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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. |
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