Time-dependent finite-dimensional dynamical system representation of breather solutions

TL;DR

This paper introduces a finite-dimensional dynamical system approach to analyze breather solutions of nonlinear Klein-Gordon equations, revealing geometric and rotational features underlying these solutions.

Contribution

It presents a novel finite-dimensional representation of breather solutions, enabling geometric and dynamical analysis of their behavior in nonlinear Klein-Gordon equations.

Findings

Breather solutions form a geometric object in finite-dimensional systems.

Rotational motion around fixed points is key to breather dynamics.

The approach helps understand coexistence of positive and negative parts in nonlinear systems.

Abstract

A concept of finite-dimensional dynamical system representation is introduced. Since the solution trajectory of partial differential equations are usually represented within infinite-dimensional dynamical systems, the proposed finite-dimensional representation provides decomposed snapshots of time evolution. Here we focus on analyzing the breather solutions of nonlinear Klein-Gordon equations, and such a solution is shown to form a geometrical object within finite-dimensional dynamical systems. In this paper, based on high-precision numerical scheme, we represent the breather solutions of the nonlinear Klein-Gordon equation as the time evolving trajectory on a finite-dimensional dynamical system. Consequently, with respect to the evolution of finite-dimensional dynamical systems, we confirm that the rotational motion around multiple fixed points plays a role in realizing the breather solutions. Also, such a specific feature of breather solution provides us to understand mathematical mechanism of realizing the coexistence of positive and negative parts in nonlinear systems.