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

Abstract

The existence of 2-dimensional KAM tori is proved for the perturbed generalized nonlinear vibrating string equation with singularities \begin{document} $u_{tt}=((1-x^2)u_x)_x-mu-u^3$ \end{document} 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.