On two-sided estimates for the nonlinear Fourier transform of KdV

Abstract

The KdV-equation ut = −uxxx + 6uux on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearizing the KdV flow. The regularity properties of u are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we obtain two-sided polynomial estimates of all integer Sobolev norms ||u||m, m 0, in terms of the weighted norms of the nonlinear Fourier transformed, which are linear in the highest order. We further obtain quantitative estimates of the nonlinear Fourier transformed in arbitrary weighted Sobolev spaces.

On the wellposedness of the KdV/KdV2 equations and their frequency maps

“…In this section we review results from [14,16,17,18,19,21,28]. In addition, we prove asymptotics of spectral quantities for potentials in Fourier Lebesgue spaces.…”

New Estimates of the Nonlinear Fourier Transform for the Defocusing NLS Equation

“…Korotyaev [17,19] obtained polynomial bounds of the Sobolev norms u m in terms of the action variables where the order of the polynomials grows factorial in m. Note that the bound in (ii) of Theorem 2 is of order 1 in the Sobolev norm ϕ m and the order of the remainder grows linearly in m. It turns out that our method can also be applied to the KdV-equation. In [25] we improve on the bounds obtained by Korotyaev in [17,19]. For NLS in weighted Sobolev spaces the qualitative relationship…”

On the Convexity of the KdV Hamiltonian

“…Standard -see e.g. [19,Lemma 1]. v Next we define for any q ∈ W and λ ∈ C \ γn =0 n 0 G n the improper integral…”

“…It is well known that the coefficients a k can be explicitly computed -see e.g. [19] for a self-contained proof.…”