Recurrent motions in certain continuum dynamical systems

Recent observations of Yuen and Ferguson on the dynamical structure of nonlinearly saturated, spatially periodic solutions of the nonlinear Schrödinger equation are given a simple analytical explanation. It is shown that many nonlinear systems can be described as effectively possessing a finite number of degrees of freedom even though the evolution equations formally relate to a continuum. Powerful theorems in general dynamics then lead to the existence of generally recurrent motions which need not, however, be quasi-periodic or even almost periodic. An explicit example is given to show that a system need not be conservative in order to exhibit recurrence. Explicit estimates of the effective number of degrees of freedom are given for the important nonlinear Schrödinger equation and the Korteweg–de Vries equation.