On the Cauchy problem for the XFEL Schrödinger equation

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

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

* Corresponding author: Binhua Feng

Received August 2017 Published 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.
[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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