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Two-component integrable extension of general heavenly equation

Wojciech Kryński, Artur Sergyeyev

Abstract

We introduce an integrable two-component extension of the general heavenly equation and prove that the solutions of this extension are in one-to-one correspondence with 4-dimensional hyper-para-Hermitian metrics. Furthermore, we demonstrate that if the metrics in question are hyper-para-Kähler, then our system reduces to the general heavenly equation. We also present an infinite hierarchy of nonlocal symmetries and a recursion operator for the system under study.

Two-component integrable extension of general heavenly equation

Abstract

We introduce an integrable two-component extension of the general heavenly equation and prove that the solutions of this extension are in one-to-one correspondence with 4-dimensional hyper-para-Hermitian metrics. Furthermore, we demonstrate that if the metrics in question are hyper-para-Kähler, then our system reduces to the general heavenly equation. We also present an infinite hierarchy of nonlocal symmetries and a recursion operator for the system under study.
Paper Structure (3 sections, 5 theorems, 47 equations)

This paper contains 3 sections, 5 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Hyper-para-hermitian structures
  3. Symmetries and Recursion Operator

Key Result

Theorem 1

Any hyper-para-Hermitian metric $[g]$ a 4-dimensional manifold can be locally put in the form where 1-forms $\omega^i$, $i=1,\ldots,4$ are defined as and functions $u$ and $v$ satisfy the following system where $\lambda_i\in\mathbb{R}$, $i=1,\ldots,4$, are fixed, arbitrary pairwise distinct constants. Moreover, the above system is integrable and admits an isospectral Lax pair with the Lax opera

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Definition
  • Lemma 1
  • proof
  • Remark
  • Proposition 1
  • Proposition 2