On traveling waves in lattices: The case of Riccati lattices

TL;DR

This paper analyzes Riccati lattices, a class of differential-difference equations, using the simplest equation method to find exact traveling wave solutions, revealing that many Holling lattices are Riccati lattices.

Contribution

It identifies Riccati lattices within generalized Lotka-Volterra and Holling classes and constructs exact solutions for select Holling lattices.

Findings

Wadati lattice is the only Lotka-Volterra Riccati lattice.

Many Holling lattices are Riccati lattices.

Exact traveling wave solutions are constructed for three Holling lattices.

Abstract

The method of simplest equation is applied for analysis of a class of lattices described by differential-difference equations that admit traveling-wave solutions constructed on the basis of the solution of the Riccati equation. We denote such lattices as Riccati lattices. We search for Riccati lattices within two classes of lattices: generalized Lotka - Volterra lattices and generalized Holling lattices. We show that from the class of generalized Lotka - Volterra lattices only the Wadati lattice belongs to the class of Riccati lattices. Opposite to this many lattices from the Holling class are Riccati lattices. We construct exact traveling wave solutions on the basis of the solution of Riccati equation for three members of the class of generalized Holing lattices.