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The inverse scattering transform and squared eigenfunctions for a degenerate $3×3$ operator. (English) Zbl 1178.35382
Summary: We present the covering set of the squared eigenfunctions for a degenerate $3×3$ eigenvalue problem. The derivation follows the approach recently outlined by us on this same equation [J. Math. Phys. 50, No. 2, Paper No. 023504 (2009; Zbl 1202.35275)]. This eigenvalue problem is important since it serves as the spectral problem for the inverse scattering transform (IST) of the vector NLS equation, the Sasa-Satsuma equation, and a degenerate two-level Maxwell-Bloch system. The use of this covering set allows one to treat the linear perturbations of these equations in a common and systematic manner. Comparison with previous results on the perturbed continuous spectrum of the Sasa-Satsuma equation is made.
##### MSC:
 35R30 Inverse problems for PDE 35P25 Scattering theory (PDE) 81U40 Inverse scattering problems (quantum theory)