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September  2020, 9(3): 865-889. doi: 10.3934/eect.2020037

On the management fourth-order Schrödinger-Hartree equation

Carlos Banquet 1, and  Élder J. Villamizar-Roa 2,,

1. 

Universidad de Córdoba, Departamento de Matemáticas y Estadística, A.A. 354, Montería, Colombia
2. 

Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia

* Corresponding author: Élder J. Villamizar-Roa

Received  May 2019 Revised  October 2019 Published  March 2020

Fund Project: The second author has been supported by the the Vicerrectoría de Investigación y Extensión-UIS and Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, contrato Colciencias FP 44842-157-2016

We consider the Cauchy problem associated to the fourth-order nonlinear Schrödinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise constant in time functions. We prove local and global well-posedness results for initial data in $ H^s $-spaces. We also analyze the scaling limit of the fast dispersion management and the convergence to a model with averaged dispersions.

Keywords: Dispersion management, fourth-order nonlinear Schrödinger-Hartree equation.
Mathematics Subject Classification: Primary: 35Q55; 35A01; Secondary: 35B40; 35G25.
Citation: Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2020, 9 (3) : 865-889. doi: 10.3934/eect.2020037
References:
[1]

F. K. Abdullaev, B. B. Bakhtiyor and M. Salerno, Stable two-dimensional dispersion- managed soliton, Phys. Rev. E, 68 (2003), 066605-066609.   Google Scholar

[2]

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[3]

P. Antonelli, J.-C. Saut and C. Sparber, Well-posedness and averaging of NLS with time-periodic dispersion management, Adv. Differential Equations, 18 (2013), 49-68.   Google Scholar

[4]

P. Antonelli, A. Athanassoulis, H. Hajaiej and P. Markowich, On the XFEL Schrödinger equation: Highly oscillatory magnetic potentials and time averaging, Arch. Ration. Mech. Anal, 211 (2014), 711-732.  doi: 10.1007/s00205-013-0715-8.  Google Scholar

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J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252.  doi: 10.3934/dcds.2009.23.1241.  Google Scholar

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T. Cazenave and M. Scialom, A Schrödinger equation with time-oscillating nonlinearity, Rev. Mat. Univ. Complut. Madrid, 23 (2010), 321-339.  doi: 10.1007/s13163-009-0018-7.  Google Scholar

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X. Carvajal, M. Panthee and M. Scialom, On well-posedness of the third-order nonlinear Schrödinger equation with time-dependent coefficients, Commun. Contemp. Math., 17 (2015), 1450031, 24 pp. doi: 10.1142/S021919971450031X.  Google Scholar

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S. B. Cui, Pointwise estimates for oscillatory integrals and related $L^p-L^q$ estimates Ⅱ: Multidimensional case, J. Fourier Anal. Appl., 12 (2006), 605-627.  doi: 10.1007/s00041-005-5025-6.  Google Scholar

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A. Elgart and B. Schlein, Mean field dynamics of Boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.  Google Scholar

[10]

B. H. Feng, D. Zhao and C. Y. Sun, Homogenization for nonlinear Schrödinger equation with periodic nonlinearity and dissipation in fractional order spaces, Acta Math. Sci. Ser. B, 35 (2015), 567-582.  doi: 10.1016/S0252-9602(15)30004-7.  Google Scholar

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B. H. Feng, Averaging of the nonlinear Schrödinger equation with highly oscillatory magnetic potentials, Nonlinear Anal., 156 (2017), 275-285.  doi: 10.1016/j.na.2017.02.028.  Google Scholar

[12]

G. Fibich, The Nonlinear Schrödinger Equation. Singular Solutions and Optical Collapse, Applied Mathematical Sciences, 192. Springer, Cham, 2015. doi: 10.1007/978-3-319-12748-4.  Google Scholar

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G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.  Google Scholar

[14]

Y. F. Gao, Blow-up for the focusing $\dot{H}^{1/2}$-critical Hartree equation with radial data, J. Differential Equations, 255 (2013), 2801-2825.  doi: 10.1016/j.jde.2013.07.014.  Google Scholar

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C. H. Guo and S. B. Cui, Global existence of solutions for a fourth-order nonlinear Schrödinger equation, Appl. Math. Lett., 25 (2006), 706-711.  doi: 10.1016/j.aml.2005.10.002.  Google Scholar

[16]

A. Guo and S. B. Cui, On the Cauchy problem of fourth-order nonlinear Schrödinger equations, Nonlinear Anal., 66 (2007), 2911-2930.  doi: 10.1016/j.na.2006.04.020.  Google Scholar

[17]

C. H. Guo, Global existence of solutions for a fourth-order nonlinear Schrödinger equation in $n+1$ dimensions, Nonlinear Anal., 73 (2010), 555-563.  doi: 10.1016/j.na.2010.03.052.  Google Scholar

