Quad-graph Equations Research Articles

Classification of Integrable Boundary Equations for Integrable Systems on Quad-Graphs

AbstractIn the context of integrable systems on quad-graphs, the boundary consistency around a half of a rhombic dodecahedron, as a companion notion to the three-dimensional consistency around a cube, was introduced as a criterion for defining integrable boundary conditions for quad-graph systems with a boundary. In this paper, we formalize the notions of boundary equations as boundary conditions for quad-graph systems, and provide a systematic method for solving the boundary consistency, which results in a classification of integrable boundary equations for quad-graph equations in the Adler–Bobenko–Suris classification. This relies on factorizing, first the quad-graph equations into pairs of dual boundary equations, and then the consistency on a rhombic dodecahedron into two equivalent boundary consistencies. Generalization of the method to rhombic-symmetric equations is also considered.

A discrete Darboux–Lax scheme for integrable difference equations

We propose a discrete Darboux–Lax scheme for deriving auto-Bäcklund transformations and constructing solutions to quad-graph equations that do not necessarily possess the 3D consistency property. As an illustrative example we use the Adler–Yamilov type system which is related to the nonlinear Schrödinger (NLS) equation [7]. In particular, we construct an auto-Bäcklund transformation for this discrete system, its superposition principle, and we employ them in the construction of the one- and two-soliton solutions of the Adler–Yamilov system. PACS numbers02.30.Ik, 02.90.+p, 03.65.Fd Mathematics subject classification 202037K60, 39A36, 35Q55, 16T25.

The non-autonomous YdKN equation and generalized symmetries of Boll equations

In this paper, we study the integrability of a class of nonlinear non-autonomous quad graph equations compatible around the cube introduced by Boll in the framework of the generalized Adler, Bobenko, and Suris (ABS) classification. We show that all these equations possess three-point generalized symmetries which are subcases of either the Yamilov discretization of the Krichever–Novikov equation or of its non-autonomous extension. We also prove that all those symmetries are integrable as they pass the algebraic entropy test.

Algebraic entropy, symmetries and linearization of quad equations consistent on the cube

We discuss the non–autonomous nonlinear partial difference equations belonging to Boll classification of quad graph equations consistent around the cube. We show how starting from the compatible equations on a cell we can construct the lattice equations, its Bäcklund transformations and Lax pairs. By carrying out the algebraic entropy calculations we show that the H4 trapezoidal and the H6 families are linearizable and in a few examples we show how we can effectively linearize them.

Linearizability and a fake Lax pair for a nonlinear nonautonomous quad-graph equation consistent around the cube

We discuss the linearization of a nonautonomous nonlinear partial difference equation belonging to the Boll classification of quad-graph equations consistent around the cube. We show that its Lax pair is fake. We present its generalized symmetries, which turn out to be nonautonomous and dependent on an arbitrary function of the dependent variables defined at two lattice points. These generalized symmetries are differential–difference equations, which admit peculiar Backlund transformations in some cases.

A Lax pair of a lattice equation whose entropy vanishes

In this paper, we present multi-parametric quad-graph equations which are consistent around the cube. These equations are obtained by applying a ‘double twist’ to known integrable equations. Furthermore, we perform a limit to one of these equations to derive the non-symmetric equation which is equation (19) in Hietarinta and Viallet (2007 J. Phys. A: Math. Gen. 40 12629). As a result, we obtain a Lax pair for this equation.

Non-Point Invertible Transformations and Integrability of Partial Difference Equations

This article is devoted to the partial difference quad-graph equations that can be represented in the form $\varphi (u(i+1,j),u(i+1,j+1))=\psi (u(i,j),u(i,j+1))$, where the map $(w,z) \rightarrow (\varphi(w,z),\psi(w,z))$ is injective. The transformation $v(i,j)=\varphi (u(i,j),u(i,j+1))$ relates any of such equations to a quad-graph equation. It is proved that this transformation maps Darboux integrable equations of the above form into Darboux integrable equations again and decreases the orders of the transformed integrals by one in the $j$-direction. As an application of this fact, the Darboux integrable equations possessing integrals of the second order in the $j$-direction are described under an additional assumption. The transformation also maps symmetries of the original equations into symmetries of the transformed equations (i.e. preserves the integrability in the sense of the symmetry approach) and acts as a difference substitution for symmetries of a special form. The latter fact allows us to derive necessary conditions of Darboux integrability for the equations defined in the first sentence of the abstract.

Darboux integrable discrete equations possessing an autonomous first-order integral

Darboux integrable difference equations on the quad-graph are completely described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is obtained from a subclass of this equation via a non-point transformation. The detailed proof of the general proposition on the symmetry structure for the quad-graph equations is given as an auxiliary result.

Formal diagonalization of a discrete lax operator and conservation laws and symmetries of dynamical systems

An alternative method of constructing the formal diagonalization for the discrete Lax operators is proposed which can be used to calculate conservation laws and in some cases generalized symmetries for discrete dynamical systems. Discrete potential KdV equation, lattice derivative nonlinear Schr\"odinger equation, dressing chain, Toda lattice are considered as illustrative examples. For the Toda lattice on a quad graph corresponding to the Lie algebra $A_1^{(1)}$ infinite series of conservation laws are described. Systems of quad graph equations are represented including lattice versions of the "matrix" NLS and "vector" derivative NLS equations.

Method for searching higher symmetries for quad-graph equations

A generalized symmetry integrability test for discrete equations on the square lattice is studied. Integrability conditions are discussed. A method for searching higher symmetries (including non-autonomous ones) for quad-graph equations is suggested based on characteristic vector fields.

Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method

The application of the Gardner method for the generation of conservation laws to all the ABS equations is considered. It is shown that all the necessary information for the application of the Gardner method, namely Bäcklund transformations and initial conservation laws, follows from the multidimensional consistency of ABS equations. We also apply the Gardner method to an asymmetric equation which is not included in the ABS classification. An analog of the Gardner method for the generation of symmetries is developed and applied to the discrete Korteweg–de Vries equation. It can also be applied to all the other ABS equations.

Asymmetric integrable quad-graph equations

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 order to find candidates for integrability. It turns out that all such candidates have an equivalent affine form. These are tested by calculating their algebraic entropy. In this way, we have found several types of integrable equations, one of which seems to be entirely unrelated to any known discrete integrable system. We also list all single-tile conservation laws for the integrable equations in the above class.

Integrability conditions for nonautonomous quad-graph equations

This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between these techniques for partial difference equations.