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The inverse scattering transform and squared eigenfunctions for the nondegenerate $3 \times 3$ operator and its soliton structure. (English) Zbl 1204.47011
The authors develop a soliton perturbation theory for the non-degenerate $3\times 3$ eigenvalue operator, with obvious applications to the three-wave resonant interaction. The key elements of an inverse scattering perturbation theory for integrable systems are the squared eigenfunctions and their adjoints. These functions serve as a mapping between variations in the potentials and variations in the scattering data. The authors also address the problem of the normalization of the Jost functions, how this affects the structure and solvability of the inverse scattering equations and the definition of the scattering data. They explicitly provide the construction of the covering set of squared eigenfunctions and their adjoints, in terms of the Jost functions of the original eigenvalue problem. The authors also obtain, by a new and direct method due to {\it J.-K.\thinspace Yang} and {\it D. J.\thinspace Kaup} [J. Math. Phys. 50, No. 2, Paper No. 023504 (2009; Zbl 1202.35275)], the inner products and closure relations for these squared eigenfunctions and their adjoints. With this universal covering group, one would have tools to study the perturbations for any integrable system whose Lax pair contained the non-degenerate $3\times 3$ eigenvalue operator, such as that found in the Lax pair of the integrable three-wave resonant interaction.

##### MSC:
 47A40 Scattering theory of linear operators 35P25 Scattering theory (PDE) 58J50 Spectral problems; spectral geometry; scattering theory 37K40 Soliton theory, asymptotic behavior of solutions 47A75 Eigenvalue problems (linear operators) 34L25 Scattering theory, inverse scattering (ODE) 35Q51 Soliton-like equations
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