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 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:

[math]\displaystyle{ a_n'=a_n(a_{n+1}+a_{n-1}) }[/math]

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

[math]\displaystyle{ R_n'=e^{R_{n-1}}+e^{R_{n+1}} }[/math]

References

• Kac, M.; van Moerbeke, P. (1975), "Some probabilistic aspects of scattering theory", in Arthurs, A.M., Functional integration and its applications (Proc. Internat. Conf., London, 1974), Oxford: Clarendon Press, pp. 87–96, ISBN 978-0198533467, https://archive.org/details/functionalintegr0000inte/page/87 
• Kac, M.; van Moerbeke, Pierre (1975), "On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices.", Advances in Mathematics 16 (2): 160–169, doi:10.1016/0001-8708(75)90148-6 
• Moser, Jürgen (1975), "Finitely many mass points on the line under the influence of an exponential potential–an integrable system.", Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., 38, Berlin: Springer, pp. 467–497, doi:10.1007/3-540-07171-7_12, ISBN 978-3-540-07171-6