Lotka--Volterra Type Equations and their Explicit Integration

In the present note we give an explicit integration of some two--dimensionalised Lotka--Volterra type equations associated with simple Lie algebras, other than the familiar $A_n$ case, possessing a representation without branching. This allows us, in particular, to treat the first fundamental representations of $A_r$, $B_r$, $C_r$, and $G_2$ on the same footing.Comment: 3 pages LATEX fil

Riccati-type equations, generalised WZNW equations, and multidimensional Toda systems

We associate to an arbitrary $\mathbb Z$-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 Redheffer--Reid 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.Comment: LaTeX2e, 18 page

Continuous approximation of binomial lattices

A systematic analysis of a continuous version of a binomial lattice, containing a real parameter $\gamma$ and covering the Toda field equation as $\gamma\to\infty$, is carried out in the framework of group theory. The symmetry algebra of the equation is derived. Reductions by one-dimensional and two-dimensional subalgebras of the symmetry algebra and their corresponding subgroups, yield notable field equations in lower dimensions whose solutions allow to find exact solutions to the original equation. Some reduced equations turn out to be related to potentials of physical interest, such as the Fermi-Pasta-Ulam and the Killingbeck potentials, and others. An instanton-like approximate solution is also obtained which reproduces the Eguchi-Hanson instanton configuration for $\gamma\to\infty$. Furthermore, the equation under consideration is extended to $(n+1)$--dimensions. A spherically symmetric form of this equation, studied by means of the symmetry approach, provides conformally invariant classes of field equations comprising remarkable special cases. One of these $(n=4)$ enables us to establish a connection with the Euclidean Yang-Mills equations, another appears in the context of Differential Geometry in relation to the socalled Yamabe problem. All the properties of the reduced equations are shared by the spherically symmetric generalized field equation.Comment: 30 pages, LaTeX, no figures. Submitted to Annals of Physic

Topological gravity on plumbed V-cobordisms

An ensemble of cosmological models based on generalized BF-theory is constructed where the role of vacuum (zero-level) coupling constants is played by topologically invariant rational intersection forms (cosmological-constant matrices) of 4-dimensional plumbed V-cobordisms which are interpreted as Euclidean spacetime regions. For these regions describing topology changes, the rational and integer intersection matrices are calculated. A relation is found between the hierarchy of certain elements of these matrices and the hierarchy of coupling constants of the universal (low-energy) interactions. PACS numbers: 0420G, 0240, 0460Comment: 29 page