18 research outputs found
On application of Liouville type equations to constructing B\"acklund transformations
It is shown how pseudoconstants of the Liouville-type equations can be
exploited as a tool for construction of the B\"acklund transformations. Several
new examples of such transformations are found. In particular we obtained the
B\"acklund transformations for a pair of three-component analogs of the
dispersive water wave system, and auto-B\"acklund transformations for coupled
three-component KdV-type systems.Comment: 11 pages, no figure
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 equations
Algebraic properties of Gardner's deformations for integrable systems
An algebraic definition of Gardner's deformations for completely integrable
bi-Hamiltonian evolutionary systems is formulated. The proposed approach
extends the class of deformable equations and yields new integrable
evolutionary and hyperbolic Liouville-type systems. An exactly solvable
two-component extension of the Liouville equation is found.Comment: Proc. conf. "Nonlinear Physics: Theory and Experiment IV" (Gallipoli,
2006); Theor. Math. Phys. (2007) 151:3/152:1-2, 16p. (to appear
Representations of sl(2,?) in category O and master symmetries
We show that the indecomposable sl(2,?)-modules in the Bernstein-Gelfand-Gelfand category O naturally arise for homogeneous integrable nonlinear evolution systems. We then develop a new approach called the O scheme to construct master symmetries for such integrable systems. This method naturally allows computing the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For new integrable equations such as a Benjamin-Ono-type equation, a new integrable Davey-Stewartson-type equation, and two different versions of (2+1)-dimensional generalized Volterra chains, we generate their conserved densities using their master symmetries
Some integrable maps and their Hirota bilinear forms
We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case