In mathematics, the Volterra lattice, also known as the discrete KdV equation, the Kac–van 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 Marc Kac and Pierre van Moerbeke (1975) and Jürgen Moser (1975) and is named after Vito Volterra. The Volterra lattice is a special case of the generalized Lotka–Volterra equation describing predator–prey 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.

Definition

The Volterra lattice is the set of ordinary differential equations for functions a_n:

${\displaystyle a_{n}'=a_{n}(a_{n+1}+a_{n-1})}$

where n is an integer. Usually one adds boundary conditions: for example, the functions a_n could be periodic: a_n = a_n+N for some N, or could vanish for n ≤ 0 and n ≥ N.

The Volterra lattice was originally stated in terms of the variables R_n = –log a_n in which case the equations are

${\displaystyle R_{n}'=e^{R_{n-1))+e^{R_{n+1))}$