[18]

C. H. Guo and S. B. Cui, Well-posedness of the Cauchy problem of high dimension non-isotropic fourth-order Schrödinger equations in Sobolev spaces, Nonlinear Anal., 70 (2009), 3761-3772.  doi: 10.1016/j.na.2008.07.032.  Google Scholar

[19]

R. Hirota, Direct Methods in Soliton Theory, Springer, Berlin, 1980. Google Scholar

[20]

B. Ivano and A. Kosevich, Stable three-dimensional small-amplitude soliton in magnetic materials, Sov. J. Low Temp. Phys., 9 (1983), 439-442.   Google Scholar

[21]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E., 53 (1996), R1336–R1339. Google Scholar

[22]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.  Google Scholar

[23]

T. Kato, On nonlinear Schrödinger equations II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794.  Google Scholar

[24]

T. Kato, Perturbation Theory of Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[25]

C. Kurtzke, Suppression of fiber nonlinearities by appropriate dispersion management. IEEE, Phot. Tech. Lett., 5 (1993), 1250-1253.   Google Scholar

[26]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.  Google Scholar

[27]

E. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[28]

P. Lushnikov, Dispersion-managed soliton in a strong dispersion map limit, Optics Lett., 26 (2001), 1535-1537.  doi: 10.1364/OL.26.001535.  Google Scholar

[29]

P. Lushnikov, Oscillating tails of dispersion-managed soliton, J. Opt. Soc. Am. B, 21 (2004), 1913-1918.  doi: 10.1364/JOSAB.21.001913.  Google Scholar

[30]

C. X. Miao, G. X. Xu and L. F. Zhao, The Cauchy problem of the Hartree equation, J. Partial Differ. Equ., 21 (2008), 22-44.   Google Scholar

[31]

C. Sulem and P.-L. Sulem, The Non-linear Schrödinger Equation. Self-Focusing and Wave Collapse, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001.  Google Scholar

[32]

P. A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 81 (1975), 477-478.  doi: 10.1090/S0002-9904-1975-13790-6.  Google Scholar

[33]

E. J. Villamizar-Roa and C. Banquet, On the Schrödinger equations with isotropic and anisotropic fourth-order dispersion, Electron. J. Differential Equations, 2016 (2016), 20 pp.  Google Scholar

[34]

B. Yu, K. Gaididei, O. Rasmussen and P. Christiansen, Nonlinear excitations in two-dimensional molecular structures with impureties, Phys. Rev. E., 52 (1995), 2951-2962.   Google Scholar

[35]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863.  Google Scholar

[36]

V. Zharnitsky, E. Grenier, C. K. R. T. Jones and S. K. Turitsyn, Stabilizing effects of dispersion management, Phys. D, 152/153 (2001), 794-817.  doi: 10.1016/S0167-2789(01)00213-5.  Google Scholar

show all references

References:
[1]

F. K. Abdullaev, B. B. Bakhtiyor and M. Salerno, Stable two-dimensional dispersion- managed soliton, Phys. Rev. E, 68 (2003), 066605-066609.   Google Scholar

[2]

G. Agrawal, Nonlinear Fiber Opticss, Second Edition, Academic Press, San Diego, 1995. Google Scholar

[3]

P. Antonelli, J.-C. Saut and C. Sparber, Well-posedness and averaging of NLS with time-periodic dispersion management, Adv. Differential Equations, 18 (2013), 49-68.   Google Scholar

[4]

P. Antonelli, A. Athanassoulis, H. Hajaiej and P. Markowich, On the XFEL Schrödinger equation: Highly oscillatory magnetic potentials and time averaging, Arch. Ration. Mech. Anal, 211 (2014), 711-732.  doi: 10.1007/s00205-013-0715-8.  Google Scholar

[5]

J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252.  doi: 10.3934/dcds.2009.23.1241.  Google Scholar

[6]

T. Cazenave and M. Scialom, A Schrödinger equation with time-oscillating nonlinearity, Rev. Mat. Univ. Complut. Madrid, 23 (2010), 321-339.  doi: 10.1007/s13163-009-0018-7.  Google Scholar

[7]

X. Carvajal, M. Panthee and M. Scialom, On well-posedness of the third-order nonlinear Schrödinger equation with time-dependent coefficients, Commun. Contemp. Math., 17 (2015), 1450031, 24 pp. doi: 10.1142/S021919971450031X.  Google Scholar

[8]

S. B. Cui, Pointwise estimates for oscillatory integrals and related $L^p-L^q$ estimates Ⅱ: Multidimensional case, J. Fourier Anal. Appl., 12 (2006), 605-627.  doi: 10.1007/s00041-005-5025-6.  Google Scholar

[9]

A. Elgart and B. Schlein, Mean field dynamics of Boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.  Google Scholar

