CONSTRUCTION OF APPROXIMATIVE AND ALMOST-PERIODIC SOLUTIONS OF PERTURBED LINEAR SCHRODINGER AND WAVE-EQUATIONS

Consider 1D nonlinear Schrodinger equation iu(t) - u(xx) + V (x)u + (e )psilon partial derivative h/partial derivative (U) over bar = 0 and n onlinear wave equation y(tt) - y(xx) + rho y + epsilon F'(y) = 0 under Dirichlet boundary conditions. We assume here H (u, (u) over bar) and F(y) polynomials. It is proved that for ''typical'' periodic potentia l V in (0.1) and typical rho is an element of R in (0.2) the following is true. Let u(0) (resp. y(0), y'(0)) be smooth initial data for t = 0. Then the corresponding solution u(t) of (0.1) (resp. u(t) of (0.2)) will be epsilon(1/2)-close to the unperturbed solution (with appropri ate frequency adjustment), for times \t\ epsilon(-M) where M may be an y chosen number (letting epsilon --> 0) (See Prop. 4.18 and Prop. 5.13 ). This result may be seen as a Nekhoroshev type result (cf. [N]) for Hamiltonian PDE, in the nonresonant regime (which is the easiest to st udy.) In this spirit, results in finite dimensional phase space have b een obtained by various authors but for different interactions, essent ially of finite range, which does not cover natural PDE models. See fo r instance [BFG]. We started here to investigate this phenomenon in th e PDE context. In the second part of the paper, we use the technique f rom [Bo] (see also relevant references [Bo] on the earlier work such a s [CrW]) to construct almost periodic (in time) solutions of say a wav e equation y(tt) - y(xx) + V(x)y +epsilon F'(y) = 0 under Dirichlet bo undary conditions. Here V is a ''typical'' real analytic periodic pote ntial. The frequencies of these solutions form a full set, i.e. lambda '(j) approximate to lambda(j) = root u(j) where {u(j)} is the Dirichle t spectrum of -d2/dr2 + V(x). However, they are obtained starting from an unperturbed solution u(o)(x,t) = Sigma(j=1)(infinity) a(j) cos lam bda(j)t.phi(j)(x), subject to a strong decay assumption \a(j)\ --> 0 o n the initial amplitudes {a(j)}. The argument would need to be conside rably refined to reach a more realistic decay. Again, the construction of invariant tori of infinite dimension (via usual KAM techniques) is achieved for certain models with finite range interaction (see [FSW]) . There are also the results of [CP], but they require a very rapidly increasing frequency sequence {lambda(j)}.