Massive Cantor families of periodic solutions of resonant Klein-Gordon equation on $\mathbb{S}^3$

TL;DR

This paper establishes the existence of numerous small amplitude, time-periodic solutions for a resonant cubic Klein-Gordon equation on the 3-sphere, using advanced analytical methods to handle resonance and small divisors.

Contribution

It introduces a novel approach combining Lyapunov-Schmidt and Nash-Moser schemes to construct Cantor families of solutions for the resonant Klein-Gordon equation on 3, with results valid for a large set of frequencies.

Findings

Proved existence of Cantor families of solutions.

Constructed solutions using Lyapunov-Schmidt and Nash-Moser methods.

Achieved results for a full measure set of frequencies near 1.

Abstract

We prove existence and multiplicity of Cantor families of small amplitude analytic in time periodic solutions of the completely resonant cubic nonlinear Klein-Gordon equation on $\mathbb{S}^3$ for an asymptotically full measure set of frequencies close to 1. The solutions are constructed by a Lyapunov-Schmidt decomposition and a Nash-Moser iterative scheme. We first find non-degenerate solutions of the Kernel equation. Then we solve the Range equation with a Nash-Moser iterative scheme to overcome small divisors problems.