On the symplectic phase space of KdV

We prove that the Birkhoff map \Om for KdV constructed on H^{-1}_0(\T) can be interpolated between H^{-1}_0(\T) and L^2_0(\T). In particular, the symplectic phase space H^{1/2}_0(\T) can be described in terms of Birkhoff coordinates. As an application, we characterize the regularity of a potential q\in H^{-1}(\T) in terms of the decay of the gap lengths of the periodic spectrum of Hill's operator on the interval $[0,2]$

Solutions of mKdV in classes of functions unbounded at infinity

In 1974 P. Lax introduced an algebro-analytic mechanism similar to the Lax L-A pair. Using it we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and may even include functions which tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schr{\"o}dinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.Comment: 35 pages, new results about spectra and eigenfunctions of Schr\"odinger operators added, new references adde

Interpolation of nonlinear maps

Let $(X_0, X_1)$ and $(Y_0, Y_1)$ be complex Banach couples and assume that $X_1\subseteq X_0$ with norms satisfying $\|x\|_{X_0} \le c\|x\|_{X_1}$ for some $c > 0$. For any $0<\theta <1$, denote by $X_\theta = [X_0, X_1]_\theta$ and $Y_\theta = [Y_0, Y_1]_\theta$ the complex interpolation spaces and by $B(r, X_\theta)$, $0 \le \theta \le 1,$ the open ball of radius $r>0$ in $X_\theta$, centered at zero. Then for any analytic map $\Phi: B(r, X_0) \to Y_0+ Y_1$ such that $\Phi: B(r, X_0)\to Y_0$ and $\Phi: B(c^{-1}r, X_1)\to Y_1$ are continuous and bounded by constants $M_0$ and $M_1$, respectively, the restriction of $\Phi$ to $B(c^{-\theta}r, X_\theta)$, $0 < \theta < 1,$ is shown to be a map with values in $Y_\theta$ which is analytic and bounded by $M_0^{1-\theta} M_1^\theta$

Birkhoff Coordinates for the Focusing NLS Equation

In this paper we construct Birkhoff coordinates for the focusing nonlinear Schrödinger equation near the zero solutio