Stability and instability of standing-wave solutions to one dimensional quadratic-cubic Klein-Gordon equations

TL;DR

This paper investigates the stability properties of standing-wave solutions in a one-dimensional quadratic-cubic Klein-Gordon equation, analyzing how these solutions behave under perturbations using semigroup methods.

Contribution

It provides a detailed analysis of the stability and instability of standing-waves for this specific nonlinear Klein-Gordon model, introducing new techniques for orbit construction.

Findings

Identification of conditions for stability and instability.

Characterization of the orbit structure of solutions.

Application of semigroup methods to analyze solution dynamics.

Abstract

We study the stability of standing-waves solutions to a scalar non-linear Klein-Gordon equation in dimension one with a quadratic-cubic non-linearity. Orbits are obtained by applying the semigroup generated by the negative complex unit multiplication on a critical point of the energy constrained to the charge.