On the Cauchy problem for the XFEL Schrödinger equation

Binhua Feng; Dun Zhao

doi:10.3934/dcdsb.2018131

Article Contents

2018, Volume 23, Issue 10: 4171-4186. Doi: 10.3934/dcdsb.2018131

This issue Previous Article Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ) Next Article A perturbed fourth order elliptic equation with negative exponent

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
* Corresponding author: Binhua Feng

Received: July 31, 2017

Early access: April 2018

Published: September 2018

This work is supported by NSFC Grants (No. 11601435, No. 11475073), Gansu Provincial Natural Science Foundation (1606RJZA010) and NWNU-LKQN-14-6.

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Abstract

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.

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Mathematics Subject Classification: 35Q51, 35Q55.

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References

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