Theory of an Equation with the Fermi-Pasta-Ulam Nonlinearity

We develop a full theory of an equation with the Fermi-Pasta-Ulam nonlinearity. It includes finding families of periodic solutions and of traveling solutions, creating the infinite-dimensional analog of the Kolmogorov-Moser-Arnol'd theory (the KAM) theory for families of quasiperiodic solutions with continual and countable and finite number of rational independent frequencies. We also prove stability of stationary solutions. The last statement contradicts to the Arnol'd hipothesis on unstability in Hamiltonian systems.