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.

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. AntonelliA. AthanassoulisH. 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. AntonelliA. AthanassoulisZ. 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. FengD. 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. AntonelliA. AthanassoulisH. 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. AntonelliA. AthanassoulisZ. 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. FengD. 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]

Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705

[2]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639

[3]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903

[4]

Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240

[5]

Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 865-900. doi: 10.3934/dcdss.2018053

[6]

Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141

[7]

Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969

[8]

Alexander Zlotnik. The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis. Kinetic & Related Models, 2015, 8 (3) : 587-613. doi: 10.3934/krm.2015.8.587

[9]

Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911

[10]

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 & Applied Analysis, 2014, 13 (6) : 2253-2272. doi: 10.3934/cpaa.2014.13.2253

[11]

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

[12]

Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041

[13]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307

[14]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1

[15]

Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic & Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011

[16]

Wen-Hung Wu, Yunqiang Yin, Wen-Hsiang Wu, Chin-Chia Wu, Peng-Hsiang Hsu. A time-dependent scheduling problem to minimize the sum of the total weighted tardiness among two agents. Journal of Industrial & Management Optimization, 2014, 10 (2) : 591-611. doi: 10.3934/jimo.2014.10.591

[17]

Mourad Choulli, Yavar Kian. Stability of the determination of a time-dependent coefficient in parabolic equations. Mathematical Control & Related Fields, 2013, 3 (2) : 143-160. doi: 10.3934/mcrf.2013.3.143

[18]

Feng Zhou, Chunyou Sun, Xin Li. Dynamics for the damped wave equations on time-dependent domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1645-1674. doi: 10.3934/dcdsb.2018068

[19]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations & Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015

[20]

Stephen Anco, Maria Rosa, Maria Luz Gandarias. Conservation laws and symmetries of time-dependent generalized KdV equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 607-615. doi: 10.3934/dcdss.2018035

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (46)
  • HTML views (521)
  • Cited by (0)

Other articles
by authors

[Back to Top]