Related papers: Classification of 3D consistent quad-equations
Rubik's Cube is one of the most famous combinatorial puzzles involving nearly $4.3 \times 10^{19}$ possible configurations. Its mathematical description is expressed by the Rubik's group, whose elements define how its layers rotate. We…
A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P\Delta Es) is reviewed. The method assumes that the P\Delta Es are defined on a quadrilateral, and consistent around the…
For every smooth (irreducible) cubic surface $S$ we give an explicit construction of a representative for each of the 72 equivalence classes of determinantal representations. Equivalence classes (under $\GL_3\times \GL_3$ action by left and…
Integrable difference equations commonly have more low-order conservation laws than occur for nonintegrable difference equations of similar complexity. We use this empirical observation to sift a large class of difference equations, in…
Solvable via Bethe Ansatz (BA) anisotropic statistical model on cubic lattice consisting of locally interacting 6-vertex planes, is studied. Symmetries of BA lead to infinite hierarchy of possible phases, which is further restricted by…
We prove that a face of a cube can be optimally partitioned into connected 193 sets on which the cut locus, or ridge tree, is constant up to isomorphism as a labeled graph. These are 60 connected open sets, curves bounding them, and…
A previously introduced scheme for describing integrable deformations of of algebraic curves is completed. Lenard relations are used to characterize and classify these deformations in terms of hydrodynamic type systems. A general solution…
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 a class of third-order evolution equations of the form \begin{equation*} \left\{ \begin{array}{l} \displaystyle u_{t}=F\left(x,t,u,u_x,u_{xx},u_{xxx},v,v_x,v_{xx},v_{xxx}\right), \displaystyle…
We present a discrete inverse scattering transform for all ABS equations excluding Q4. The nonlinear partial difference equations presented in the ABS hierarchy represent a comprehensive class of scalar affine-linear lattice equations which…
In the paper we derive rational solutions for the lattice potential modified Korteweg-de Vries equation, and Q2, Q1($\delta$), H3($\delta$), H2 and H1 in the Adler-Bobenko-Suris list. B\"acklund transformations between these lattice…
In this paper, we examine the space of initial values for integrable lattice equations, which are lattice equations classified by Adler {\em et al} (2003), known as ABS equations. By considering the map which iterates the solution along…
Construction and classification of 2D superintegrable systems (i.e. systems admitting, in addition to two global integrals of motion guaranteeing the Liouville integrability, the third global and independent one) defined on 2D spaces of…
One of the most fascinating and technically demanding parts of the theory of two-dimensional integrable systems constitute the models with the spectral parameter on an elliptic curve, including Landau-Lifshitz and Krichever-Novikov…
We study quotients of quadratic forms and associated polar lines in the projective plane. Our results, applied pointwise to quadratic differential forms, shed some light on classical binary differential equations (BDEs) associated to…
We will give a short introduction to discrete or lattice soliton equations, with the particular example of the Korteweg-de Vries as illustration. We will discuss briefly how B\"acklund transformations lead to equations that can be…
For two-dimensional lattice equations one definition of integrability is that the model can be naturally and consistently extended to three dimensions, i.e., that it is "consistent around a cube" (CAC). As a consequence of CAC one can…
In this paper, we study relations between automorphism groups of cubic fourfolds and Kuznetsov components. Firstly, we characterize automorphism groups of cubic fourfolds as subgroups of autoequivalence groups of Kuznetsov components using…
We study finite foldable cubical complexes of nonpositive curvature (in the sense of A.D. Alexandrov). We show that such a complex X admits a graph of spaces decomposition. It is also shown that when dim X=3, X contains a closed rank one…
This paper proposes a method for identifying and classifying integrable nonlinear equations with three independent variables, one of which is discrete and the other two are continuous. A characteristic property of this class of equations,…