American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Nonlocal final value problem governed by semilinear anomalous diffusion equations
  • EECT Home
  • This Issue
  • Next Article
    Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability
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  September 2020 Early access  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 and 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. 

[2]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[28]

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

[29]

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

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

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. 

[2]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[28]

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

[29]

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

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

Figure 1.  Sketch of the dispersion functions
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113
[2]

Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431
[3]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284
[4]

Kelin Li, Huafei Di. On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4293-4320. doi: 10.3934/dcdss.2021122
[5]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205
[6]

Yingying Xie, Jian Su, Liquan Mei. Blowup results and concentration in focusing Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 5001-5017. doi: 10.3934/dcds.2020209
[7]

Jinxing Liu, Xiongrui Wang, Jun Zhou, Huan Zhang. Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4321-4335. doi: 10.3934/dcdss.2021108
[8]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
[9]

Benoît Pausader. The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1275-1292. doi: 10.3934/dcds.2009.24.1275
[10]

Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2721-2757. doi: 10.3934/dcdsb.2021156
[11]

José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1
[12]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225
[13]

Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009
[14]

Edcarlos D. Silva, Marcos L. M. Carvalho, Claudiney Goulart. Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1039-1065. doi: 10.3934/dcds.2021146
[15]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027
[16]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[17]

Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure and Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[18]

Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174
[19]

Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093
[20]

Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control and Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (213)
  • HTML views (321)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy