Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables

We consider the KdV equation on the Sobolev space of periodic distributions. We obtain estimates of the solution of the KdV in terms of action variables.


Introduction and main results
Consider the KdV equation on the Sobolev space of zero-meanvalue 1-periodic distributions H −1 = {ψ = q ′ : q ∈ H}, where the real Hilbert space H = H 0 consists of zero-meanvalue functions q ∈ L 2 (T), T = R/Z. The space H −1 is equipped with the norm ψ 2 −1 = q 2 = 1 0 q 2 (x)dx for ψ = q ′ ∈ H −1 . The initial value problem for KdV in the phase space of periodic distributions was solved by Kappeler and Topalov [KT], see also [B], [CT]. That problem in various Sobolev spaces was studied by many authors, see references in [B], [CT], [KT]. The action-angle variables for the periodic KdV are studied by Veselov-Novikov [VN], Kuksin [Ku], Kappeler-Pöschel [KP]. The action-angle variables for the case ψ ∈ H −1 were constructed by Kappeler-Möhr-Topalov [KMT] and were essentially used in [KT]. In Sobolev spaces estimates for the potential ψ and for the KdV-Hamiltonian in terms of the action variables were obtained by Korotyaev [K2].
We describe the motivation of the present paper. Introduce the real Hilbert spaces ℓ 2 m , m ∈ R, of sequences (f n ) ∞ 1 , equipped with the norm f 2 m = n 1 (2πn) 2m f 2 n . Recall that the KDV equation on H −1 admits action-angle variables A n 0, φ n ∈ [0, 2π), n 1 such that (see [KT]): (1) for each q ∈ H −1 there exist actions A n 0 such that n 1 An n < ∞ and angles φ n ∈ [0, 2π), n 1.
For the case of periodic distributions no estimates for the potential q in terms of action variables A n were known. Our main goal in this paper is to obtain them.
We formulate our main result.
Corollary 1.2. Let ψ(x, t) be a solution of (1.1) such that ψ(·, 0) ∈ H −1 . Then for all time t the following estimates hold true: Example. We now discuss relation of estimates (1.6) with the inverse cascade of energy in the KdV equation. Let an initial condition ψ(·, 0) ∈ H 0 satisfies Then for any N 1 and every t estimate (1.6) yields ψ(·, t) −1 6ε, P N ψ(·, t) 6(2πN)ε, (1.9) where P N f, f ∈ H 0 is given by P N f = |n| N e i2πnx 1 0 f (s)e −i2πns ds. Let in addition, δ = 6(2πN)ε be small enough. Then (1.9) gives (1.10) Thus we deduce that in our case the inverse cascade of energy is impossible. It means that if the initial condition is such that ψ(·, 0) = 1 and P N 0 ψ(·, 0) = 0 for some N 0 ≫ N, then P N ψ(·, t) 6N N 0 will be small for all time t, since ψ(·, 0) −1 1 2πN . That is, if the energy of a solution was initially concentrated in high modes, then a substantial part of the energy cannot flow to low modes.
Note that the function ψ(x, 0) with the property (1.8) may be a finite trigonometric polynomial.
5) The function k(z) maps a horizontal cut (a "gap" ) g n onto a vertical cut Γ n and a spectral band σ n onto the segment [π(n − 1), πn] for all ±n ∈ N.
The heights h n , n 1 are so-called Marchenko-Ostrovski parameters [MO1]. In spirit, such result goes back to the classical Hilbert Theorem (for a finite number of cuts, see e.g. [J]) in the conformal mapping theory. A similar theorem for the Hill operator is technically more complicated (there is a infinite number of cuts) and was proved by Marchenko-Ostrovski [MO1] for the case ψ ∈ H. For additional properties of the conformal mapping we also refer to our previous papers [K1]- [K6]. Note that the inverse problems for the operator H with ψ ∈ H −1 in terms of the Marchenko-Ostrovski parameters h n , n 1 and gap-lengths were solved by Korootyev in [K3].
Theorem 2.1. The Riccati map R : H → H given by p → q = R(p), q ′ = p ′ (x) + p 2 (x) − p 2 is a real analytic isomorphism of H onto itself. Moreover, the following estimates hold true: A n πn = P −1 . (2.12)