On spectral and inverse spectral problems related to non-smooth solitons of Novikov equation

As non-smooth solitons, peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. One natural concept of peakons is dictated by the distributional compatibility of the Lax pairs. Due to the Lax integrability, the inverse spectral method is a powerful tool for the study of these solutions. In this talk, I will introduce some known results for the Camassa-Holm equation as well as our recent progress on Novikov equation.