On traveling waves in lattices: The case of Riccati lattices

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.

💡 Research Summary

The paper investigates traveling‑wave solutions of differential‑difference lattice equations by employing the “method of simplest equation” together with the Riccati equation. The authors first define a “Riccati lattice” as a lattice whose governing equation can be reduced to a form that admits solutions constructed from the known solution of the Riccati ordinary differential equation
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