Related papers: Backlund transformations for integrable lattice eq…
The B\"acklund transformation (BT) for the "good" Boussinesq equation and its superposition principles are presented and applied. Unlike many other standard integrable equations, the Boussinesq equation does not have a strictly algebraic…
B\"acklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here we give an improved method of constructing BTs for hierarchies of ODEs. This…
We present a geometric construction of Backlund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Backlund transformations, which are naturally…
We construct integrable discrete nonautonomous quad-equations as B\"acklund auto-transformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by…
Binary symmetry constraints are applied to constructing B\"acklund transformations of soliton systems, both continuous and discrete. Construction of solutions to soliton systems is split into finding solutions to lower-dimensional Liouville…
We consider GL(K|M)-invariant integrable supersymmetric spin chains with twisted boundary conditions and elucidate the role of Backlund transformations in solving the difference Hirota equation for eigenvalues of their transfer matrices.…
In this work we give a mechanical (Hamiltonian) interpretation of the so called spectrality property introduced by Sklyanin and Kuznetsov in the context of B\"acklund transformations (BTs) for finite dimensional integrable systems. The…
An auto-B\"acklund transformation for the quad equation $\mathrm{Q1}_1$ is considered as a discrete equation, called $\mathrm{H2}^a$, which is a so called torqued version of $\mathrm{H2}$. The equations $\mathrm{H2}^a$ and $\mathrm{Q1}_1$…
A new integrable lattice system is introduced, and its integrable discretizations are obtained. A B\"acklund transformation between this new system and the Toda lattice, as well as between their discretizations, is established.
The usual superposition formulas for Baecklund transformations of (2+1)-dimensional integrable systems include quadratures unlike the well known case of (1+1)-dimensional inegrable systems where the fourth solution is found with algebraic…
Backlund transformations of admissible curves in the Galilean 3-space and pseudo-Galilean 3-space and also spatial Backlund transformations of space curves in Galilean 4-space preserve the torsions under certain assumptions.
The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The…
The B\"acklund transformations for the relativistic lattices of the Toda type and their discrete analogues can be obtained as the composition of two duality transformations. The condition of invariance under this composition allows to…
We wish to explore a link between the Lax integrability of the $q$-Painlev\'e equations and the symmetries of the $q$-Painlev\'e equations. We shall demonstrate that the connection preserving deformations that give rise to the…
Addition formulae of trigonometric and elliptic functions are used to generate B\"acklund transformations together with their connecting quadrilateral equations. As a result we obtain periodic solutions for a number of multidimensionally…
Integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. These equation generate the corresponding continuous hierarchy of…
Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Backlund transformation, and the method of solving a Riccati map by…
It is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations, members of which are compatible between one another on a higher dimensional lattice. There turn out to be two such families, a…
We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund…
We consider the algebraic setting of classical defects in discrete and continuous integrable theories. We derive the "equations of motion" on the defect point via the space-like and time-like description. We then exploit the structural…