Contributions to the theory of peaked solitons

Abstract: The aim of this work is to present some new contributions to the theory of peaked solitons. The thesis contains two papers, named "Lie symmetry analysis of the Novikov and Geng-Xue equations, and new peakon-like unbounded solutions to the Camassa-Holm, Degasperis-Procesi and Novikov equations'' and "Peakon-antipeakon solutions of the Novikov equation'' respectively.In the first paper, a new kind of peakon-like solutions to the Novikov equation is obtained, by transforming the one-peakon solution via a Lie symmetry transformation. This new kind of solution is unbounded as x tends to infinity and/or minus infinity. It also has a peak, though only for some interval of time. We make sure that the peakon-like function is still a solution in the weak sense for those times where the function is non-differentiable. We find that similar solutions, with peaks living only for some interval in time, are valid weak solutions to the Camassa-Holm equation, though these can not be obtained via a symmetry transformation.The second paper covers peakon-antipeakon solutions of the Novikov equation, on the basis of known solution formulas from the pure peakon case. A priori, these formulas are valid only for some interval of time and only for some initial values. The aim of the article is to study the Novikov multipeakon solution formulas in detail, to overcome these problems. We find that the formulas for locations and heights of the peakons are valid for all times at least in an ODE sense. Also, we suggest a procedure of how to deal with multipeakons where the initial conditions are such that the usual spectral data are not well-defined as residues of single poles of a Weyl function. In particular we cover the interaction between one peakon and one antipeakon, revealing some unexpected properties. For example, with complex spectral data, the solution is shown to be periodic, except for a translation, with an infinite number of collisions between the peakon and the antipeakon. Also, plotting solution formulas for larger number of peakons shows that there are similarities to the phenomenon called "waltzing peakons''.