Stabilization of small solutions of discrete NLS with potential having two eigenvalues

ABSTRACT

We study the long-time behavior of small (in $l^2$) solutions of discrete nonlinear Schrödinger equations with potential. In particular, we are interested in the case that the corresponding discrete Schrödinger operator has exactly two eigenvalues. We show that under the nondegeneracy condition of Fermi Golden Rule, all small solutions decompose into a nonlinear bound state and dispersive wave. We further show the instability of excited states and generalized equipartition property.

Keywords:

• Discrete Schrodinger equation
• asymptotic stability of soliton
• nonlinear Fermi golden rule

2010 Mathematics Subject Classification:

• 35Q55

Acknowledgments

The author thank valuable suggestions from Scipio Cuccagna and Kenji Nakanishi. Further, he is grateful for helpful comments given by the anonymous referees to improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The author was supported by the Japan Society for the Promotion of Science (JSPS KAKENHI) [grant numbers JP15K17568, JP17H02851 and JP17H02853].