American Institute of Mathematical Sciences

Advanced
December  2018, 23(10): 4171-4186. doi: 10.3934/dcdsb.2018131

On the Cauchy problem for the XFEL Schrödinger equation

Binhua Feng 1,, and  Dun Zhao 2,

1. 

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
2. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

* Corresponding author: Binhua Feng

Received  August 2017 Published  December 2018 Early access  April 2018

Fund Project: This work is supported by NSFC Grants (No. 11601435, No. 11475073), Gansu Provincial Natural Science Foundation (1606RJZA010) and NWNU-LKQN-14-6.

In this paper, we consider the Cauchy problem for the nonlinear Schrödinger equation with a time-dependent electromagnetic field and a Coulomb potential, which arises as an effective single particle model in X-ray free electron lasers(XFEL). We firstly show the local and global well-posedness for the Cauchy problem under the assumption that the magnetic potential is unbounded and time-dependent, and then obtain the regularity by a fixed point argument.

Keywords: The Cauchy problem, time-dependent electromagnetic field, time-dependent Coulomb potential, XFEL Schrödinger equation.
Mathematics Subject Classification: 35Q51, 35Q55.
Citation: Binhua Feng, Dun Zhao. On the Cauchy problem for the XFEL Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4171-4186. doi: 10.3934/dcdsb.2018131
References:
[1]

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

[2]

P. Antonelli, A. Athanassoulis, Z. Y. Huang and P. Markowich, Numerical Simulations of X-Ray Free Electron Lasers (XFEL), Multiscale Model. Simul., 12 (2014), 1607-1621.  doi: 10.1137/130927838.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.  Google Scholar

[4]

T. Cazenave and M. J. Esteban, On the stability of stationary states for non-linear Schrödinger equations with an external magnetic field, Mat. Apl. Comput., 7 (1988), 155-168.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[5]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998. Google Scholar

[6]

T. Cazenave and F. B. Weissler, The Cauchy problem for the non-linear Schrödinger equation in $H^1$, Manuscripta Math., 61 (1988), 477-494.  doi: 10.1007/BF01258601.  Google Scholar

[7]

H. N. Chapman, Femtosecond time-delay X-ray holography, Nature, 61 (2007), 676-679.   Google Scholar

[8]

A. De Bouard, Non-linear Schrödinger equations with magnetic fields, Differential Integral Equations, 4 (1991), 73-88.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[9]

B. 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

[10]

B. Feng and X. Yuan, On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control Theory, 4 (2015), 431-445.  doi: 10.3934/eect.2015.4.431.  Google Scholar

[11]

B. Feng and D. Zhao, Optimal bilinear control of Gross-Pitaevskii equations with Coulombian potentials, J. Differential Equations, 260 (2016), 2973-2993.  doi: 10.1016/j.jde.2015.10.026.  Google Scholar

[12]

B. Feng, D. Zhao and C. Sun, On the Cauchy problem for the nonlinear Schrödinger equations with time-dependent linear loss/gain, J. Math. Anal. Appl., 416 (2014), 901-923.  doi: 10.1016/j.jmaa.2014.03.019.  Google Scholar

[13]

A. Fratalocchi and G. Ruocco, Single-molecule imaging with X-ray free electron lasers: Dream or reality? Phys. Rev. Lett., 106 (2011), 105504. doi: 10.1103/PhysRevLett.106.105504.  Google Scholar

[14]

T. Kato, On nonlinear Schrödinger equations, Ann. IHP (Phys. Theor.), 46 (1987), 113-129.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[15]

L. Michel, Remarks on non-linear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 33 (2008), 1198-1215.  doi: 10.1080/03605300801891927.  Google Scholar

[16]

Y. Nakamura and A. Shimomura, Local well-posedness and smoothing effects of strong solutions for non-linear Schrödinger equations with potentials and magnetic fields, Hokkaido Math. J., 34 (2005), 37-63.  doi: 10.14492/hokmj/1285766208.  Google Scholar

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. Google Scholar

[18]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, Springer-Verlag, New York, 1999. Google Scholar

[19]

K. Yajima, Schrödinger evolution equations with magnetic fields, J. Analyse Math., 56 (1991), 29-76.  doi: 10.1007/BF02820459.  Google Scholar

show all references

References:
[1]

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

[2]

