Integration of the modified Korteweg–de Vries equation of negative order in the case of multiple eigenvalues

Abstract The modified Korteweg–de Vries equation of negative order is investigated by the inverse scattering transform method in the case where the corresponding non-self-adjoint Dirac-type operator has a finite number of discrete multiple eigenvalues of a given multiplicity. The evolution laws of the scattering data, including the normalization chains corresponding to the multiple eigenvalues, are derived. A representation of the solution to the Cauchy problem is constructed based on the Gelfand–Levitan–Marchenko integral equations adapted to the degenerate spectral structure.

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