In mathematics, the Volterra lattice, also known as the discrete KdV equation, the KacâÂÂvan Moerbeke lattice, and the Langmuir lattice, is a system of ordinary differential equations with variables indexed by some of the points of a 1-dimensional lattice. It was introduced by and and is named after Vito Volterra. The Volterra lattice is a special case of the generalized LotkaâÂÂVolterra equation describing predatorâÂÂprey interactions, for a sequence of species with each species preying on the next in the sequence. The Volterra lattice also behaves like a discrete version of the KdV equation. The Volterra lattice is an integrable system, and is related to the Toda lattice. It is also used as a model for Langmuir waves in plasmas.
The Volterra lattice is the set of ordinary differential equations for functions a<sub>n</sub>:
where n is an integer. Usually one adds boundary conditions: for example, the functions a<sub>n</sub> could be periodic: a<sub>n</sub> = a<sub>n+N</sub> for some N, or could vanish for n ⤠0 and n âÂÂ¥ N.
The Volterra lattice was originally stated in terms of the variables R<sub>n</sub> = -log a<sub>n</sub> in which case the equations are