Perturbative, spatially–periodic solutions of the Korteweg–deVries, the modified Korteweg–deVries, and the nonlinear Schro¨dinger equations are shown to be recurrent and nonstochastic, densely covering parts of the phase space bounded by level surfaces of the constants of motion. The connection of this result with the numerical phenomena of recurrences and the slow randomization of nonlinear systems is discussed.