American Institute of Mathematical Sciences

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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  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

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