Periodic dynamics of a derivative nonlinear Schrödinger equation with variable coefficients

ABSTRACT

We study the existence and multiplicity of periodic waves with a nontrivial phase on the derivative nonlinear Schrödinger equation with a periodic coefficient. The existence of infinitely many periodic solutions with a nontrivial phase is proved by using the Poincaré–Birkhoff twist theorem and the method of averaging. The sequence of rotation numbers for large amplitude periodic solutions tends to infinity, while the one for small amplitude periodic solutions tends to a certain constant. Additionally, exact expressions of small amplitude periodic solutions are obtained by introducing a small parameter.

COMMUNICATED BY:

• Alexander Pankov

KEYWORDS:

• Derivative nonlinear Schrödinger equation
• variable coefficient
• coherent state
• periodic solution
• nontrivial phase

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35Q55
• 35B10

Acknowledgements

The authors wish to express their thanks to Professor Peter W. Bates, Michigan State University, for reading and improving this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is partially supported by National Nature Science Foundation of China Grant Nos. 11771105, 11662001, Guangxi Natural Science Foundation Grant Nos. 2017GXNSFFA198012, 2016GXNSFDA380031, Innovation Project of Guet Graduate Education No. 2018YJCX59 and MINECO Grant with FEDER funds MTM2017-82348-C2-1-P.