KAM theory for the Hamiltonian derivative wave equation (Q2854289)

The paper addresses the KAM theory for the Hamiltonian wave equation in the form \(y_{tt} + D^2 y + f(Dy) = 0\), where \(D = \sqrt{m^2 - \partial_x^2}\) for a constant \(m\) and a real analytic nonlinearity \(f\), which contains the cubic term and the higher-order terms (quintic or higher). The variable \(x\) is defined on the torus subject to periodic boundary conditions. The linear version of the wave equation has a family of quasi-periodic solutions. The authors prove that the nonlinear wave equation admits families of small-amplitude analytic quasi-periodic solutions with zero Lyapunov exponents along the Cantor sets in the parameter space. Such Cantor families have asymptotically full measure at the origin in the parameter space. Also, the linearized equations of motion at the families of quasi-periodic solutions are reducible to equations with constant coefficients.NEWLINENEWLINEThis work develops the ideas of the previous studies of J. Bourgain, W. Craig, and G. Wayne. Technical simplifications are performed with the use of the momentum conservation, like in the work of J. Geng and J. You. The estimates are achieved by exploiting the quasi-Toplitz property of the perturbation, similar to the recent works of M. Procesi and X. Xu as well as L. H. Eliasson and S. B. Kuksin.