Almost Periodic Behavior of Nonlinear Waves

Abstract

This chapter discusses the periodic behaviour of nonlinear waves. Theoretical results concerning three nonlinear systems have a bearing on the questions raised by FPU. Each of the systems discussed has an unusually large number of integrals, that is, functional that are conserved during motion; this might explain why some numerically computed trajectories of these systems seem to be confined to such an unexpectedly small portion of phase space. It should be added that it is not known whether the FPU system has any integrals other than total momentum and energy, although the contrary has not been demonstrated either. This shows that there must be an additional mechanism at work. The first of the systems is Hamiltonian and completely integrable; accumulating evidence indicates that so is the second example, with infinitely many degrees of freedom. The chapter also discusses a method for constructing nonlinear systems with many integrals. In any concrete representation of the operator U(t)-1L(t) U(t)=L(0), L appears as an integral or differential operator, described in terms of coefficients. If the operators L are symmetric or hermitean symmetric, then similarity implies unitary equivalence.