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Polynomial tau-functions of the n-th Sawada-Kotera hierarchy

Victor Kac, Johan Van de Leur

Abstract

We find all polynomial tau-functions of the n-th reduced BKP hierarchy (=n-th Sawada-Kotera hierarchy). The name comes from the fact that for n=3 the simplest equation of the hierarchy is the famous Sawada-Kotera equation.

Polynomial tau-functions of the n-th Sawada-Kotera hierarchy

Abstract

We find all polynomial tau-functions of the n-th reduced BKP hierarchy (=n-th Sawada-Kotera hierarchy). The name comes from the fact that for n=3 the simplest equation of the hierarchy is the famous Sawada-Kotera equation.
Paper Structure (3 sections, 4 theorems, 84 equations)

This paper contains 3 sections, 4 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. The BKP hierarchy and its polynomial tau-functions
  3. The $n$-th Sawada-Kotera hierarchy and its polynomial tau-functions

Key Result

Theorem 2

KvdLB2 All polynomial tau-functions of the BKP hierarchy Sb, up to a scalar multiple, are equal to where $Pf$ is the Pfaffiann of a skew-symmetric matrix, $\lambda=(\lambda_1,\lambda_2,\ldots ,\lambda_{2n})$ is an extended strict partition, i.e. $\lambda_1>\lambda_2>\cdots >\lambda_{2n}\ge 0$, $\tilde{t}=(t_1,0,t_3,0,\ldots)$, $c_i=(c_{1i},c_{2i}, c_{3i}, \ldots)$, $c_{ij}\in\mathbb C$.

Theorems & Definitions (13)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Proposition 4
  • Remark 5
  • Definition 6
  • Example 7
  • Remark 8
  • Theorem 9
  • Remark 10
  • ...and 3 more