Dynamical transitions between equilibria in a dissipative Klein-Gordon lattice

TL;DR

This paper investigates the energy landscape and bifurcation structure of a dissipative Klein-Gordon lattice with a  potential, demonstrating convergence to equilibria and revealing complex topological states through analysis and simulations.

Contribution

It introduces a combined analytical and numerical framework to study equilibrium convergence, bifurcations, and topological states in a dissipative Klein-Gordon lattice.

Findings

Convergence to a single equilibrium for all initial conditions.

Existence of non-trivial topological (kink-shaped) equilibria.

Rich bifurcation structure influenced by discreteness, nonlinearity, and dissipation.

Abstract

We consider the energy landscape of a dissipative Klein-Gordon lattice with a $\phi^4$ on-site potential. Our analysis is based on suitable energy arguments, combined with a discrete version of the \L{}ojasiewicz inequality, in order to justify the convergence to a single, nontrivial equilibrium for all initial configurations of the lattice. Then, global bifurcation theory is explored, to illustrate that in the discrete regime all linear states lead to nonlinear generalizations of equilibrium states. Direct numerical simulations reveal the rich structure of the equilibrium set, consisting of non-trivial topological (kink-shaped) interpolations between the adjacent minima of the on-site potential, and the wealth of dynamical convergence possibilities. These dynamical evolution results also provide insight on the potential stability of the equilibrium branches, and glimpses of the emerging global bifurcation structure, elucidating the role of the interplay between discreteness, nonlinearity and dissipation.