Multipeakons and the Classical Moment Problem

Classical results of Stieltjes are used to obtain explicit formulas for the peakon–antipeakon solutions of the Camassa–Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon–antipeakon pairs, and the details of the collisions are analyzed using results from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly.