On invariant tori of full dimension for 1D periodic NLS

Abstract

Consider the NLS with periodic boundary conditions in 1D (0.1)iut+Δu+Mu±ɛu|u|4=0, where M is a random Fourier multiplier defined by (0.2)Mu^(n)=Vnu^(n) and (Vn)n∈Z are independently chosen in [-1,1]. The quintic nonlinearity in (0.1) is unimportant and may be replaced by u|u|p-2,p∈2Z,p⩾4. We give a proof of the following fact. Theorem. For appropriate M, (0.1) has an invariant tori T (of full dimension) satisfying 12e-r|n|<|qn|<2e-r|n|(n∈Z,q∈T) (r>0 is arbitrary). Remark. The statement holds in fact for most (Vn)n∈Z∈[-1,1]Z, although not explicitly proven here. Written in Fourier modes (qn)n∈Z, the Hamiltonian corresponding to (0.1) is given by (0.3)H(q,q¯)=∑(n2+Vn)|qn|2+ɛ∑n1-n2+n3-n4+n5-n6=0qn1q¯n2qn3q¯n4qn5q¯n6. The proof of Theorem 1 will proceed along the ‘usual’ KAM scheme where the perturbation is eventually removed by consecutive canonical transformations of phase space. The most relevant literature in the present context of an infinite dimensional phase space are the papers of Frohlich et al. [Frohlich, Spencer, Wayne, Localization in disordered, nonlinear dynamical systems, J. Statist. Phys. 42 (1986) 247–274] and especially Poschel [Poschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393] on disordered systems. Both [Frohlich, Spencer, Wayne, Localization in disordered, nonlinear dynamical systems, J. Statist. Phys. 42 (1986) 247–274, Poschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393] consider Hamiltonians with short-range interactions and hence these results do not apply to our problem. It turns out, however that the scheme, as elaborated on in great detail in [Poschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393], is still applicable to (0.3), due to special arithmetical features as will be explained in the next section. Roughly speaking, the key point is the following observation. Let (ni) be a finite set of modes, |n1|⩾|n2|⩾⋯ and (0.4)n1-n2+n3-⋯=0. In the case of a ‘near’ resonance, there is also a relation (0.5)n12-n22+n32-⋯=o(1). Unless n1=n2, one may then control |n1|+|n2| from (0.4), (0.5) by ∑j⩾3|nj|. This feature is specifically 1-dimensional and we do not know at this time how to prove a 2D-analogue of Theorem 1, considering for instance the cubic NLS iut+Δu±u|u|2=0 on T2. It should also be pointed out that almost periodic solutions on a full set of frequencies for NLS and NLW in 1D were constructed in earlier works (see [Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrodinger and wave equations, GAFA 6 (2) (1996) 201–230] and [Poschel, On the construction of almost periodic solutions for nonlinear Schrodinger equations, Ergodic Theory Dynamical Systems 22 (5) (2002) 1537–1559]). These invariant tori (of full dimension) were obtained by successive small perturbations of finite-dimensional tori, resulting in very strong compactness properties and in fact a nonexplicit decay rate of the action variables In for n→∞. On the other hand, the construction in this paper (similarly to [Poschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393]) treats all Fourier modes at once and requires explicit and realistic decay conditions. The multiplier M=(Vn) in (0.3) is to be considered as a parameter and (0.1) a parameter-dependent equation. The role of this parameter is essential to ensure appropriate nonresonance properties of the (modulated) frequencies along the iteration. In the absence of exterior parameters, these conditions need to be realized from amplitude–frequency modulation and suitable restriction of the action-variables. This problem is harder. Indeed, a fast decay of the action-variables (enhancing convergence of the process) allows less frequency modulation and worse small divisors (cf. [Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDE, J. Anal. Math. 80 (2000) 1–35]).