Multiple normalized standing-waves solutions to the scalar non-linear Klein-Gordon equation with two competing powers

TL;DR

This paper proves the existence and stability of multiple standing-wave solutions for a one-dimensional scalar non-linear Klein-Gordon equation with competing power nonlinearities, including ground-states and symmetric minima.

Contribution

It establishes the existence of multiple normalized standing-wave solutions and analyzes their stability properties in the presence of competing nonlinearities.

Findings

Existence of standing-wave solutions in the Klein-Gordon equation.

Orbital stability of ground-state solutions.

Presence of degenerate and symmetric minima with the same charge.

Abstract

In this work we prove the existence of standing-wave solutions to the scalar non-linear Klein-Gordon equation in dimension one and the stability of the ground-state, the set which contains all the minima of the energy constrained to the manifold of the states sharing a fixed charge. For non-linearities which are combinations of two competing powers we prove that standing-waves in the ground-state are orbitally stable. We also show the existence of a degenerate minimum and the existence of two positive and radially symmetric minima having the same charge.