1,068 research outputs found
Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle
Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for
the non-selfadjoint Zakharov-Shabat operator on the circle are derived using an
exact WKB method. The conditions are given in terms of the action associated
with the unit circle or the action associated with turning points following the
absence or presence of real turning points.Comment: 3 figures. Included additional references in the introductio
A unified approach to Darboux transformations
We analyze a certain class of integral equations related to Marchenko
equations and Gel'fand-Levitan equations associated with various systems of
ordinary differential operators. When the integral operator is perturbed by a
finite-rank perturbation, we explicitly evaluate the change in the solution. We
show how this result provides a unified approach to Darboux transformations
associated with various systems of ordinary differential operators. We
illustrate our theory by deriving the Darboux transformation for the
Zakharov-Shabat system and show how the potential and wave function change when
a discrete eigenvalue is added to the spectrum.Comment: final version that will appear in Inverse Problem
N-wave interactions related to simple Lie algebras. Z_2- reductions and soliton solutions
The reductions of the integrable N-wave type equations solvable by the
inverse scattering method with the generalized Zakharov-Shabat systems L and
related to some simple Lie algebra g are analyzed. The Zakharov- Shabat
dressing method is extended to the case when g is an orthogonal algebra.
Several types of one soliton solutions of the corresponding N- wave equations
and their reductions are studied. We show that to each soliton solution one can
relate a (semi-)simple subalgebra of g. We illustrate our results by 4-wave
equations related to so(5) which find applications in Stockes-anti-Stockes wave
generation.Comment: 18 pages, 1 figure, LaTeX 2e, IOP-style; More clear exposition.
Introduction and Section 5 revised. Some typos are correcte
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