Ghostpeakons

Sammanfattning: In this thesis we study peakons (peaked solitons), a class of solutions which occur in certain wave equations, such as the Camassa–Holm shallow water equation and its mathematical relatives, the Degasperis–Procesi, Novikov and Geng– Xue equations. These four non-linear partial differential equations are all integrable systems in the sense of having Lax pairs, infinitely many conservation laws, and multipeakon solutions given by explicitly known formulas.In the first paper, we develop a method which uses so-called ghostpeakons (peakons with amplitude zero) to find explicit formulas for the characteristic curves associated with the multipeakon solutions of the Camassa–Holm, Degasperis– Procesi and Novikov equations.In the second paper, we use the ghostpeakon method to derive explicit formulas for arbitrary multipeakon solutions of the two-componentGeng–Xue equation. The general case involves many inequivalent peakons configurations, depending on the order in which the peakons occur in the two components of the solution, and previously the solution was known only in the so-called interlacing case where the peakons lie alternatingly in one component and in the other. To obtain the solution formulas for an arbitrary configuration, we introduce auxiliary peakons to make the configuration interlacing. By taking suitable limits, we then drive the amplitudes of the auxiliary peakons to zero, leaving the solution formulas for the remaining ordinary peakons.