The work is devoted to the study of the following two perturbations of the Schrödinger and wave equations, respectively:
under Dirichlet boundary conditions. Here and are assumed to be polynomials, while is a periodic potential. The main goal of the work is to find almost periodic solutions of these partial differential equations and evaluate their asymptotic behaviour as . This fact is established for suitable potentials satisfying nonresonance properties. More precisely, for generic it is shown that
for all , , , and sufficiently large. Here is the Dirichlet spectrum for the operator . Moreover, for generic one has
for , is large, , . For these typical potentials, i.e. potentials satisfying (3) and (4), the first main result states the following.
Theorem. Suppose that is an even real periodic potential and is a polynomial of the form . Let be a smooth initial function for . Then the solution of (1) will be, for times , an -perturbation of the unperturbed solution with appropriate frequency adjustment. Here may be taken to be any fixed number.
For the case of the wave equation (2) the nonlinear term is assumed to be an odd polynomial function of , . Denote by and the Dirichlet spectrum and the eigenfunctions of . Setting , the author looks for a solution of (2) in the form
This solution is constructed by the meth;od developed before by the same author as a small -perturbation of the (unperturbed) solution
Under the natural assumption that tends to 0 sufficiently rapidly, for typical real analytic potentials , the existence of an almost periodic solution of (2) is established. In addition this solution satisfies the properties , (uniformly in ) is the perturbed frequency, (-unit vector in , being the space of finite sequences of integers).