Symmetry algebra of discrete KdV equations and corresponding differential-difference equations of Volterra type

A sequence of canonical conservation laws for all the Adler-Bobenko-Suris equations is derived and is employed in the construction of a hierarchy of master symmetries for equations H1-H3, Q1-Q3. For the discrete potential and Schwarzian KdV equations it is shown that their local generalized symmetries and non-local master symmetries in each lattice direction form centerless Virasoro type algebras. In particular, for the discrete potential KdV, the structure of its symmetry algebra is explicitly given. Interpreting the hierarchies of symmetries of equations H1-H3, Q1-Q3 as differential-difference equations of Yamilov's discretization of Krichever-Novikov equation, corresponding hierarchies of isospectral and non-isospectral zero curvature representations are derived for all of them.Comment: 22 page

Continuous symmetric reductions of the Adler-Bobenko-Suris equations

Continuously symmetric solutions of the Adler-Bobenko-Suris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three point generalized symmetries admitted by the corresponding equations, these solutions are shown to be determined by an integrable system of partial differential equations. The connection of this system to the Nijhoff-Hone-Joshi "generating partial differential equations" is established and an auto-Backlund transformation and a Lax pair for it are constructed. Applied to the H1 and Q1$_{\delta=0}$ members of the Adler-Bobenko-Suris family, the method of continuously symmetric reductions yields explicit solutions determined by the Painleve trancendents.Comment: 28 pages, submitted to J. Phys. A: Math. Theo

On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations

Recently, the first-named author gave a classification of 3D consistent 6-tuples of quad-equations with the tetrahedron property; several novel asymmetric 6-tuples have been found. Due to 3D consistency, these 6-tuples can be extended to discrete integrable systems on Z^m. We establish Lagrangian structures and flip-invariance of the action functional for the class of discrete integrable systems involving equations for which some of the biquadratics are non-degenerate and some are degenerate. This class covers, among others, some of the above mentioned novel systems.Comment: 21 pp, pdfLaTe

Cosymmetries and Nijenhuis recursion operators for difference equations

In this paper we discuss the concept of cosymmetries and co--recursion operators for difference equations and present a co--recursion operator for the Viallet equation. We also discover a new type of factorisation for the recursion operators of difference equations. This factorisation enables us to give an elegant proof that the recursion operator given in arXiv:1004.5346 is indeed a recursion operator for the Viallet equation. Moreover, we show that this operator is Nijenhuis and thus generates infinitely many commuting local symmetries. This recursion operator and its factorisation into Hamiltonian and symplectic operators can be applied to Yamilov's discretisation of the Krichever-Novikov equation

Classification of integrable discrete Klein-Gordon models

The Lie algebraic integrability test is applied to the problem of classification of integrable Klein-Gordon type equations on quad-graphs. The list of equations passing the test is presented containing several well-known integrable models. A new integrable example is found, its higher symmetry is presented.Comment: 12 pages, submitted to Physica Script

Linear quadrilateral lattice equations and multidimensional consistency

It is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations, members of which are compatible between one another on a higher dimensional lattice. There turn out to be two such families, a natural parametrisation is given for each.Comment: 7 pages, 1 figur

An integrable multicomponent quad equation and its Lagrangian formulation

We present a hierarchy of discrete systems whose first members are the lattice modified Korteweg-de Vries equation, and the lattice modified Boussinesq equation. The N-th member in the hierarchy is an N-component system defined on an elementary plaquette in the 2-dimensional lattice. The system is multidimensionally consistent and a Lagrangian which respects this feature, i.e., which has the desirable closure property, is obtained.Comment: 10 page

Integrable lattice equations with vertex and bond variables

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 equations as non-autonomous "Yang-Baxter maps". We also present a model in which the vertex and bond variables are fully coupled. Integrability is tested with algebraic entropy as well as multidimensional consistencyComment: 15 pages, remarks added, other minor change

Association between age at disease onset of anti-neutrophil cytoplasmic antibody-associated vasculitis and clinical presentation and short-term outcomes

Objectives: ANCA-associated vasculitis (AAV) can affect all age groups. We aimed to show that differences in disease presentation and 6 month outcome between younger- A nd older-onset patients are still incompletely understood. Methods: We included patients enrolled in the Diagnostic and Classification Criteria for Primary Systemic Vasculitis (DCVAS) study between October 2010 and January 2017 with a diagnosis of AAV. We divided the population according to age at diagnosis: <65 years or ≥65 years. We adjusted associations for the type of AAV and the type of ANCA (anti-MPO, anti-PR3 or negative). Results: A total of 1338 patients with AAV were included: 66% had disease onset at <65 years of age [female 50%; mean age 48.4 years (s.d. 12.6)] and 34% had disease onset at ≥65 years [female 54%; mean age 73.6 years (s.d. 6)]. ANCA (MPO) positivity was more frequent in the older group (48% vs 27%; P = 0.001). Younger patients had higher rates of musculoskeletal, cutaneous and ENT manifestations compared with older patients. Systemic, neurologic,cardiovascular involvement and worsening renal function were more frequent in the older-onset group. Damage accrual, measured with the Vasculitis Damage Index (VDI), was significantly higher in older patients, 12% of whom had a 6 month VDI ≥5, compared with 7% of younger patients (P = 0.01). Older age was an independent risk factor for early death within 6 months from diagnosis [hazard ratio 2.06 (95% CI 1.07, 3.97); P = 0.03]. Conclusion: Within 6 months of diagnosis of AAV, patients >65 years of age display a different pattern of organ involvement and an increased risk of significant damage and mortality compared with younger patients

Symmetries and integrability of discrete equations defined on a black–white lattice

We study the deformations of the H equations, presented recently by Adler, Bobenko and Suris, which are naturally defined on a black–white lattice. For each one of these equations, two different three-leg forms are constructed, leading to two different discrete Toda-type equations. Their multidimensional consistency leads to Bäcklund transformations relating different members of this class as well as to Lax pairs. Their symmetry analysis is presented yielding infinite hierarchies of generalized symmetries