Baofeng Feng (Department of Mathematics, The University of Texas-Pan American)
Bilinear forms and multi-loop solutions for the Dapasperis-Procesi equation and its short-wave model
In this talk, we propose the bilinear equations for the Dapasperis-Procesi (DP) equation, a shallow water wave model and its short wave model, the Vakhnenko equation, a model which describes high-frequency waves in a relaxing medium. First, we show that the Vakhnenko equation can be derived from a 3-reduction of BKP-or CKP-Toda equations through a hodograph transformation. The situation of the Dapasperis-Procesi equation is complicated. To deduce the DP equation, we need a pseudo 3-reduction of the CKP-Toda equation. One of the tau function is believed to be the product of two pfa.ans. Then, by intro-ducing a hodograph transformation, and dependent variable transformation, we can show that the DP equation is derived from a pair of bilinear equations. As a by-product, the multi-soliton solutions including multi-loop solutions for the Dapasperis-Procesi equation and the Vakhnenko equation are given. This is a joint work with Dr. Maruno at the University of Texas-Pan American and Dr. Ohta at Kobe University.
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