Singular solitons and spectral meromorphy

Along with regular solutions of soliton equations one usually can also construct interesting classes of singular solutions. The conditions for the compatibility of their singularities with the dynamics prescribed by the equation impose stringent restrictions on the form of the singular points. For instance, the known meromorphic solutions of the Korteweg-de Vries equation have second-order poles with respect to the space variable, and the leading coefficient is always a triangular number. Singular finite-gap solutions are an important example of this type of solution. In the case of one space dimension the eigenfunctions of the auxiliary linear operators with pole singularities that are compatible with the dynamics turn out to be also locally meromorphic for all values of the spectral parameter. This property, which will be called spectral meromorphy, makes it possible to define a natural indefinite metric on the space spanned by the eigenfunctions, and the number of negative squares of this metric is a new integral of motion. Also discussed are analogues of these results for two-dimensional problems.

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