Related papers: Geometry of B\"acklund Transformations I: Generali…
General Lagrangian theory of discrete one-dimensional integrable systems is illustrated by a detailed study of B\"acklund transformations for Toda-type systems. Commutativity of B\"acklund transformations is shown to be equivalent to…
We present a B\"acklund transformation (a discrete symmetry transformation) for the self-duality equations for supersymmetric gauge theories in N-extended super-Minkowski space ${\cal M}^{4|4N}$ for an arbitrary semisimple gauge group. For…
We consider a long--range homogeneous chain where the local variables are the generators of the direct sum of $N$ $\mathfrak{e}(3)$ interacting Lagrange tops. We call this classical integrable model rational ``Lagrange chain'' showing how…
We give B\"acklund transformations for first and second Painlev\'e hierarchies. These B\"acklund transformations are generalization of known B\"acklund transformations of the first and second Painlev\'e equations and they relate the…
The B\"acklund problem is solved for both the compact and noncompact versions of the Ishimori (2+1)-dimensional nonlinear spin model. In particular, a realization of the arising B\"acklund algebra in the form of an infinite-dimensional loop…
We prove the consistency of the B\"{a}cklund transformation (BT) for the spin Calogero-Moser (sCM) system in the rational, trigonometric, and hyperbolic cases. The BT for the sCM system consists of an overdetermined system of ODEs; to…
We consider a canonical transformation of parabolic coordinates on the plain and suppose that this transformation together with some additional relations may be considered as a counterpart of the auto and hetero B\"acklund transformations…
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…
In this paper, from the algebraic reductions from the Lie algebra $gl(n,\mathbb C)$ to its commutative subalgebra $Z_n$, we construct the general $Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon systems which contain many multi-component Sine-Gordon…
For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant…
Elie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and…
We generalize the geometrical model of transformation optics to Rieman-Cartan space with torsion by introducing topological defects in physical space. By relaxing the integrable condition, we show explicitly that the generalized equivalent…
Let $M$ be a closed orientable surface of negative curvature. A connection is said to be transparent if its parallel transport along closed geodesics is the identity. We describe all transparent SU(2)-connections and we show that they can…
The construction of Miura and B\"acklund transformations for $A_n$ mKdV and KdV hierarchies are presented in terms of gauge transformations acting upon the zero curvature representation. As in the well known $sl(2)$ case, we derive and…
We study the elliptic sinh-Gordon and sine-Gordon equations on the real plane and we introduce new families of solutions. We use a Backlund transformation that connects the elliptic versions of sinh-Gordon and sine-Gordon equations. As an…
In classical mechanics well-known cryptographic algorithms and protocols can be very useful for construction canonical transformations preserving form of Hamiltonians. We consider application of a standard generic divisor doubling for…
This paper is a continuation of Part I where the general setup was developed. Here we discuss the general equivalence problem for geometric structures and provide criteria for the equivalence, local and global, of transitive structures.…
We establish an explicit form of the Backlund transformation for the most known integrable systems.
In this paper we obtain a general expression for the n-defect matrix for the sinh-Gordon model. This in turn generate the general B\"acklund transformations (BT) for a system with $n$ type-I defects, through a gauge transformation.
This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the…