Riccati-Type Equations, Generalised WZNW Equations, and Multidimensional Toda Systems

Abstract

We associate to an arbitrary Z-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati equations. The multidimensional extension of these equations is given. The generalisation of the associated RedhefferReid differential systems appears in a natural way. The connection between the Toda systems and the Riccati-type equations in lower and higher dimensions is established. Within this context the integrability problem for those equations is studied. As an illustration, some examples of the integrable multidimensional Riccati-type equations related to the maximally nonabelian Toda systems are given.

On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications

“…If we restrict the expansion of φ α j to the case where its multi-indices satisfy |K| ≤ 2, then the right-hand side of (8.2) becomes a second-order polynomial, which suggests that we impose conditions on the a α j K (x) to match the form of the so-called partial differential matrix Riccati equations [5,6,16], which is an integrable first-order system of PDEs (in the sense of satisfying the compatibility condition) of the form…”

Quasi-Lie schemes for PDEs

“…Riccati partial differential equations appear, for instance, in the study of Toda lattices and in particular cases of Wess-Zumino-Novikov-Witten (WZNW) (cf. [39]). A particular Riccati partial differential equation satisfying the ZCC condition is given by 1…”

“…[58,59]), relevant differential equations can be brought into the form (1.1), e.g. linear spectral problems and soliton surfaces in Lie algebras [41,42], the Von Misses transformation for studying Navier-Stockes equations [53,70], Toda lattices [39], Bäcklund transformations to analyse heat equations and modified KdV equations through Burguers equations and KdV equations respectively [58], and others [28].…”

Lie symmetries for Lie systems: Applications to systems of ODEs and PDEs

“…with the t-dependent coefficients satisfying the appropriate integrability condition (6.5). Such systems appear in the study of WZNW equations and multidimensional Toda systems [20]. Observe that the partials ∂x/∂t 1 and ∂x/∂t 2 are related to the t-dependent vector fields X pRic span a Vessiot-Guldberg Lie algebra V pRic ≃ sl(2, R).…”

“…As before, this permits us to calculate simultaneously Lie symmetries for all PDE Lie systems possessing isomorphic Vessiot-Guldberg Lie algebras by solving another PDE Lie system. To illustrate our results, we analyze PDE Lie systems of relevance in physics and mathematics, e.g., multidimensional Riccati equations and PDEs describing certain flat connection forms [20]. It is remarkable that the literature on PDE Lie systems is scarce (specially concerning their applications) and this paper contributes to enlarge their applications and to understand their properties [12].…”