Almost-periodic Response Solutions for a forced quasi-linear Airy equation

TL;DR

This paper proves the existence of almost-periodic solutions for a quasi-linear Airy equation, demonstrating analyticity in time and space, using advanced mathematical techniques like KAM theory and pseudo-differential calculus.

Contribution

First demonstration of almost-periodic solutions for a quasi-linear PDE, combining Craig-Wayne approach with KAM reducibility and pseudo-differential calculus.

Findings

Existence of almost-periodic solutions for the quasi-linear Airy equation.

Solutions are analytic in both time and space.

Innovative use of KAM and pseudo-differential calculus techniques.

Abstract

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig-Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on ${\mathbb T}^\infty$.