Polynomial tau-functions of the symplectic KP, orthogonal KP and BUC hierarchies

TL;DR

This paper constructs polynomial tau-functions for symplectic KP, orthogonal KP, and BUC hierarchies, expressing them as determinants and Pfaffians, and explores their soliton solutions.

Contribution

It introduces a new method to explicitly construct polynomial tau-functions for these hierarchies using generating functions and Pfaffian/determinant formulas.

Findings

Polynomial tau-functions are coefficients of specific generating functions.

Tau-functions can be expressed as determinants and Pfaffians.

Soliton solutions for SKP and OKP hierarchies are discussed.

Abstract

This paper is concerned with the construction of the polynomial tau-functions of the symplectic KP (SKP), orthogonal KP (OKP) hierarchies and universal character hierarchy of B-type (BUC hierarchy), which are proved as zero modes of certain combinations of the generating functions. By applying the strategy of carrying out the action of the quantum fields on vacuum vector, the generating functions for symplectic Schur function, orthogonal Schur function and generalized Q-function have been presented. The remarkable feature is that polynomial tau-functions are the coefficients of certain family of generating functions. Furthermore, in terms of the Vandermonde-like identity and properties of Pfaffian, it is showed that the polynomial tau-functions of the SKP, OKP and BUC hierarchies canbe written as determinant and Pfaffian forms, respectively. In addition, the soliton solutions of the SKP and OKP hierarchies have been discussed.