Existence of quasi-periodic solutions for elliptic equations on a cylindrical domain (Q851100)

Summary: The elliptic equation \(\partial_{tt}u=-\partial_{xx}u-\alpha u-g(u)\), \(\alpha>0\), is ill-posed and `most' initial conditions lead to no solutions. Nevertheless, we show that for almost every \(\alpha\) there exist smooth solutions which are quasi-periodic. These solutions are anti-symmetric in space, and hence they are not traveling waves. Our approach uses the existence of an invariant center manifold, and the solutions are obtained from a KAM-type theorem for the restriction of the equation to that manifold.