Volterra Lattice

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 and 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'':

:''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
: ''R''_''n''' = e^{−''R''_''n''–1} – e^{−''R''_''n''+1}

References

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Integrable systems

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