Instability of standing waves for a weakly coupled non-linear Schrödinger system

Abstract

This article discusses the weakly coupled non-linear Schrödinger equations. With the variational characterization of the ground state solutions, the potential well argument and the concavity method, we derive a sharp condition for blow-up and global existence to the solutions of the Cauchy problem. At the same time, we also prove the instability of standing waves.

Keywords:

• weakly coupled non-linear Schrödinger equations
• standing wave
• ground state
• global existence
• blow-up

2000 AMS Subject Classifications::

• 35Q55
• 35Q60

Acknowledgements

The authors would like to thank the referee for his helpful comments. The main part of this work was done while the first author was visiting Brigham Young University in 2008. He would like to thank Professor Kening Lu and other people there for the kind hospitality during the stay. This work was partially supported by National Natural Science Foundation of China (No. 10747148, No. 10771151) and Scientific Research Fund of Sichuan Provincial Education Department.