Exact Static Solutions of a Generalized Discrete \phi^4 Model Including Short-Periodic Solutions

TL;DR

This paper derives a comprehensive set of exact static solutions for a five-parameter discrete  model, including elliptic, hyperbolic, and trigonometric functions, revealing new solution types and stability properties.

Contribution

It introduces new exact static solutions for a generalized discrete  model, including short-periodic and staggered solutions, and explores their translational invariance and stability.

Findings

Derived exact solutions in elliptic, hyperbolic, and trigonometric forms.

Identified seven cases where solutions are admitted, including new cases.

Revealed four new types of cubic nonlinearity with translational invariance.

Abstract

For a five-parameter discrete $\phi^4$ model, we derive various exact static solutions, including the staggered ones, in the form of the basic Jacobi elliptic functions $\sn$, $\cn$, and $\dn$, and also in the form of their hyperbolic function limits such as kink ($\tanh$) and single-humped pulse ($\sech$) solutions. Such solutions are admitted by the considered model in seven cases, two of which have been discussed in the literature, and the remaining five cases are addressed here. We also obtain $\sin$e, staggered $\sin$e as well as a large number of short-periodic static solutions of the generalized 5-parameter model. All the Jacobi elliptic, hyperbolic and trigonometric function solutions (including the staggered ones) are translationally invariant (TI), i.e., they can be shifted along the lattice by an arbitrary $x_0$, but among the short-periodic solutions there are both TI and non-TI solutions. The stability of these solutions is also investigated. Finally, the constructed Jacobi elliptic function solutions reveal four new types of cubic nonlinearity with the TI property.