Janossy Densities. II. Pfaffian Ensembles

We extend the main result of the companion paper J. Stat. Phys. 113:595–610 to the case of the pfaffian ensembles.     

A pfaffian version of the modified discrete KP equation

We first present Casorati determinant and Gram-type determinant solutions to a modified discrete KP equation and then produce a pfaffianized version of modified discrete KP equations by using Hirota and Ohta's pfaffianization procedure. The solutions to a coupled modified discrete KP equation are expressed by two types of pfaffians.     

Eynard–Mehta Theorem, Schur Process, and their Pfaffian Analogs

We give simple linear algebraic proofs of the Eynard–Mehta theorem, the Okounkov-Reshetikhin formula for the correlation kernel of the Schur process, and Pfaffian analogs of these results. We also discuss certain general properties of the spaces of all determinantal and Pfaffian processes on a given finite set     

SOLITON SOLUTIONS TO AN EQUATION IN 2+1 DIMENSIONS

This paper considers a 2+1 dimensional equation. A bilinear form for the equation and its 3-soliton solutions are obtained by the Hirota method. The N-soliton solution is given in the form of pfaffian. At the same time the proof of the solutions are given.     

Extended Determinant Solution of a （3 ＋ 1）-Dimensional KP Equation

In this paper, new extended Grammian determinant solutions to a （3 ＋ 1）-dimensional KP equation are presented by using Hirora＇s bilinear method, and a broad set of suftlcient conditions of systems of linear partial differential equations is given. Moreover, some special solutions of the representative systems are obtained through a systematic analysis.     

COMMUTATIVITY OF PFAFFIANIZATION AND BÄCKLUND TRANSFORMATIONS: THE LEZNOV LATTICE

In this paper, we first obtain Wronskian solutions to the Bäcklund transformation of the Leznov lattice and then derive the coupled system for the Bäcklund transformation through Pfaffianization. It is shown the coupled system is nothing but the Bäcklund transformation for the coupled Leznov lattice introduced by J. Zhao etc. [1]. This implies that Pfaffianization and Bäcklund transformation is commutative for the Leznov lattice. Moreover, since the two-dimensional Toda lattice constitutes the Leznov lattice, it is obvious that the commutativity is also valid for it.     

Contact Structure with Basic Potentials

This paper defines basic potentials for contact structure. Contact manifolds that admit a basic potential are shown to have an additional foliated structure of co-dimension 1. The properties of this new foliation, and its relation to the characteristic vector field, are explored.     

Pfaffianization of the generalized variable-coefficient Kadomtsev-Petviashvili equation

In this paper, the pfaffianization procedure is applied to the generalized variable-coefficient Kadomtsev-Petviashvili (vcKP) equation which can describe the realistic nonlinear phenomena in the fluid dynamics and plasmas. Using the pfaffianization procedure, the coupled system for the generalized vcKP equation is derived together with the Wronski-type pfaffian solution for this generalized coupled vcKP system under certain coefficient constraint. Furthermore, the Gramm-type pfaffian solution for such a coupled system is presented and verified by virtue of the pfaffian identities.     

Descent properties of biquaternion algebras with involution

We use the pfaffian to study some descent properties of biquaternion algebras with involution of the first kind in arbitrary characteristic.