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For a five-parameter discrete $\phi^4$ model, we derive various exact staticsolutions, including the staggered ones, in the form of the basic Jacobielliptic functions $\sn$, $\cn$, and $\dn$, and also in the form of theirhyperbolic function limits such as kink ($\tanh$) and single-humped pulse($\sech$) solutions. Such solutions are admitted by the considered model inseven cases, two of which have been discussed in the literature, and theremaining 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 thegeneralized 5-parameter model. All the Jacobi elliptic, hyperbolic andtrigonometric function solutions (including the staggered ones) aretranslationally invariant (TI), i.e., they can be shifted along the lattice byan arbitrary $x_0$, but among the short-periodic solutions there are both TIand non-TI solutions. The stability of these solutions is also investigated.Finally, the constructed Jacobi elliptic function solutions reveal four newtypes of cubic nonlinearity with the TI property.

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