Dmitry K. Demskoi

On recursion operators for elliptic models

New quasilocal recursion and Hamiltonian operators for the Krichever–Novikov and the Landau–Lifshitz equations are found. It is shown that the associative algebra of quasilocal recursion operators for these models is generated by a couple of operators related by an elliptic curve equation. A theoretical explanation of this fact for the Landau–Lifshitz equation is given in terms of multiplicators of the corresponding Lax structure.

Zero-curvature representation for a chiral-type three-field system

The matrix 4 × 4 zero-curvature representation for a two-dimensional chiral-type system with three fields is constructed. The system under consideration belongs to the class of scalar fields with the Lagrangian L = 1/2gij (u)uxi u tj + f(u), where gij is the metric tensor of the three-dimensional reducible Riemann space. This system was found by the authors earlier in the frame of the symmetry method. The zero-curvature representation is computed with the help of the third order symmetry ut = S(u). This was possible because the hyperbolic system is a nonlocal member in the hierarchy of the evolution systems and the matrix U of the zero-curvature representation is the common one for the whole hierarchy. As the test for non-triviality of the representation the recursion relations for the conserved currents are found.

A novel nth order difference equation that may be integrable

We derive an nth order difference equation as a dual of a very simple periodic equation, and construct ⌊(n + 1)/2⌋ explicit integrals and integrating factors of this equation in terms of multi-sums of products. We also present a generating function for the degrees of its iterates, exhibiting polynomial growth. In conclusion we demonstrate how the equation in question arises as a reduction of a system of lattice equations related to an integrable equation of Levi and Yamilov. These three facts combine to suggest the integrability of the nth order difference equation.

On application of Liouville type equations to constructing Bäcklund transformations

Abstract It is shown how pseudoconstants of the Liouville-type equations can be exploited as a tool for construction of the Bäcklund transformations. Several new examples of such transformations are found. In particular we obtained the Bäcklund transformations for a pair of three-component analogs of the dispersive water wave system, and auto- Bäcklund transformations for coupled three-component KdV-type systems.

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 = [gij(u)uxiutj]/2 + f(u), where gij 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 ut = 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.

Algebraic entropy for semi-discrete equations

We extend the definition of algebraic entropy to semi-discrete (difference-differential) equations. Calculating the entropy for a number of integrable and non integrable systems, we show that its vanishing is a characteristic feature of integrability for this type of equation.

Auto-Bäcklund transformations and superposition formulas for solutions of Drinfeld–Sokolov systems

Abstract The paper is devoted to constructing Auto-Backlund transformations (ABT) and superposition formulas for the solutions of the Drinfeld–Sokolov (DS) systems. The transformations are derived from pairs of differential substitutions relating different systems of the DS type. The nonlinear superposition formulas for solutions of the DS systems are obtained from the assumption of commutativity of the Bianchi diagram. We indicate a seed solution for each system which can be used to generate multi-soliton solutions. As an application of the superposition formulas we construct two-soliton solutions for each of the DS systems.

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.

Darboux integrability of determinant and equations for principal minors

We consider equations that represent a constancy condition for a 2D Wronskian, mixed Wronskian-Casoratian and 2D Casoratian. These determinantal equations are shown to have the number of independent integrals equal to their order - this implies Darboux integrability. On the other hand, the recurrent formulas for the leading principal minors are equivalent to the 2D Toda equation and its semi-discrete and lattice analogues with particular boundary conditions (cut-off constraints). This connection is used to obtain recurrent formulas and closed-form expressions for integrals of the Toda-type equations from the integrals of the determinantal equations. General solutions of the equations corresponding to vanishing determinants are given explicitly while in the non-vanishing case they are given in terms of solutions of ordinary linear equations.