On an inverse problem associated with an integrable equation of Camassa–Holm type: explicit formulas on the real axis

Abstract

The inverse problem which arises in the study of the integrable PDE proposed by V Novikov is solved for a class of discrete densities. The method of solution relies on the use of Cauchy biorthogonal polynomials. The explicit formulas are obtained directly from the analysis on the real axis without any additional transformations to the ‘string’-type boundary value problems known from prior works.