Jacek Szmigielski

Nonlinear Waves and an Extension of the Theory of Orthogonal Polynomials

There exists a very interesting connection between inverse problems which one poses as part of the Lax formulation of certain nonlinear wave equations, like Camassa-Holm (CH), or Degasperis-Procesi (DP), and the classical theory of orthogonal polynomials. This connection, in the case of the CH equation, is essentially equivalent to an approach to orthogonal polynomials which originated in the work of T. Stieltjes on continued fractions. The case of the DP equation eventually leads to a new class of polynomials, called Cauchy biorthogonal polynomials. They appear not only in the treatment of the DP equation but also in the new, recently discovered, generalizations. Two such generalizations will be discussed along with a discussion of peakon solutions which can be constructed and analyzed in terms of this new class of polynomials. The crucial aspect of the theory is total positivity, a concept known from the work of M. G. Krein, F. Gantmacher and S. Karlin.
This talk is based on an ongoing work with H. Lundmark as well as a series of joint papers with M. Bertola and M. Gekhtman.