391 research outputs found
Spectral conservation laws for periodic nonlinear equations of the Melnikov type
We consider the nonlinear equations obtained from soliton equations by adding
self-consistent sources. We demonstrate by using as an example the
Kadomtsev-Petviashvili equation that such equations on periodic functions are
not isospectral. They deform the spectral curve but preserve the multipliers of
the Floquet functions. The latter property implies that the conservation laws,
for soliton equations, which may be described in terms of the Floquet
multipliers give rise to conservation laws for the corresponding equations with
self-consistent sources. Such a property was first observed by us for some
geometrical flow which appears in the conformal geometry of tori in three- and
four-dimensional Euclidean spaces (math/0611215).Comment: 16 page
Faddeev eigenfunctions for two-dimensional Schrodinger operators via the Moutard transformation
We demonstrate how the Moutard transformation of two-dimensional Schrodinger
operators acts on the Faddeev eigenfunctions on the zero energy level and
present some explicitly computed examples of such eigenfunctions for smooth
fast decaying potentials of operators with non-trivial kernel and for deformed
potentials which correspond to blowing up solutions of the Novikov-Veselov
equation.Comment: 11 pages, final remarks are adde
The Novikov-Veselov Equation and the Inverse Scattering Method, Part I: Analysis
The Novikov-Veselov (NV) equation is a
(2+1)-dimensional nonlinear evolution equation that generalizes the
(1+1)-dimensional Korteweg-deVries (KdV) equation. Solution of the NV
equation using the inverse scattering method has been discussed in the
literature, but only formally (or with smallness assumptions in case of nonzero
energy) because of the possibility of exceptional points, or singularities in
the scattering data. In this work, absence of exceptional points is proved at
zero energy for evolutions with compactly supported, smooth and rotationally
symmetric initial data of the conductivity type:
with a strictly positive function
. The inverse scattering evolution is shown to be well-defined,
real-valued, and preserving conductivity-type. There is no smallness assumption
on the initial data
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