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.
Citation: |
[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, et al., 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.![]() ![]() ![]() |