Dispersive estimate for quasi-periodic Klein-Gordon equation on 1-d lattices

TL;DR

This paper proves dispersive estimates for solutions to the 1-d lattice Klein-Gordon equation with quasi-periodic potentials, which is essential for understanding long-term behavior and well-posedness of the nonlinear system.

Contribution

It establishes dispersive estimates for the Klein-Gordon equation on 1-d lattices with quasi-periodic potentials, extending understanding to Diophantine frequency conditions.

Findings

Dispersive estimates hold under quasi-periodic potentials with Diophantine frequencies.

Results are valid when the potential is close to positive constants.

The work advances the analysis of long-term behavior for lattice Klein-Gordon equations.

Abstract

The dispersive estimate plays a pivotal role in establishing the long-term behavior of solutions to the nonlinear equation, thereby being crucial for investigating the well-posedness of the equation.In this work we prove that the solutions to Klein-Gordon equation on 1-d lattices follow the dispersive estimate provided that potential is quasi-periodic with Diophantine frequencies and closed to positive constants.