Integrability and Symmetries of Difference Equations: the Adler-Bobenko-Suris Case

We consider the partial difference equations of the Adler-Bobenko-Suris classification, which are characterized as multidimensionally consistent. The latter property leads naturally to the construction of auto-B{\"a}cklund transformations and Lax pairs for all the equations in this class. Their symmetry analysis is presented and infinite hierarchies of generalized symmetries are explicitly constructed.Comment: 16 pages, for the proceedings of the 4th Workshop "Group Analysis of Differential Equations and Integrable Systems", Cyprus, 200

Darboux transformations with tetrahedral reduction group and related integrable systems

In this paper, we derive new two-component integrable differential difference and partial difference systems by applying a Lax-Darboux scheme to an operator formed from a

Darboux transformation with dihedral reduction group

We construct the Darboux transformation with Dihedral reduction group for the 2-dimensional generalisation of the periodic Volterra lattice. The resulting Bäcklund transformation can be viewed as a nonevolutionary integrable differential difference equation. We also find its generalised symmetry and the Lax representation for this symmetry. Using formal diagonalisation of the Darboux matrix, we obtain local conservation laws of the system

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

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

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

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

Development and Validation of the Learning Disabilities Needs Assessment Tool (LDNAT), a HoNOS-based needs assessment tool for use with people with intellectual disability

Background: In meeting the needs of individuals with intellectual disabilities (ID) who access health services, a brief, holistic assessment of need is useful. This study outlines the development and testing of the Learning Disabilities Needs Assessment Tool (LDNAT), a tool intended for this purpose. Method: An existing mental health (MH) tool was extended by a multidisciplinary group of ID practitioners. Additional scales were drafted to capture needs across six ID treatment domains that the group identified. LDNAT ratings were analysed for the following: item redundancy, relevance, construct validity and internal consistency (n =1692); test–retest reliability (n = 27); and concurrent validity (n =160). Results: All LDNAT scales were deemed clinically relevant with little redundancy apparent. Principal component analysis indicated three components (developmental needs, challenging behaviour, MH and well-being). Internal consistency was good (Cronbach alpha 0.80). Individual item test–retest reliability was substantial-near perfect for 20 scales and slight-fair for three scales. Overall reliability was near perfect (intra-class correlation =0.91). There were significant associations with five of six conditionspecific measures, i.e. the Waisman Activities of Daily Living Scale (general ability/disability), Threshold Assessment Grid (risk), Behaviour Problems Inventory for Individuals with Intellectual Disabilities-Short Form (challenging behaviour) Social Communication Questionnaire (autism) and a bespoke physical health questionnaire. Additionally, the statistically significant correlations between these tools and the LDNAT components made sense clinically. There were no statistically significant correlations with the Psychiatric Assessment Schedules for Adults with Developmental Disabilities (a measure of MH symptoms in people with ID). Conclusions: The LDNAT had clinically utility when rating the needs of people with ID prior to condition-specific assessment(s). Analyses of internal and external validity were promising. Further evaluation of its sensitivity to changes in needs is now required