Quasi-periodic solutions for Schrödinger equation with derivative nonlinearity (|u|^2pu)_x

Abstract

In this paper, one-dimensional derivative Schrödinger equation, with periodic boundary condition is considered. It is proved that the above equation admits a Whitney smooth family of small amplitude, quasi-periodic solutions with two-dimensional Diophantine frequencies. The proof is based on infinite-dimensional Kolmogorov-Arnold-Moser (KAM) theory, partial Birkhoff normalization and scaling skills.

Keywords:

• Schrödinger equation
• quasi-periodic solution
• normal form
• KAM theory

MSC:

• 37K55
• 37N10
• 35Q35

Acknowledgements

We would like to thank the referees for their invaluable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is partially supported by NSFJS [grant number BK 20131285]; the Scientific Research Foundation of Xuzhou Institute of Technology [grant number XKY2012205]. The third author is supported by the NSFC [grant number 11371090]. The fourth author is supported by NSFC [grant number 11301072].