Exact static solutions of a generalized discrete φ⁴ model including short-periodic solutions

Abstract

For a five-parameter discrete φ⁴ 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 sine, staggered sine 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 x₀, 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.