Article

# Dissipative lattice model with exact traveling discrete kink-soliton solutions: discrete breather generation and reaction diffusion regime.

Laboratoire d'Electronique, Informatique et Image (LE21) Université de Bourgogne, Aile des Sciences de l'Ingénieur, BP 47870, 21078 Dijon Cedex, France.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 01/2000; 60(6 Pt B):7484-9. DOI: 10.1103/PhysRevE.60.7484 Source: PubMed

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**ABSTRACT:**In this paper, the dynamic behaviour of the "click" mechanism is analysed. A more accurate model is used than in the past, in which the limits of movement due to the geometry of the flight mechanism are imposed. Moreover, the effects of different damping models are investigated. In previous work, the damping model was assumed to be of the linear viscous type for simplicity, but it is likely that the damping due to drag forces is nonlinear. Accordingly, a model of damping in which the damping force is proportional to the square of the velocity is used, and the results are compared with the simpler model of linear viscous damping. Because of the complexity of the model an analytical approach is not possible so the problem has been cast in terms of non-dimensional variables and solved numerically. The peak kinetic energy of the wing root per energy input in one cycle is chosen to study the effectiveness of the "click" mechanism compared with a linear resonant mechanism. It is shown that, the "click" mechanism has distinct advantages when it is driven below its resonant frequency. When the damping is quadratic, there are some further advantages compared to when the damping is linear and viscous, provided that the amplitude of the excitation force is large enough to avoid the erratic behaviour of the mechanism that occurs for small forces.Journal of Theoretical Biology 09/2011; 289:173-80. · 2.35 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The method of simplest equation is applied for analysis of a class of lattices described by differential-difference equations that admit traveling-wave solutions constructed on the basis of the solution of the Riccati equation. We denote such lattices as Riccati lattices. We search for Riccati lattices within two classes of lattices: generalized Lotka - Volterra lattices and generalized Holling lattices. We show that from the class of generalized Lotka - Volterra lattices only the Wadati lattice belongs to the class of Riccati lattices. Opposite to this many lattices from the Holling class are Riccati lattices. We construct exact traveling wave solutions on the basis of the solution of Riccati equation for three members of the class of generalized Holing lattices.Journal of Theoretical and Applied Mechanics. 08/2012; 42(3). -
##### Article: Exact static solutions of a generalized discrete phi4 model including short-periodic solutions

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**ABSTRACT:**We carry out a comprehensive analysis of a generalized discrete phi4 model, of which virtually all phi4 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.Journal of Physics A Mathematical and Theoretical 01/2009; 42(14). · 1.77 Impact Factor

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