The Bilinear Identities For An Extended BKP Hierarchy And The Extension Of The Non-abelian Discrete KP Equation

Integrable systems and soliton equations with self-consistent sources have important applications in mathematics and physics. In recent years, the extensions of integrable systems and soliton equations attract much interests. One of the methods to extend integrable systems is the squared eigenfunction symmetry procedure or so-called the ”ghost” symmetry method, which has been used on many integrable systems. For example, extended B-type Kadomtsev-Petviashvili hierarchy(extended BKP hierarchy for short) has been constructed via squared eigenfunction symmetry procedure. Darboux transformation and binary Darboux transformation, as an important method for solving soliton equations,also play important roles in squared eigenfunction symmetry method. In discrete integrable systems, such as discrete KP equation, discrete potential constructed by eigenfunction and adjoint eigenfunction is highly needed in the derivation of binary Darboux transformation. Also the Hirota’s bilinear method is a useful tool to solve soliton equations with self-consistent sources.The first part of this thesis is concerned with the bilinear identities for an extended BKP hierarchy. By introducing an auxiliary time flow of squared eigenfunction symmetry, first we construct bilinear residue identities for the original BKP hierarchy with an auxiliary time flow. Then we combine the auxiliary flow with a specific time flow to get the extended BKP hierarchy and finally derive the bilinear identities of the extended BKP hierarchy. After finding the τ-function for extended BKP hierarchy, we get the Hirota’s bilinear equations for the zero-curvature forms of the extended BKP hierarchy, and then give some examples under specific indexes, including two types of(2+1)-dimensional Sawada-Kotera equation with a self-consistent source. The Hirota’s bilinear equations for the second type of(2+1)-dimensional Sawada-Kotera equation with a self-consistent source(2d-SKw S-II) is a new result. We give a method to go back from Hirota’s bilinear equations to nonlinear partial di?erential equations and also give some examples, which verify the correctness of our results. These bilinear equations serve as basis for the further studies on n-soliton solutions or quasi-periodic solutions of extended BKP hierarchy.In the second part of this thesis, we construct the extended non-Abelian discrete KP equation. We use discrete potential to derive the explicit expressions of binary Darboux transformation for the discrete non-Abelian discrete KP equation and the iteration of binary Darboux transformation in quasideterminant form. We utilize the squared eigenfunction symmetry method and binary Darboux transformation, modify the shift operator of a specific discrete variable, and then extend the non-Abelian discrete KP equation. We give the explicit form of the linear problem and adjoint linear problem of non-Abelian discrete KP equation with self-consistent sources, and give a method to take non-Abelian case back to Abelian case(discrete KP equation with self-consistent sources). The nonAbelian discrete KP equation with self-consistent sources we derived in this thesis is not in the literatures before according to our knowledge. In further studies we may concentrate on finding exact solutions of the extended equation we get.