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Involutive scroll structures on solutions of 4D dispersionless integrable hierarchies

Evgeny Ferapontov, Boris Kruglikov

Abstract

A rational normal scroll structure on an $(n+1)$-dimensional manifold $M$ is defined as a field of rational normal scrolls of degree $n-1$ in the projectivised cotangent bundle $\mathbb{P}T^*M$. We show that geometry of this kind naturally arises on solutions of various 4D dispersionless integrable hierarchies of heavenly type equations. In this context, rational normal scrolls coincide with the characteristic varieties (principal symbols) of the hierarchy. Furthermore, such structures automatically satisfy an additional property of involutivity. Our main result states that involutive scroll structures are themselves governed by a dispersionless integrable hierarchy, namely, the hierarchy of conformal self-duality equations.

Involutive scroll structures on solutions of 4D dispersionless integrable hierarchies

Abstract

A rational normal scroll structure on an -dimensional manifold is defined as a field of rational normal scrolls of degree in the projectivised cotangent bundle . We show that geometry of this kind naturally arises on solutions of various 4D dispersionless integrable hierarchies of heavenly type equations. In this context, rational normal scrolls coincide with the characteristic varieties (principal symbols) of the hierarchy. Furthermore, such structures automatically satisfy an additional property of involutivity. Our main result states that involutive scroll structures are themselves governed by a dispersionless integrable hierarchy, namely, the hierarchy of conformal self-duality equations.

Paper Structure

This paper contains 25 sections, 13 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Rational normal scrolls
  3. Scroll structures on manifolds
  4. Involutive scroll structures (ISS)
  5. Examples of ISS via the general heavenly hierarchy
  6. Main results on general ISS in dimensions five and six
  7. Examples of ISS in low dimensions
  8. Two constructions in dimension four
  9. Modified heavenly hierarchy and $S_{1,2}$ scroll structures
  10. Hierarchy of conformal self-duality equations and $S_{1,2}$ scroll structures
  11. Construction 1
  12. Construction 2
  13. Hierarchy of conformal self-duality equations and $S_{2,2}$ scroll structures
  14. Construction 1
  15. Construction 2
  16. ...and 10 more sections

Key Result

Theorem 1

System gh is involutive and has involutive scroll structure on every solution. Truncation of the hierarchy to dimension $n+1$ yields a family of ISS depending on $n-1$ functions of 3 variables.

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 6 more