Periodic and quasi-periodic solutions of a derivative nonlinear Schrödinger equation

Abstract

This paper is concerned with a one-dimensional (1D) derivative nonlinear Schrödinger equation with periodic boundary conditions

We show that above equation admits a family of real analytic quasi-periodic solutions with two Diophantine frequencies and a family of real analytic periodic solutions. The proof is based on a partial Birkhoff normal form and KAM method.

Keywords:

• derivative nonlinear Schrödinger equation
• periodic solution
• quasi-periodic solution
• KAM theory

AMS Subject Classifications:

• 37K55
• 35B10
• 35B15
• 35Q55

Acknowledgements

The author thanks the editor and the referees for several kinds of suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by the NNSF of China [11171185] and the NSF of Shandong Province [ZR2010AM013].