Quasi-periodic solutions for a class of wave equation system

Abstract

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the 1D wave equation system $\{\begin{array}{l}{u}_{1\mathrm{tt}}-u_{1\mathrm{xx}}+\sigma u_1+u_1{u}_2^2=0\\ u_{2\mathrm{tt}}-u_{2\mathrm{xx}}+\mu u_2+u_1^2{u}_2=0\end{array}$under Dirichlet boundary conditions, where $0<\sigma \in [{\sigma }_1,{\sigma }_2]\subset [0,1],$ $0<\mu \in [{\mu }_1,{\mu }_2]\subset [0,1]$ are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.

Keywords:

• Wave equation system
• quasi-periodic solution
• infinite dimensional KAM theory
• normal form

MSC:

• 37K55
• 35G30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author is partially supported by National Natural Science Foundation of China [grant number 11801492].