P. Antonelli, A. Athanassoulis, Z. Y. Huang and P. Markowich, Numerical Simulations of X-Ray Free Electron Lasers (XFEL), Multiscale Model. Simul., 12 (2014), 1607-1621.  doi: 10.1137/130927838.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.  Google Scholar

[4]

T. Cazenave and M. J. Esteban, On the stability of stationary states for non-linear Schrödinger equations with an external magnetic field, Mat. Apl. Comput., 7 (1988), 155-168.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[5]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998. Google Scholar

[6]

T. Cazenave and F. B. Weissler, The Cauchy problem for the non-linear Schrödinger equation in $H^1$, Manuscripta Math., 61 (1988), 477-494.  doi: 10.1007/BF01258601.  Google Scholar

[7]

H. N. Chapman, Femtosecond time-delay X-ray holography, Nature, 61 (2007), 676-679.   Google Scholar

[8]

A. De Bouard, Non-linear Schrödinger equations with magnetic fields, Differential Integral Equations, 4 (1991), 73-88.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[9]

B. 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

[10]

B. Feng and X. Yuan, On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control Theory, 4 (2015), 431-445.  doi: 10.3934/eect.2015.4.431.  Google Scholar

[11]

B. Feng and D. Zhao, Optimal bilinear control of Gross-Pitaevskii equations with Coulombian potentials, J. Differential Equations, 260 (2016), 2973-2993.  doi: 10.1016/j.jde.2015.10.026.  Google Scholar

[12]

B. Feng, D. Zhao and C. Sun, On the Cauchy problem for the nonlinear Schrödinger equations with time-dependent linear loss/gain, J. Math. Anal. Appl., 416 (2014), 901-923.  doi: 10.1016/j.jmaa.2014.03.019.  Google Scholar

[13]

A. Fratalocchi and G. Ruocco, Single-molecule imaging with X-ray free electron lasers: Dream or reality? Phys. Rev. Lett., 106 (2011), 105504. doi: 10.1103/PhysRevLett.106.105504.  Google Scholar

[14]

T. Kato, On nonlinear Schrödinger equations, Ann. IHP (Phys. Theor.), 46 (1987), 113-129.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[15]

L. Michel, Remarks on non-linear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 33 (2008), 1198-1215.  doi: 10.1080/03605300801891927.  Google Scholar

[16]

Y. Nakamura and A. Shimomura, Local well-posedness and smoothing effects of strong solutions for non-linear Schrödinger equations with potentials and magnetic fields, Hokkaido Math. J., 34 (2005), 37-63.  doi: 10.14492/hokmj/1285766208.  Google Scholar

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. Google Scholar

[18]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, Springer-Verlag, New York, 1999. Google Scholar

[19]

K. Yajima, Schrödinger evolution equations with magnetic fields, J. Analyse Math., 56 (1991), 29-76.  doi: 10.1007/BF02820459.  Google Scholar

[1]

Yavar Kian, Alexander Tetlow. Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation. Inverse Problems & Imaging, 2020, 14 (5) : 819-839. doi: 10.3934/ipi.2020038
[2]

Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
[3]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639
[4]

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

Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems & Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039
[6]

Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240
[7]

Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 865-900. doi: 10.3934/dcdss.2018053
[8]

Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141
[9]

Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969
[10]

Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911
[11]

Alexander Zlotnik. The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis. Kinetic & Related Models, 2015, 8 (3) : 587-613. doi: 10.3934/krm.2015.8.587
[12]

P. Cerejeiras, U. Kähler, M. M. Rodrigues, N. Vieira. Hodge type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2253-2272. doi: 10.3934/cpaa.2014.13.2253
[13]

María Teresa Cao-Rial, Peregrina Quintela, Carlos Moreno. Numerical solution of a time-dependent Signorini contact problem. Conference Publications, 2007, 2007 (Special) : 201-211. doi: 10.3934/proc.2007.2007.201
[14]

Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041
[15]

Chan Liu, Jin Wen, Zhidong Zhang. Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation. Inverse Problems & Imaging, 2020, 14 (6) : 1001-1024. doi: 10.3934/ipi.2020053
[16]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307
[17]

Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279
[18]

Haixia Li. Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity. Evolution Equations & Control Theory, 2021, 10 (4) : 723-732. doi: 10.3934/eect.2020088
[19]

Morteza Fotouhi, Mohsen Yousefnezhad. Homogenization of a locally periodic time-dependent domain. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1669-1695. doi: 10.3934/cpaa.2020061
[20]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (251)
  • HTML views (616)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy