1,003 research outputs found
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
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 and be complex Banach couples and assume that
with norms satisfying for
some . For any , denote by
and the complex interpolation spaces and by
, the open ball of radius in
, centered at zero. Then for any analytic map such that and
are continuous and bounded by constants and , respectively, the
restriction of to , is
shown to be a map with values in which is analytic and bounded by
Birkhoff Coordinates for the Focusing NLS Equation
In this paper we construct Birkhoff coordinates for the focusing nonlinear Schrödinger equation near the zero solutio
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