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.
In recent years there has been a growing interest in the analysis of new discrete nonlinear models since they play a very important role in many physical applications. For example, the question of mobility of solitonic excitations in discrete media is a key issue in many physical contexts; for example the mobility of dislocations, a kind of topological solitons, is of importance in the physics of plastic deformation of metals and other crystalline bodies [1]. Similar questions arise in optics for light pulses moving in optical waveguides or in photorefractive crystal lattices (see e.g., [2] for a relevant recent discussion) and in atomic physics for Bose-Einstein condensates moving through optical lattice potentials (see e.g., [3] for a recent review). These issues may prove critical in aspects related to the guidance and manipulation of coherent, nonlinear wavepackets in solid-state, atomic and optical physics applications.
In particular, the translationally invariant (TI) discrete models [4] have received considerable attention since they admit static solutions that can be placed anywhere with respect to the lattice. Such discretizations have been constructed and investigated for the Klein-Gordon field [4,5,6,7,8,9,10,11,12,13,14,15] and for the nonlinear Schrödinger equation [16,17,18,19,20,21]. For the Hamiltonian TI lattices [5,8], this can be interpreted as the absence of the Peierls-Nabarro (PN) potential [1]. For the non-Hamiltonian lattices, the height of the Peierls-Nabarro barrier is path-dependent but there exists a continuous path along which the work required for a quasi-static shift of the solution along the lattice is zero [12].
In general, one can state that coherent structures in the TI models are not trapped by the lattice and they can be accelerated by even a weak external field. This particular property makes the TI discrete models potentially interesting for physical applications and one such physically meaningful model has been recently reported [13].
For some of the TI models it has been demonstrated that they conserve momentum [4] or energy (Hamiltonian) [5,8] (see also [18,19]). However, we do not know a TI model conserving both momentum and Hamiltonian and, for the Klein-Gordon lattices with classically defined momentum, it was proved that these two conservation laws are mutually exclusive [10].
It may be noted that the TI discrete models can support even moving solutions, but only for selected propagation velocities [19]. In some cases, the exact static or even moving solutions to the TI models can be expressed explicitly in terms of the Jacobi elliptic functions (JEF). Even in the cases when JEF solutions are impossible, static solutions to a TI model can always be obtained iteratively from a nonlinear map (first integral), solving at each step an algebraic equation.
While there has been no universally acceptable definition of TI models, it is fairly easy to describe what a TI solution is. It is a static solution which can be placed anywhere with respect to the lattice. In particular, if there is an analytic TI solution with an arbitrary shift x 0 along the chain or if one can show that there is a corresponding Goldstone mode with zero frequency for any x 0 . As far as TI models are concerned, it is believed that they should possess following properties (i) they admit static solutions which can be placed anywhere with respect to the lattice, which can be associated with the absence of PN barrier. Note that in case analytical static solutions can be constructed with an arbitrary shift along the chain, x 0 , that would automatically imply the absence of the PN barrier (ii) static version of TI discrete models are integrable, i.e. the static problems are reducible to a first-order difference equation which can be viewed as a nonlinear map from which static solutions can be constructed iteratively (in this study we will show that non-integrable three-point static problems also can have particular TI solutions derivable from a set of two lower-order finite-difference equations, and one of those equations is a two-point one while another is a three-point one) (iii) static solutions in TI models possess the translational Goldstone mode with zero frequency for any x 0 .
A prototype class of discrete models, relevant to a variety of applications are the so called discrete φ 4 models which feature a cubic nonlinearity. The purpose of this paper is to study in detail several issues related to TI models. In particular we consider a rather general discrete φ 4 model with cubic nonlinearity which is invariant under the interchange of φ n+1 and φ n-1
with the model parameters satisfying the constraint
In Eq. ( 1), φ n (t) is the unknown function defined on the lattice x n = hn with the lattice spacing h > 0 and overdot means derivative with respect to time t. Without the loss of generality it is sufficient to consider λ = 1 or λ = -1.
If model parameters A k are constant (i.e. indepen
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