Related papers: Integrable deformations of a polygon
Simple deformations, with a parameter $\epsilon$, of classical $R$-matrices which follow from decomposition of appropriate Lie algebras, are considered. As a result nonstandard Lax representations for some well known integrable systems are…
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…
A complete classification of isotropic vector equations of the geometric type that possess higher symmetries is proposed. New examples of integrable multi-component systems of the geometric type and their auto-Backlund transformations are…
It is shown that each integrable mapping is connected with a hierarchical completely integrable sytem of equations of evolution type which are invariant with respect to the transformation described by this mapping.
We provide a geometrical interpretation for the series of transformations used by Sakovich to map the third-order nonlinear evolution equation obtained by Chou and Qu to the mKdV equation. We also discuss its bi-Hamiltonian integrability as…
We present interpretation of known results in the theory of discrete asymptotic and discrete conjugate nets from the "discretization by B\"{a}cklund transformations" point of view. We collect both classical formulas of XIXth century…
A couple of applications of B\"acklund transformations in the study of nonlinear evolution equations is here given. Specifically, we are concerned about third order nonlinear evolution equations. Our attention is focussed on one side, on…
We prove that a generic $4$-dimensional integrable rolling distribution of contact elements with the symmetry of the tangency configuration (excluding developable seed and isotropic developable leaves) splits into an $1$-dimensional family…
We establish an explicit form of the Backlund transformation for the most known integrable systems.
A new method of determining B\"acklund transformations for nonlinear partial differential equations of the evolution type is introduced. Using the Hilbert space approach the problem of finding B\"acklund transformations is brought down to…
The geometry of an admissible B\"acklund transformation for an exterior differential system is described by an admissible Cartan connection for a geometric structure on a tower with infinite--dimensional skeleton which is the universal…
We begin by considering several properties commonly (but not universally) possessed by B\"acklund transformations between hyperbolic Monge-Amp\`ere equations: wavelike nature of the underlying equations, preservation of independent…
We propose a general algebraic analytic scheme for the spectral transform of solutions of nonlinear evolution equations. This allows us to give the general integrable evolution corresponding to an arbitrary time and space dependence of the…
This paper introduces a notion of decompositions of integral varifolds into countably many integral varifolds, and the existence of such decomposition of integral varifolds whose first variation is representable by integration is…
The existence of decompositions of the nonlinear integrable systems not only permits us to establish so-called linear superposition solutions but also to derive new nonlinear integrable coupled systems. Restricting our attention to the…
In trying to generalize Bianchi's B\"acklund transformation of quadrics to B\"acklund transformations of isometric deformations of other (classes of) surfaces, we investigate basic features of the isometric deformation of surfaces via the…
Nonlocal symmetries for exactly integrable two-field evolutionary systems of the third order have been computed. Differentiation of the nonlocal symmetries with respect to spatial variable gives a few nonevolutionary systems for each…
We study, theoretically and experimentally, a 1-parameter family of transformations and their limiting vector field on the space of plane polygons. These transformations are discrete analogs of completely integrable transformation on closed…
A very natural construction of integrable extensions of soliton systems is presented. The extension is made on the level of evolution equations by a modification of the algebra of dynamical fields. The paper is motivated by recent works of…
A set of infinitely many nonlocal conservation laws are revealed for (1+1)-dimensional evolution equations. For some special known integrable systems, say, the KdV and Dym equations, it is found that different nonlocal conservation laws can…