On construction of symmetries from integrals of hyperbolic partial differential systems

An algorithm is proposed which allows one to construct higher symmetries of arbitrary order for some special classes of hyperbolic systems possessing integrals. The Pohlmeyer-Lund-Regge system and the open two-dimensional Toda lattices are shown to belong to the class of systems where our algorithm is applicable. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.     

Lax representations for triplets of two-dimensional scalar fields of the chiral type

We consider two-dimensional relativistically invariant systems with a three-dimensional reducible configuration space and a chiral-type Lagrangian that admit higher symmetries given by polynomials in derivatives up to the fifth order. Nine such systems are known: two are Liouville-type systems, and zero-curvature representations for two others have previously been found. We here give zero-curvature representations for the remaining five systems. We show how infinite series of conservation laws can be derived from the established zero-curvature representations. We give the simplest higher symmetries; others can be constructed from the conserved densities using the Hamiltonian operator. We find scalar formulations of the spectral problems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 189–205, August, 2006.     

Integrals of open two-dimensional lattices

We present an explicit formula for integrals of the open two-dimensional Toda lattice of type A_n. This formula is applicable for various reductions of this lattice. As an illustration, we find integrals of the G₂ Toda lattice. We also reveal a connection between the open An Toda and Shabat-Yamilov lattices.     

Lax Representation for a Triplet of Scalar Fields

We construct a 3×3 matrix zero-curvature representation for the system of three two-dimensional relativistically invariant scalar fields. This system belongs to the class described by the Lagrangian L = [g _ij(u)u _x ⁱ u _t ^j]/2+f(u), where g _ij is the metric tensor of a three-dimensional reducible Riemannian space. We previously found all systems of this class that have higher polynomial symmetries of the orders 2, 3, 4, or 5. In this paper, we find a zero-curvature representation for one of these systems. The calculation is based on the analysis of an evolutionary system u _t = S(u), where S is one of the higher symmetries. This approach can also be applied to other hyperbolic systems. We also find recursion relations for a sequence of conserved currents of the triplet of scalar fields under consideration.     

One Class of Liouville-Type Systems

We prove that a class of systems with the Lagrangian of the form is of the Liouville type. We construct new integrable Hamiltonian systems related to the symmetries of the hyperbolic systems under consideration by substitutions of the Miura transformation type. For one of the systems obtained, we construct the second-order recursion operator.     

Factorisation of recursion operators of some Lagrangian systems

We observe that recursion operator of an S-integrable hyperbolic equation that degenerates into a Liouvile-type equation admits a particular factorisation. This observation simplifies the construction of such operators. We use it to find a new quasi-local recursion operator for a triplet of scalar fields. The method is also illustrated with examples of the sinh-Gordon, the Tzitzeica and the Lund-Regge equations.