Stable and unstable periodic orbits in the one dimensional lattice $\phi^4$ theory

TL;DR

This paper systematically constructs and analyzes the stability of periodic orbits in the one-dimensional $\, ext{ extphi}^4$ lattice theory, clarifying which normal modes extend to the non-linear regime and linking stability to resonance and Lyapunov spectra.

Contribution

It provides a systematic method to construct and analyze the stability of periodic orbits in $\, ext{ extphi}^4$ lattice theory, extending normal modes from linear to non-linear regimes.

Findings

Identified which normal modes extend to non-linear theory.

Linked stability of orbits to parametric resonance and Lyapunov spectra.

Applied analysis to theories with different on-site potentials.

Abstract

Periodic orbits for the classical $\phi^4$ theory on the one dimensional lattice are systematically constructed by extending the normal modes of the harmonic theory, for periodic, fixed and free boundary conditions. Through the process, we clarify, in general, which normal modes of the linear theory can or can not be extended to the full non-linear theory. We then analyze the stability of these orbits, clarifying the link between the stability, parametric resonance and the Lyapunov spectra for these orbits.The construction of the periodic orbits and the stability analysis is applicable to theories governed by Hamiltonians with quadratic inter-site potentials and a general on-site potential. We also apply the analysis to theories with on-site potentials that have qualitatively different behavior from the $\phi^4$ theory, with some concrete examples.