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For the nonlinear Klein-Gordon type models, we describe a general method of discretization in which the static kink can be placed anywhere with respect to the lattice. These discrete models are therefore free of the static Peierls-Nabarro potential. Previously reported models of this type are shown to belong to a wider class of models derived by means of the proposed method. A relevant physical consequence of our findings is the existence of a wide class of discrete Klein-Gordon models where slow kinks practically do not experience the action of the Peierls-Nabarro potential. Such kinks are not trapped by the lattice and they can be accelerated by even weak external fields.

We propose a generalization of the discrete Klein-Gordon models free of the Peierls-Nabarro barrier derived in Nonlinearity 12, 1373Nonlinearity 12, (1999 and Phys. Rev. E 72, 035602(R) (2005), such that they support not only kinks but a one-parameter set of exact static solutions. These solutions can be obtained iteratively from a two-point nonlinear map whose role is played by the discretized first integral of the static Klein-Gordon field, as suggested in J. Phys. A 38, 7617 (2005). We then discuss some discrete φ 4 models free of the Peierls-Nabarro barrier and identify for them the full space of available static solutions, including those derived recently in Phys. Rev. E 72 036605 (2005) but not limited to them. These findings are also relevant to standing wave solutions of discrete nonlinear Schrödinger models. We also study stability of the obtained solutions. As an interesting aside, we derive the list of solutions to the continuum φ 4 equation that fill the entire two-dimensional space of parameters obtained as the continuum limit of the corresponding space of the discrete models.

A wrinkle formation mechanism with cutaneous aging is addressed through a mechanical calculation of linear buckling. Skin is divided into five mechanically distinct layers in this study. In general, the outer layer is stiffer than the inner layer, so buckling occurs in the outer layer against the uniform compression caused by muscle contraction. This buckling damages the skin and affects the formation of permanent wrinkles. We propose a multistage buckling theory for evaluation of the wrinkle property, namely, the specific wrinkle size and critical strain in three stages. The specific wrinkle size is derived as the wavelength of the minimum-buckling mode for infinite-length skin. A sensitivity analysis is carried out to investigate the effect of age-related changes of the mechanical parameters on the wrinkle property. We employ some aging hypotheses and prepare two sets of mechanical parameters, one for young and one for aged skin. The numerical results show that the buckling mode switch from Stage I to Stage II is the main reason why pronounced wrinkles suddenly appear in aged skin.

We consider nonlinear Klein-Gordon wave equations and illustrate that standard discretizations thereof (involving nearest neighbors) may preserve either standardly defined linear momentum or total energy but not both. This has a variety of intriguing implications for the "non-potential" discretizations that preserve only the linear momentum, such as the self-accelerating or self-decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices. On the other hand, spatially discrete systems (of coupled nonlinear ordinary differential equations) are also relevant as discretizations and computational implementations of the corresponding continuum field theories that are applicable to a variety of contexts such as statistical mechanics [10], solid state physics [11], fluid mechanics [12] and particle physics [13] (see also references therein). Nonlinear Klein-Gordon type equations are a prototypical example among such wave models and their variants span a variety of applications including Josephson junctions in superconductivity, cosmic domain walls in cosmology, elementary particles in particle physics and denaturation bubbles in the DNA, among others.In this communication, we examine some of the key properties that ensue when discretizing nonlinear KleinGordon (KG) equations, using nearest neighbor approximations (which are the most standard ones implemented in the literature; see e.g., [1]). In particular, we focus on the physically relevant invariances of the continuum equation (more specifically, the conservation of the linear momentum and of the total energy of the system) and illustrate the surprising result that if we demand that the energy be conserved, then the momentum cannot be conserved, while if we demand that the momentum be conserved then the energy cannot be conserved (resulting in a so-called non-potential model [14]). Our presentation will be structured as follows. First, we will provide the general mathematical setting of KG equations and study their discretizations that conserve linear momentum and energy, comparing and constructing the properties of the two. Then, we are going to give an application of our

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From a wide class of translationally invariant discrete nonlinear Schrödinger (DNLS) equations, we extract a two-parameter subclass corresponding to Kerr nonlinearity for which any stationary solution can be derived recurrently from a quadratic equation. This subclass, which incorporates the integrable (Ablowitz–Ladik) lattice as a special case, admits exact stationary solutions that are derived in terms of the Jacobi elliptic functions. Exact moving solutions for the discrete equations are also obtained. In the continuum limit, the constructed stationary solutions reduce to the exact moving solutions to the continuum NLS equation with Kerr nonlinearity. Numerical results are also presented for the special case of localized solutions, including sech (pulse, or bright soliton), tanh (kink, or dark soliton) and 1/tanh (called here inverted kink) profiles. For these solutions, we discuss their linearization spectra and their mobility. Particularly, we demonstrate that discrete dark solitons are dynamically stable for a wide range of lattice spacings, contrary to what is the case for their standard DNLS counterparts. Furthermore, the bright and dark solitons in the non-integrable, translationally invariant lattices can propagate at slow speed without any noticeable radiation.