[10]

B. H. Feng, D. Zhao and C. Y. Sun, Homogenization for nonlinear Schrödinger equation with periodic nonlinearity and dissipation in fractional order spaces, Acta Math. Sci. Ser. B, 35 (2015), 567-582.  doi: 10.1016/S0252-9602(15)30004-7.  Google Scholar

[11]

B. H. Feng, Averaging of the nonlinear Schrödinger equation with highly oscillatory magnetic potentials, Nonlinear Anal., 156 (2017), 275-285.  doi: 10.1016/j.na.2017.02.028.  Google Scholar

[12]

G. Fibich, The Nonlinear Schrödinger Equation. Singular Solutions and Optical Collapse, Applied Mathematical Sciences, 192. Springer, Cham, 2015. doi: 10.1007/978-3-319-12748-4.  Google Scholar

[13]

G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.  Google Scholar

[14]

Y. F. Gao, Blow-up for the focusing $\dot{H}^{1/2}$-critical Hartree equation with radial data, J. Differential Equations, 255 (2013), 2801-2825.  doi: 10.1016/j.jde.2013.07.014.  Google Scholar

[15]

C. H. Guo and S. B. Cui, Global existence of solutions for a fourth-order nonlinear Schrödinger equation, Appl. Math. Lett., 25 (2006), 706-711.  doi: 10.1016/j.aml.2005.10.002.  Google Scholar

[16]

A. Guo and S. B. Cui, On the Cauchy problem of fourth-order nonlinear Schrödinger equations, Nonlinear Anal., 66 (2007), 2911-2930.  doi: 10.1016/j.na.2006.04.020.  Google Scholar

[17]

C. H. Guo, Global existence of solutions for a fourth-order nonlinear Schrödinger equation in $n+1$ dimensions, Nonlinear Anal., 73 (2010), 555-563.  doi: 10.1016/j.na.2010.03.052.  Google Scholar

[18]

C. H. Guo and S. B. Cui, Well-posedness of the Cauchy problem of high dimension non-isotropic fourth-order Schrödinger equations in Sobolev spaces, Nonlinear Anal., 70 (2009), 3761-3772.  doi: 10.1016/j.na.2008.07.032.  Google Scholar

[19]

R. Hirota, Direct Methods in Soliton Theory, Springer, Berlin, 1980. Google Scholar

[20]

B. Ivano and A. Kosevich, Stable three-dimensional small-amplitude soliton in magnetic materials, Sov. J. Low Temp. Phys., 9 (1983), 439-442.   Google Scholar

[21]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E., 53 (1996), R1336–R1339. Google Scholar

[22]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.  Google Scholar

[23]

T. Kato, On nonlinear Schrödinger equations II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794.  Google Scholar

[24]

T. Kato, Perturbation Theory of Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[25]

C. Kurtzke, Suppression of fiber nonlinearities by appropriate dispersion management. IEEE, Phot. Tech. Lett., 5 (1993), 1250-1253.   Google Scholar

[26]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.  Google Scholar

[27]

E. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[28]

P. Lushnikov, Dispersion-managed soliton in a strong dispersion map limit, Optics Lett., 26 (2001), 1535-1537.  doi: 10.1364/OL.26.001535.  Google Scholar

[29]

P. Lushnikov, Oscillating tails of dispersion-managed soliton, J. Opt. Soc. Am. B, 21 (2004), 1913-1918.  doi: 10.1364/JOSAB.21.001913.  Google Scholar

[30]

C. X. Miao, G. X. Xu and L. F. Zhao, The Cauchy problem of the Hartree equation, J. Partial Differ. Equ., 21 (2008), 22-44.   Google Scholar

[31]

C. Sulem and P.-L. Sulem, The Non-linear Schrödinger Equation. Self-Focusing and Wave Collapse, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001.  Google Scholar

[32]

P. A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 81 (1975), 477-478.  doi: 10.1090/S0002-9904-1975-13790-6.  Google Scholar

[33]

E. J. Villamizar-Roa and C. Banquet, On the Schrödinger equations with isotropic and anisotropic fourth-order dispersion, Electron. J. Differential Equations, 2016 (2016), 20 pp.  Google Scholar

[34]

B. Yu, K. Gaididei, O. Rasmussen and P. Christiansen, Nonlinear excitations in two-dimensional molecular structures with impureties, Phys. Rev. E., 52 (1995), 2951-2962.   Google Scholar

[35]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863.  Google Scholar

[36]

V. Zharnitsky, E. Grenier, C. K. R. T. Jones and S. K. Turitsyn, Stabilizing effects of dispersion management, Phys. D, 152/153 (2001), 794-817.  doi: 10.1016/S0167-2789(01)00213-5.  Google Scholar

Figure 1.  Sketch of the dispersion functions
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