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Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability
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 |
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.
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. |

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