-
Previous Article
A perturbed fourth order elliptic equation with negative exponent
- DCDS-B Home
- This Issue
-
Next Article
Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ)
On the Cauchy problem for the XFEL Schrödinger equation
| 1. | Department of Mathematics, Northwest Normal University, Lanzhou 730070, China |
| 2. | School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China |
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.
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. |
| [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. |
| [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. |
| [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. |
| [5] |
T. Cazenave and A. Haraux,
An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998. |
| [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. |
| [7] |
H. N. Chapman,
Femtosecond time-delay X-ray holography, Nature, 61 (2007), 676-679.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
T. Kato,
On nonlinear Schrödinger equations, Ann. IHP (Phys. Theor.), 46 (1987), 113-129.
doi: 10.1016/j.jde.2003.12.002. |
| [15] |
L. Michel,
Remarks on non-linear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 33 (2008), 1198-1215.
doi: 10.1080/03605300801891927. |
| [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. |
| [17] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. |
| [18] |
C. Sulem and P. L. Sulem,
The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, Springer-Verlag, New York, 1999. |
| [19] |
K. Yajima,
Schrödinger evolution equations with magnetic fields, J. Analyse Math., 56 (1991), 29-76.
doi: 10.1007/BF02820459. |
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. |
| [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. |
| [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. |
| [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. |
| [5] |
T. Cazenave and A. Haraux,
An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998. |
| [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. |
| [7] |
H. N. Chapman,
Femtosecond time-delay X-ray holography, Nature, 61 (2007), 676-679.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
T. Kato,
On nonlinear Schrödinger equations, Ann. IHP (Phys. Theor.), 46 (1987), 113-129.
doi: 10.1016/j.jde.2003.12.002. |
| [15] |
L. Michel,
Remarks on non-linear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 33 (2008), 1198-1215.
doi: 10.1080/03605300801891927. |
| [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. |
| [17] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. |
| [18] |
C. Sulem and P. L. Sulem,
The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, Springer-Verlag, New York, 1999. |
| [19] |
K. Yajima,
Schrödinger evolution equations with magnetic fields, J. Analyse Math., 56 (1991), 29-76.
doi: 10.1007/BF02820459. |
| [1] |
Yavar Kian, Alexander Tetlow. Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation. Inverse Problems and Imaging, 2020, 14 (5) : 819-839. doi: 10.3934/ipi.2020038 |
| [2] |
Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]