  We study the question of existence of time-periodic, spatially periodic
solutions for dispersive evolution equations, and in particular, we introduce a
framework for demonstrating the nonexistence of such solutions. We formulate
the problem so that doubly periodic solutions correspond to fixed points of a
certain operator. We prove that this operator is locally contracting, for
almost every temporal period, if the Duhamel integral associated to the
evolution exhibits a weak smoothing property. This implies the nonexistence of
nontrivial, small-amplitude time-periodic solutions for almost every period if
the smoothing property holds. This can be viewed as a partial analogue of
scattering for dispersive equations on periodic intervals, since scattering in
free space implies the nonexistence of small coherent structures. We use a
normal form to demonstrate the smoothing property on specific examples, so that
it can be seen that there are indeed equations for which the hypotheses of the
general theorem hold. The nonexistence result is thus established through the
novel combination of small divisor estimates and dispersive smoothing
estimates. The examples treated include the Korteweg-de Vries equation and the
Kawahara equation.
