Duality of positive and negative integrable hierarchies via relativistically invariant fields

TL;DR

This paper explores how relativistic invariance reveals dualities between positive and negative integrable hierarchies, connecting various equations like sine-Gordon, Tzitzeica, and heavenly equations through symmetry and recursion operator methods.

Contribution

It demonstrates the duality between positive and negative integrable hierarchies using relativistic invariance and formal symmetry approaches across multiple dimensions.

Findings

Relativistic invariance underpins the duality between hierarchies.

Explicit duality relations are established in (2+1)-dimensions.

Recursion operators naturally emerge in the duality analysis.

Abstract

It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.