Quasi-periodic solutions to nonlinear beam equation on Compact Lie Groups with a multiplicative potential

TL;DR

This paper proves the existence of quasi-periodic solutions for nonlinear beam equations with multiplicative potential on various compact Lie groups and manifolds, using a Nash-Moser iterative approach.

Contribution

It extends the existence results of quasi-periodic solutions to nonlinear beam equations to a broad class of compact Lie groups and manifolds, including non-torus cases.

Findings

Existence of quasi-periodic solutions on compact Lie groups and homogeneous manifolds.

Application of Nash-Moser scheme to nonlinear beam equations.

Results include standard tori, SO(d), SU(d), spheres, and Grassmannians.

Abstract

The goal of this work is to study the existence of quasi-periodic solutions in time to nonlinear beam equations with a multiplicative potential. The nonlinearities are required to only finitely differentiable and the frequency is along a pre-assigned direction. The result holds on any compact Lie group or homogenous manifold with respect to a compact Lie group, which includes the standard torus $\mathbf{T}^{d}$, the special orthogonal group $SO(d)$, the special unitary group $SU(d)$, the spheres $S^d$ and the real and complex Grassmannians. The proof is based on a differentiable Nash-Moser iteration scheme.