Integrable nonlinear wave equations, Cauchy biorthogonal polynomials and related inverse problems

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Integrable equations such the Korteweg de Vries equation arise as approximate model equations in the mathematical description of wave phenomena in fluids, plasmas and optical fibres. They are special by virtue of possessing infinitely many integrals of motion, usually accompanied by deep mathematical structure. *In the last two decades a new class of exciting models of this type has been discovered. These new models exhibit several novel features, the central of which is the existence of localized, coherent modes, called peakons, with singular behaviour in the spatial derivative of the profile. The first such model equation-the Camassa-Holm (CH) equation-has been studied extensively by many authors from many different points of view, the most pertinent of which is a realization that the machinery of the classical moment problem provides a powerful tool to analyze more delicate features of this equation. Among the features accessible by this method are steepening of the slope at the time of the collision of peakons and the long time asymptotic behaviour. Another equation from this class which has been in the center of considerable research excitement is the Degasperis-Procesi (DP) equation. It was realized that this equation has, in addition to peakon solutions, also the shock solutions and, very interestingly, the shockpeakon solutions. The accompanying boundary value problem is non-selfadjoint, resulting in a host of new problems and challenges. This boundary value problem was named the cubic string; it is a third order problem with a weight being a measure. The case corresponding to peakons requires that this measure be discrete and for such a case the spectral and the inverse spectral problem were solved by H. Lundmark and the applicant. The solution of the problem involved a combination of ideas going back to T. Stieltjes and to M.G. Krein's study of an inhomogeneous string. **The distinct feature of the solution to the inverse problem for the cubic string is the appearance of a new type of polynomials, named Cauchy biorthogonal polynomials in view of the presence of the Cauchy kernel in the biorthogonality relation. It has subsequently been clarified that this type of polynomials replaces orthogonal polynomials *when one is dealing with certain type of inverse problems tied to non-selfadjoint boundary value problems. The resulting theory of Cauchy biorthogonal polynomials has undergone a considerable development in the last funding period. **The proposed research program is a continuation of the applicant's past work. In part, it is directed at developing a comprehensive map of applications of Cauchy biorthogonal polynomials to: *(i) solving inverse problems appearing in a variety of generalizations of CH and DP;*(ii) establishing a complete mechanism for the creation of shock peakons in the DP equation and the role of Cauchy biorthogonal polynomials in the transition from * peakons to shockpeakons;*(iii) solving random two-matrix models with Cauchy kernel; *(iv) understanding the scaling laws in Cauchy two-matrix models. **Another related objective, although not likely to rely directly on Cauchy biorthogonal polynomials, is to understand the nature of peakon collisions in the b-family (a one parameter deformation of the CH and DP equations).