相关论文: Families of Integrable Equations

We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the bond variables, and one can write the…

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…

In this paper we introduce a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices. Algebraically, this commutativity is based on a very special construction of local…

The new integrable mapping with a simple geometric interpretation is presented. This mapping arise from the nonlinear superposition principle for the B\"acklund transformations of some vector evolution equation.

We study a new example of lattice equation being one of the key equations of a recent generalized symmetry classification of five-point differential-difference equations. This equation has two different continuum limits which are the…

Integrable systems appeared in physics long ago at the onset of classical dynamics with examples being Kepler's and other famous problems. Unfortunately, the majority of nonlinear problems turned out to be nonintegrable. In accelerator…

In the present paper we consider a general family of two dimensional wave equations which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated…

We introduce a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs, with $N\times N$ matrices, linear in the spectral parameter. We give a classification scheme for such Lax pairs and the associated discrete integrable systems. We present…

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals $K$ and $E$. Our methods exploit the rich structures connecting complete elliptic integrals,…

The sets of the integrable lattice equations, which generalize the Toda lattice, are considered. The hierarchies of the first integrals and infinitesimal symmetries are found. The properties of the multi-soliton solutions are discussed.

Hexagonal circle patterns are introduced, and a subclass thereof is studied in detail. It is characterized by the following property: For every circle the multi-ratio of its six intersection points with neighboring circles is equal to -1.…

The problem of construction of the boundary conditions for the Toda lattice compatible with its higher symmetries is considered. It is demonstrated that this problem is reduced to finding of the differential constraints consistent with the…

Spectral method related to Lame equation with finite-gap potential is used to study the optical cascading equations. These equations are known not to be integrable by inverse scattering method. Due to "partial integrability" two-gap…

If we take a superintegrable Stackel system and make variables "faster" or "slower", that is equivalent to a trivial transformation of the Stackel matrix and potentials, then we obtain an infinite family of superintegrable systems with…

We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary…

A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. We consider a family of nonlinear recurrences with the Laurent property, which were…

We construct a family of integrable equations of the form $v_t=f(v,v_x,v_{xx},v_{xxx})$ such that $f$ is a transcendental function in $v,v_x,v_{xx}$. This family is related to the Krichever-Novikov equation by a differential substitution.…

We explore two distinct methods to introduce integrable defects in a family of integrable sigma-models known as Yang-Baxter models. The first method invokes a modified monodromy matrix encoding an integrable defect separating two integrable…

The procedure for obtaining integrable vertex models via reflection matrices on the square lattice with open boundaries is reviewed and explicitly carried out for a number of two- and three-state vertex models. These models include the…

Let $K$ be a number field, let $A$ be a finite-dimensional $K$-algebra, let $\mathrm{J}(A)$ denote the Jacobson radical of $A$, and let $\Lambda$ be an $\mathcal{O}_{K}$-order in $A$. Suppose that each simple component of the semisimple…