. Trabelsi, Variational approach to the orbital stability of standing waves of the Gross-Pitaevskii equation, Milan J. Math., 82 (2014), 273-295.  doi: 10.1007/s00032-014-0227-5.  Google Scholar

[12]

H. Hajaiej, On the optimality of the assumptions used to prove the existence and symmetry of minimizers of some fractional constrained variational problems, Annales Henri Poincaré, 14 (2013), 1425–1433. doi: 10.1007/s00023-012-0212-x.  Google Scholar

[13]

H. HajaiejP. A. Markowich and S. Trabelsi, Minimizers of a class of constrained vectorial variational problems: Part I., Milan J. Math., 82 (2014), 81-98.  doi: 10.1007/s00032-014-0218-6.  Google Scholar

[14]

H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Sufficient and necessary conditions for the fractional Gagliardo-Nirenberg inequalities and applications to {Navier-Stokes} and generalized Boson equations, RIMS Kôkyȗroku Bessatsu, (2011), 159–175.  Google Scholar

[15]

H. Hajaiej and C. A. Stuart, Symmetrization inequalities for composition operators of Carathéodory type, Proc. London Math. Soc., 87 (2003), 396-418.  doi: 10.1112/S0024611503014473.  Google Scholar

[16]

H. Hajaiej and C. A. Stuart, On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation, Adv. Nonlinear Stud., 4 (2004), 469-501.  doi: 10.1515/ans-2004-0407.  Google Scholar

[17]

K. Kirkpatrick and Y. Zhang, Fractional Schrödinger dynamics and decoherence, Phys. D, 332 (2016), 41-54.  doi: 10.1016/j.physd.2016.05.015.  Google Scholar

[18]

C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 470 (2014), 20140364, 26 pp. doi: 10.1098/rspa.2014.0364.  Google Scholar

[19]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[20]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[21]

H. A. Rose and M. I. Weinstein, On the bound states of the nonlinear Schrödinger equation with a linear potential, Phys. D, 30 (1988), 207-218.  doi: 10.1016/0167-2789(88)90107-8.  Google Scholar

[22]

Y. Su, H. Chen, S. Liu and X. Fang, Fractional Schrödinger- Poisson systems with weighted Hardy potential and critical exponent, Electron. J. Differential Equations, 16 (2020), Paper No. 1, 17 pp.  Google Scholar

[23]

J. Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51 (2000), 498-503.  doi: 10.1007/PL00001512.  Google Scholar

[24]

F. ZhaoL. Zhao and Y. Ding, Existence and multiplicity of solutions for a non-periodic Schrödinger equation, Nonlinear Anal., 69 (2008), 3671-3678.  doi: 10.1016/j.na.2007.10.024.  Google Scholar

Figure 3.  Ground state solutions with $ \sigma = 1 $ and different $ s $
Figure 6.  Ground state solutions with non-symmetric potential
Figure 7.  Energy and stability check with $ s = 0.8 $, $ \sigma = 1 $
Figure 9.  Abosolute value of solution with different $ s $
Figure 10.  Stability test with $ \psi^s_0(x) = 0.9*u^s_0(x) $
Figure 1.  Ground state solution with $ s = 0.8 $, $ \sigma = 1 $, $ L = 10 $ and $ J = 5000 $
Figure 2.  Time dynamics of standing waves with $ s = 0.8 $, $ \sigma = 1 $, $ L = 10 $ and $ J = 5000 $
Figure 4.  $ L^2 $ distance between ground state solutions of $ s<1 $ and $ s = 1 $ with $ \sigma = 1 $
Figure 5.  Energy and $ \lambda_c $
Figure 8.  $ \|\psi^s(x, t)-u^s(x, t)\|_{\widetilde{\Sigma}_s(\mathbb{R})} $ when $ s = 0.8 $ and $ \sigma = 1 $
Figure 11.  Standing wave, ground state solution and blow up solution with $ s = 0.5 $, $ \delta = 1 $
Figure 12.  Dynamics of FNLS
Figure 13.  Evolution of $ L^\infty $
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