XXVIII Workshop on Geometric Methods in Physics 28.06-04.07.2009

Hans Lundmark

Explicit solutions of Novikov's peakon equation

Integrable PDEs with peakon (peak-shaped soliton) solutions constitute a very active area of research. The most well-known example is the Camassa-Holm (CH) shallow water equation, and another one is the Degasperis-Procesi (DP) equation, which looks very similar to CH but has a rather different underlying structure. Vladimir Novikov recently found another such equation, similar in form to CH and DP except that the nonlinear terms are cubic instead of quadratic. It turns out that its multipeakon solutions can be explicitly computed using the methods developed for the DP equation, since the spectral problems in the Lax pairs for these two equations are in a sense dual to each other. The aim of this talk is to give an introduction to the subject of peakons, and to describe the new features which arise in the solution of Novikov's equation. For example, one by-product of this study is a curious combinatorial identity involving the sum of all minors of a given size in a symmetric matrix. (This is joint work with Andrew Hone and Jacek Szmigielski.)