Asymptotic expansions and qualitative analysis of finite-dimensional models in nonlinear field theory

The methods of the qualitative theory of dynamical systems are used to establish the reasons that prevent the nonlinear wave equation $\square u =F(u)$ from having solutions that are periodic in time and self-localized in space. The correspondence between the qualitative behavior of the singular (separatrix) trajectories in the phase space and the asymptotic solutions of the nonlinear wave equation is considered.