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 {{harvs|txt|last=Kac|last2=van Moerbeke|year=1975|first=Marc|first2=Pierre|author-link=Marc Kac|authorlink2=Pierre van Moerbeke}} and {{harvs|txt|last=Moser|year=1975|first=Jürgen|author-link=Jürgen Moser}} 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

{{citation|mr=0481238|last1=Kac|first1=M.|last2=van Moerbeke|first2=P.|authorlink1=Mark Kac|authorlink2=Pierre van Moerbeke|chapter=Some probabilistic aspects of scattering theory|title=Functional integration and its applications (Proc. Internat. Conf., London, 1974)|pages=[https://archive.org/details/functionalintegr0000inte/page/87 87–96]|publisher=Clarendon Press|place=Oxford|year=1975|isbn=978-0198533467|editor-first=A.M.|editor-last=Arthurs|chapter-url=https://archive.org/details/functionalintegr0000inte/page/87}}
{{citation|mr=0369953|last1=Kac|first1= M.|last2= van Moerbeke|first2= Pierre

|title=On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices.

|journal=Advances in Mathematics|volume= 16 |year=1975|issue=2 |pages= 160–169|doi=10.1016/0001-8708(75)90148-6|doi-access=free}}

{{citation|mr=0455038|last=Moser|first= Jürgen|authorlink=Jürgen Moser

|chapter=Finitely many mass points on the line under the influence of an exponential potential–an integrable system. |title=Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974)|pages= 467–497 |series=Lecture Notes in Phys.|volume= 38|publisher= Springer|place= Berlin|year= 1975|doi=10.1007/3-540-07171-7_12|isbn=978-3-540-07171-6|doi-access=free}}

Category:Integrable systems