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

Abstract

We carry out a comprehensive analysis of a generalized discrete ϕ4 model, of which virtually all ϕ4 models discussed in the literature are particular cases. For this model we construct the exact solutions in the form of the basic Jacobi elliptic, hyperbolic and sine functions, and also give a list of short-periodic and even aperiodic solutions. Some of those solutions coincide with the known ones, others generalize the existing solutions and the rest of them are new. We then discuss the relation between the models supporting exact static solutions and the two-point maps. In particular, we show that some of the short-periodic and sine solutions can be found from factorized difference equations and even from a set of two difference equations, one of the first and another of the second order. Particular attention is paid to the discussion of the exceptional discrete (ED) models defined as models supporting the translationally invariant (TI) static solutions that can be placed arbitrarily with respect to the lattice. We show that some of the derived short-periodic solutions are TI ones while the others are not. For the TI static solutions we demonstrate the existence of the translational Goldstone mode for any location of the solution with respect to the lattice. We then analyze numerically the stability and other properties of the TI kink solutions. In conclusion, we divide the ED models into two classes: the ED I models support a two-parameter set of TI static solutions, while the ED II models support only a one-parameter set of such solutions.