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By Francesco (i-cagl-mi) Van Der Mee


Marchenko equations and norming constants of the matrix Zakharov-Shabat system. (English summary) Oper. Matrices 2 (2008), no. 1, 79–113. The Zakharov-Shabat system was introduced in 1971 in connection with the inverse scattering transform for the nonlinear Schrödinger equation. It was later generalized to matrix equations and extended to other nonlinear evolution equations solvable by inverse scattering. The authors address the inverse scattering transform for the matrix Zakharov-Shabat system, which relies on the Marchenko integral equation and on an appropriate definition of the scattering data. The scattering data include information about the discrete spectrum, which may include multiple eigenvalues, unlike in the standard case of a linear Schrödinger equation. The authors investigate proper normalization of the scattering data in the case of multiple eigenvalues. Compared to previous papers, the authors do not assume that the geometric and algebraic multiplicities are equal, nor do they assume that the potential has a compact support. The potentials are assumed to be L 1 only, and it is also assumed that the accumulation of discrete eigenvalues on the continuous spectrum does not occur. General results are extended to the focusing case with additional symmetry properties. Reviewed by Dmitry E. Pelinovsk

Year: 2010
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