Continuous approximation of binomial lattices

TL;DR

This paper analyzes a continuous binomial lattice equation using group theory, deriving symmetries, reducing to notable physical equations, and extending to higher dimensions with applications in geometry and physics.

Contribution

It introduces a continuous binomial lattice framework, derives its symmetry algebra, and connects reduced equations to important physical and geometric problems.

Findings

Derived symmetry algebra of the continuous binomial lattice.

Obtained exact solutions related to physical potentials.

Extended the equation to higher dimensions with geometric applications.

Abstract

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.