Multi-Component Extension of CAC Systems

Abstract

In this paper an approach to generate multi-dimensionally consistent N -component systems is proposed. The approach starts from scalar multi-dimensionally consistent quadrilateral systems and makes use of the cyclic group. The obtained N -component systems inherit integrable features such as Bäcklund transformations and Lax pairs, and exhibit interesting aspects, such as nonlocal reductions. Higher order single component lattice equations (on larger stencils) and multicomponent discrete Painlevé equations can also be derived in the context, and the approach extends to N -component generalizations of higher dimensional lattice equations.

On the relation between the dual AKP equation and an equation by King and Schief, and its N-soliton solution

“…Both equations ( 11) and (13) can also be derived from a two-component generalisation of the related scalar equation, BKP (5) respectively AKP (4), cf. equation (30) in [3] and section 3.2 in [10]. By a general result [10], such two-component generalisations are multi-dimensionally consistent, and moreover they provide Bäcklund transformations for their reductions, equations (11) or (13).…”

“…equation (30) in [3] and section 3.2 in [10]. By a general result [10], such two-component generalisations are multi-dimensionally consistent, and moreover they provide Bäcklund transformations for their reductions, equations (11) or (13).…”

“…The DAKP dispersion relations Q(x i , y i , z i ) = 0 (with a i → a i ) are then satisfied. Furthermore, the DAKP interaction term R ij (3), in terms of the p i , q i , is precisely (10).…”

Lax matrices for lattice equations which satisfy consistency-around-a-face-centered-cube

“…These are equations defined on four vertices of a face of the square lattice, for which multidimensional consistency takes the form of consistency-around-a-cube (CAC). Outside of the ABS list there are also several other known types of CAC equations, including Boussinesq-type equations [8,9] and other multi-component equations [10][11][12], as well as equations satisfying formulations of multidimensional consistency which are different from CAC [13,14].…”