Nonlinear Waves Seminar - Baofeng Feng

Baofeng Feng, School of Mathematical and Statistics Sciences, The University of Texas Rio Grande Valley

Elliptic analogues of discrete KP, BKP, CKP, and DKP hierarchies

The Kadomtsev--Petviashvili (KP) hierarchies of A-,B-,C-and D-type were constructed by Kyoto School in 1980s. The tau function which is the core of each hierarchy plays an important role in many applications such as the construction of soliton solution. The tau functions in the BKP and DKP hierarchies have the Pfaffian structure whereas in the AKP and CKP hierarchies they take the form of determinant.

In this talk, we will construct elliptic analogues of discrete AKP, BKP, CKP and DKP hierarchies, as well as their tau functions expressed by Weierstrass elliptic functions. By Miwa transformation, these discrete hierarchies and tau functions are converted to their continuous counterparts. By reductions, various soliton solutions for many soliton equations such as KdV, mKdV and nonlinear Schr\"odinger equations on the elliptic background can be derived systematically. This is a joint work with my Ph.D student: Changyan Shi.