# A N × N Zakharov-Shabat System with a Quadratic Spectral Parameter

Conference paper

## Abstract

We review some analytic results of the N × N Zakharov-Shabat system d*ψ*/dx = z^{2}J*ψ* + (zQ+P)*ψ*, which is a generalization of Beals-Coifman’s results on the first order system d*ψ*/dx = zJ*ψ* + Q*ψ*. We also show that for skew-Hermitian generic potentials Q,P, the scattering data has certain symmetric properties. If the scattering data has such symmetric properties, then the inverse problem is solvable. We also give several examples of evolution equations solvable by this inverse scattering transform. The global existence in time of these evolution equations is obtained if the initial data is generic and Skew-Hermitian.

## Keywords

Inverse Problem Evolution Equation Inverse Scattering Symmetric Property Alfven Wave
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