• 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

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.

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. AbdullaevB. 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. AntonelliJ.-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. AntonelliA. AthanassoulisH. 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. FengD. 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. FibichB. 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. MiaoG. 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. YuK. GaidideiO. Rasmussen and P. Christiansen, Nonlinear excitations in two-dimensional molecular structures with impureties, Phys. Rev. E., 52 (1995), 2951-2962.   Google Scholar

[35]

Y. ChoG. HwangS. 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. ZharnitskyE. GrenierC. 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. AbdullaevB. 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. AntonelliJ.-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. AntonelliA. AthanassoulisH. 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. FengD. 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. FibichB. 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. MiaoG. 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. YuK. GaidideiO. Rasmussen and P. Christiansen, Nonlinear excitations in two-dimensional molecular structures with impureties, Phys. Rev. E., 52 (1995), 2951-2962.   Google Scholar

[35]

Y. ChoG. HwangS. 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. ZharnitskyE. GrenierC. 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
[1]

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

[2]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[3]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[4]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[5]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[6]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[7]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[8]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[9]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[10]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[11]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[12]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[13]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[14]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[15]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[16]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[17]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[18]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[19]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

[20]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (79)
  • HTML views (318)
  • Cited by (0)

Other articles
by authors

[Back to Top]