The Existence of Periodic Solutions for Nonlinear Beam Equations on $\mathbb T^d$ by a Para-differential Method

TL;DR

This paper establishes the existence of periodic solutions for nonlinear beam equations on multi-dimensional tori using para-differential conjugation and iterative methods, under non-resonance conditions.

Contribution

It introduces a para-differential approach to construct periodic solutions for nonlinear beam equations on $\

Findings

Periodic solutions exist for a large set of frequencies.

Non-resonant conditions enable the construction of solutions.

The method applies to equations on multi-dimensional tori.

Abstract

This paper focuses on the construction of periodic solutions of nonlinear beam equations on the $d$-dimensional tori. For a large set of frequencies, we demonstrate that an equivalent form of the nonlinear equations can be obtained by a para-differential conjugation. Given the non-resonant conditions on each finite dimensional subspaces, it is shown that the periodic solutions can be constructed for the block diagonal equation by a classical iteration scheme.