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Integrable systems with peakons and cuspons

Abstract

In my talk, I will introduce integrable peakon and cuspon equations and present a basic approach how to get peakon solutions. Those equations include the well-known Camassa-Holm (CH), the Degasperis-Procesi (DP), and other new peakon equations. I will take the CH case as a typical example to explain the details and show that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs). In particular, the CH peakon equation is extended to the FORQ and other higher order peakon models with peakon and weak-kink solutions. In the end of my talk, some open problems are also addressed for discussion. This talk is based on my earlier work [CMP 239(2003), 309].