**Dr. Zhijun Qiao, ****Professor, ****UT Rio Grande Valley**

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 with M/W-shape solutions. I take the CH case as a typical example to explain the details. My presentation is based on my previous work (Communications in Mathematical Physics 239, 309-341). I will show that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierarchy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We see that the well-known CH equation is included in the negative order CH hierarchy while the Dym type equation is included in the positive order CH hierarchy. In particular, the CH equation, constrained to a symplectic submanifold in R^{2N}, has the parametric solutions. Moreover, solving the parametric representation of the solution on the symplectic submanifold gives a class of a new algebro-geometric solution of the CH equation. In the end of my talk, some open problems are also addressed for discussion. .

**Friday, April 14**

**4:00 – 5:00 PM**

**FLN 4.01.20**