Related papers: Vector hyperbolic equations on the sphere possessi…
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…
We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations. Next, these results are applied to Vein's Abel equation whose solutions are expressed in terms of the third order…
A complete list of nonlinear one-field hyperbolic equations having generalized integrable x- and y-symmetries of the third order is presented. The list includes both sin-Gordon type equations and equations linearizable by differential…
The examples are considered of integrable hyperbolic equations of third order with two independent variables. In particular, an equation is found which admits as evolutionary symmetries the Krichever--Novikov equation and the modified…
V.V. Sokolov's modifying symmetry approach is applied to anisotropic evolution equations of the third order on the n-dimensional sphere. The main result is a complete classification of such equations. Auto-B\"acklund transformations are…
Affine transformations in Euclidean space generates a correspondence between integrable systems on cotangent bundles to the sphere, ellipsoid and hyperboloid embedded in $R^n$. Using this correspondence and the suitable coupling constant…
We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of…
We give new Backlund transformations (BTs) for some known integrable (in the sense of being multidimensionally consistent) quadrilateral lattice equations. As opposed to the natural auto-BT inherent in every such equation, these BTs are of…
We study the correct solvability of an abstract integro-differential equations in Hilbert space generalizing integro-differential equations arising in the theory of viscoelastisity. The equations under considerations are the abstract…
New variables of separation for few integrable systems on the two-dimensional sphere with higher order integrals of motion are considered in detail. We explicitly describe canonical transformations of initial physical variables to the…
The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations…
We explore the application of generating symmetries, i.e. symmetries that depend on a parameter, to integrable hyperbolic third order equations, and in particular to consistent pairs of such equations as introduced by Adler and Shabat (AS).…
We perform a classification of third order integrable systems of evolution equations with respect to higher symmetries. Applying it, we consider polynomial systems that are 0-homogeneous under a suitable weighting of variables with main…
New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in…
This paper examines the classification of hyperbolic equations. We study a class of equations of the form $$\frac{\partial^2 u}{\partial x\partial y}=F\left(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},u\right),$$ where…
For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the…
We prove that every finite group is the orientation-preserving isometry group of the complement of a hyperbolic link in the 3-sphere.
The 2-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions…
We discuss several ways of packing a hyperbolic surface with circles (of either varying radii or all being congruent) or horocycles, and note down some observations related to their symmetries (or the absence thereof).
Symmetric hyperbolic systems of equations are explicitly constructed for a general class of tensor fields by considering their structure as r-fold forms. The hyperbolizations depend on 2r-1 arbitrary timelike vectors. The importance of the…