Duality of positive and negative integrable hierarchies via relativistically invariant fields

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.

Integrability structures of the generalized Hunter–Saxton equation

We consider integrability structures of the generalized Hunter–Saxton equation. We obtain the Lax representation with non-removable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate an infinite-dimensional Lie algebra of higher symmetries, and prove existence of infinite number of cosymmetries of higher order. Further, we give examples of employing the higher order symmetries to constructing exact globally defined solutions for the generalized Hunter–Saxton equation.

Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation

Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order α∈(0,1)∪(1,2). It is proved that the equation in question has infinite sequences of nontrivial higher-order symmetries that are generated by two local recursion operators. It is also shown that some of the obtained higher-order symmetries can be rewritten as fractional-order symmetries, and corresponding fractional-order recursion operators are presented. The proposed approach for finding higher-order symmetries is applicable for a wide class of linear fractional differential equations.

Examples of Four- or Six-Dimensional Symplectic-Haantjes Manifolds and About a Relationship with Recursion Operators

Certain ways of characterizing integrable systems with $(1,1)$-tensor field have been investigated, so far. For example, recursion operators and Haantjes operators are known. We show that geometrical examples of four- or six-dimensional symplectic Haantjes manifolds and recursion operators for several Hamiltonian systems. Through these examples, we consider the relation between recursion operators and Haantjes operators.

Geometric curve flows in low dimensional Cayley–Klein geometries

Using the method of equivariant moving frames, we derive the evolution equations for the curvature invariants of arc-length parametrized curves under arc-length preserving geometric flows in two-, three- and four-dimensional Cayley–Klein geometries. In two and three dimensions, we obtain recursion operators, which show that the curvature evolution equations obtained are completely integrable.