Existence and stability of quasi-periodic solutions for derivative wave equations (Q361766)

The note gives a short presentation of results from the preprint [the authors, ``KAM theory for the quasi-periodic solutions for reversible derivative wave equation'', Preprint, \url{arXiv:1211.2136}]. The authors obtain KAM results, namely, existence and stability of small-amplitude, analytic, quasi-periodic solutions, for a class of wave equations NEWLINE\[NEWLINE y_{tt} - y_{xx} + my = g(x, y, y_x, y_t) NEWLINE\]NEWLINE with an explicit dependence on \(x\) in the nonlinearity.NEWLINENEWLINEThe nonlinearity obeys the following conditions:NEWLINENEWLINE1) reversibility \(g(x, y, y_x, -v) = g(x, y, y_x, v)\),NEWLINENEWLINE2) parity \(g(-x, y, -y_x, v) = g(x, y, y_x, v)\),NEWLINENEWLINE3) \(g\) vanishes at least quadratically at \((y, y_x, v) = (0, 0, 0)\),NEWLINENEWLINE4) \(g\) has the form \(y y_x^2+\) higher-order terms.NEWLINENEWLINEThe mass is positive: \(m > 0\).NEWLINENEWLINEThe paper also mentions a number of non-existence results for some special cases, when these restrictions are violated.NEWLINENEWLINEThe proof is based on an abstract KAM theorem for infinite-dimensional reversible dynamical systems.