Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities

TL;DR

This paper proves the existence of 2-dimensional KAM tori for a perturbed nonlinear vibrating string equation with singularities using advanced infinite-dimensional KAM theory techniques.

Contribution

It introduces a novel application of infinite-dimensional KAM theory to a singular nonlinear PDE, including partial Birkhoff normal form and Legendre basis analysis.

Findings

Existence of 2D KAM tori for the equation

Application of partial Birkhoff normal form to singular PDEs

Development of integral estimates related to Legendre basis

Abstract

{\bf Abstract}: The existence of 2-dimensional KAM tori is proved for the perturbed generalized nonlinear vibrating string equation with singularities $u_{tt}=((1-x^2)u_x)_x-mu-u^3$ subject to certain boundary conditions by means of infinite-dimensional KAM theory with the help of partial Birkhoff normal form, the characterization of the singular function space and the estimate of the integrals related to Legendre basis.