On asymptotic behavior of solutions to cubic nonlinear Klein-Gordon systems in one space dimension

TL;DR

This paper studies the long-term behavior of solutions to cubic nonlinear Klein-Gordon systems in one dimension, classifies these systems via matrix representations, and identifies new asymptotic behaviors including solutions with slower decay rates.

Contribution

It introduces a classification framework for such systems using matrix equivalence, and explicitly reduces complex systems to model systems, revealing novel asymptotic behaviors.

Findings

Classification of systems via matrix equivalence

Explicit reduction to model systems

Discovery of solutions with slower decay rates than linear cases

Abstract

In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Klein-Gordon equations in one space dimension. We classify the systems by studying the quotient set of a suitable subset of systems by the equivalence relation naturally induced by the linear transformation of the unknowns. It is revealed that the equivalence relation is well described by an identification with a matrix. In particular, we characterize some known systems in terms of the matrix and specify all systems equivalent to them. An explicit reduction procedure from a given system in the suitable subset to a model system, i.e., to a representative, is also established. The classification also draws our attention to some model systems which admit solutions with a new kind of asymptotic behavior. Especially, we find new systems which admit a solution of which decay rate is worse than that of a solution to the linear Klein-Gordon equation by logarithmic order.