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

Chengming Cao; Xiaoping Yuan

doi:10.3934/dcds.2017079

Article Contents

2017, Volume 37, Issue 4: 1867-1901. Doi: 10.3934/dcds.2017079

This issue Previous Article Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity Next Article An existence proof of a symmetric periodic orbit in the octahedral six-body problem

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

Chengming Cao and
Xiaoping Yuan

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Author Bio: E-mail address: 12110180001@fudan.edu.cn; E-mail address: xpyuan@fudan.edu.cn

Received: September 30, 2015

Revised: October 31, 2016

Published: May 2017

Supported by NNSFC 11271076 and NNSFC 11121101.

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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.

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Mathematics Subject Classification: 37K55.

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