Construction of approximative and almost periodic solutions of perturbed linear schrödinger and wave equations

Summary

Consider 1D nonlinear Schrödinger equation

$$iu_t - u_{xx} + V(x)u + \varepsilon \frac{{\partial {\rm H}}}{{\partial \bar u}} = 0$$

((0.1))

and nonlinear wave equation

$$y_{tt} - y_{xx} + \rho y + \varepsilon F'(y) = 0$$

((0.2))

under Dirichlet boundary conditions. We assume hereH(u,ū) andF(y) polynomials.

It is proved that for “typical” periodic potentialV in (0.1) and typical ρ∈R in (0.2) the following is true. Letu(0) (resp.y(0),y′(0)) be smooth initial data fort=0. Then the corresponding solutionu(t) of (0.1) (resp.y(t) of (0.2)) will be ɛ^1/2-close to the unperturbed solution (with appropriate frequency adjustment), for times |t|<ɛ^−M whereM may be any chosen number (letting ɛ → 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 study). In this spirit, results in finite dimensional phase space have been obtained by various authors but for different interactions, essentially of finite range, which does not cover natural PDE models. See for instance [BFG]. We started here to investigate this phenomenon in the PDE context.

In the second part of the paper, we use the technique from [Bo] (see also relevant references in [Bo] on earlier work such as [CrW]) to construct almost periodic (in time) solutions of say a wave equation

$$y_{tt} - y_{xx} + V(x)y + \varepsilon F'(y) = 0$$

under Dirichlet boundary conditions. HereV is a “typical” real analytic periodic potential. The frequencies of these solutions form a full set, i.e.$\lambda '_j \approx \lambda _j = \sqrt {\mu _j }$ where {μ _j } is the Dirichlet spectrum of$- \frac{{d^2 }}{{dx^2 }} + V(x)$. However, they are obtained starting from an unperturbed solutionu ₀(x, t)=Σ^∞ _j=1 a _j cos λ _j t.ϕ _j (x), subject to a strong decay assumption |a _j | → 0 on the initial amplitudes {a _j }. The argument would need to be considerably 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 {λ _j }.