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

Abstract

The defocusing NLS equation iu t = −u xx + 2|u| 2 u on the circle admits a global nonlinear

On the Well-Posedness of the Defocusing mKdV Equation Below $L^{2}$

“…on the interval [0, 2] with periodic boundary conditions. The spectral theory of this operator is well known in the case p = 2 and has been extended to the case p > 2 in [13]. One can show that the spectrum is discrete, and consists of a sequence of pairs of complex eigenvalues λ + n (ϕ) and λ − n (ϕ), when listed with algebraic multiplicities, such that…”

“…locally uniformly on F p c for any 2 p < ∞. We also consider the Dirichlet spectrum of the Zakharov-Shabat operator -see [6] for the definition in the case p = 2 and [13] for an extension to the case 2 < p < ∞. This spectrum, referred to as the Dirichlet spectrum of ϕ, is discrete and given by a sequence of eigenvalues (µ n ) n∈Z , counted with multiplicities and ordered lexicographically so that…”

“…and for ϕ ∈ W p it is defined by continuous extension. According to [13,6], for any given k ∈ Z the action I k is a real analytic function on W p . The Dirichlet eigenvalues and the discriminant can be used to construct the angles θ k (ϕ), k ∈ Z, which are conjugated to the actions I n (ϕ), n ∈ Z.…”

“…(We point out that in [6] the meaning of z ± k is a different one.) It is shown in [13,6] that the mappings F p r \ Z k → C, ϕ → z ± k (ϕ) analytically extend to a neighborhood W p , (after possibly shrinking W p if necessary). The Birkhoff map is then defined as follows…”

“…Standard -see e.g. [13]. v By Lemma 3.2, for any ϕ ∈ W p , 2 p < ∞, the quotient (24) is analytic in both variables (λ, ψ) on (C\ m∈Z U n )×V ϕ and analytic in λ on C\ γm =0 G m .…”