Duality of positive and negative integrable hierarchies via relativistically invariant fields

Q. P. Liu,S. Y. Lou,X. B. Hu

doi:10.1007/jhep07(2021)058

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.

Highlights

For a (1+1)-dimensional integrable system, there are infinitely many local and nonlocal symmetries
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 Kortewegde Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like 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 hierarchies are local in a proper space ({x, tn}) and nonlocal in its dual space ({τ, tn}) where the negative hierarchies are local in the space {τ, τn} and nonlocal in the dual space {x, τn}

Summary

Duality of the PMKdV hierarchy and the sine-Gordon hierarchy

Is one of the most physically relevant field equations [39, 52,53,54,55,56,57]. It is important in quantum and classical field theories and in almost all the physical branches and even in other natural scientific fields. It is known [62] that K2n+1 defined in (2.6) are local symmetries and K−2n−1 defined in (2.7) are nonlocal symmetries of the PMKdV equation (2.2) for all n = 0, 1, 2,. The local symmetries K2n+1 and the nonlocal symmetries K−2n−1 are dual with the duality relation vx ↔ vτ [= ∂x−1 sin(v)]. We can say that the set of the nonlocal symmetries of the PMKdV equation can be localized with help of the duality relation (2.23). If the positive hierarchy (like the PMKdV hierarchy) is local and the negative hierarchy is nonlocal (like the sG hierarchy), the duality conjecture indicates that the local symmetries and nonlocal symmetries may be dual each other via a possible dual relation

Duality of the PFG hierarchy and the Tzitzeica hierarchy

Duality hierarchies from the two-dimensional Toda lattice

From the real second heavenly equation to dual hierarchies

Summary and discussions