Symmetric discrete AKP and BKP equations

TL;DR

This paper explores the relationship between symmetric discrete AKP and BKP equations, showing how they can share tau functions under certain symmetries and extending these results to higher dimensions with explicit solutions.

Contribution

It demonstrates that discrete BKP can be expressed as a combination of AKP and its symmetric forms, and extends this connection to higher-dimensional hierarchies with explicit tau function forms.

Findings

Discrete BKP expressed as a combination of AKP and symmetric forms

Extension of AKP-BKP connection to higher dimensions

Explicit tau functions including elliptic coefficient solutions

Abstract

We show that when KP (Kadomtsev-Petviashvili) $\tau$ functions allow special symmetries, the discrete BKP equation can be expressed as a linear combination of the discrete AKP equation and its reflected symmetric forms. Thus the discrete AKP and BKP equations can share the same $\tau$ functions with these symmetries. Such a connection is extended to 4 dimensional (i.e. higher order) discrete AKP and BKP equations in the corresponding discrete hierarchies. Various explicit forms of such $\tau$ functions, including Hirota's form, Gramian, Casoratian and polynomial, are given. Symmetric $\tau$ functions of Cauchy matrix form that are composed of Weierstrass $\sigma$ functions are investigated. As a result we obtain a discrete BKP equation with elliptic coefficients.