New solitary waves for the Klein–Gordon–Zakharov equations

This paper studies new solitary waves for the Klein–Gordon–Zakharov equations. The obtained results are diverse and some specific ones emerge as dark, bright and bell-shape. The two integration schemes lead to obtaining waves which propagate without deformation as illustrated in graphical representations. Our obtained results are more specific compared to those obtained by Refs. 8 and 31 [C. H. Zhao and Z. M. Sheng, Acta Phys. Sin. 53 (2004) 29; S. Yakada, B. Depelair, G. Betchewe and S. Y. Doka, Optik 197 (2019) 163108].

New general decay results for a viscoelastic plate equation with a logarithmic nonlinearity

In this paper, we investigate the stability of the solutions of a viscoelastic plate equation with a logarithmic nonlinearity. We assume that the relaxation function g satisfies the minimal condition $$ g^{\prime }(t)\le -\xi (t) G\bigl(g(t)\bigr), $$g′(t)≤−ξ(t)G(g(t)), where ξ and G satisfy some properties. With this very general assumption on the behavior of g, we establish explicit and general energy decay results from which we can recover the exponential and polynomial rates when $G(s) = s^{p}$G(s)=sp and p covers the full admissible range $[1, 2)$[1,2). Our new results substantially improve and generalize several earlier related results in the literature such as Gorka (Acta Phys. Pol. 40:59–66, 2009), Hiramatsu et al. (J. Cosmol. Astropart. Phys. 2010(06):008, 2010), Han and Wang (Acta Appl. Math. 110(1):195–207, 2010), Messaoudi and Al-Khulaifi (Appl. Math. Lett. 66:16–22, 2017), Mustafa (Math. Methods Appl. Sci. 41(1):192–204, 2018), and Al-Gharabli et al. (Commun. Pure Appl. Anal. 18(1):159–180, 2019).

DYNAMICS WITH EXTERNAL FORCES REVISITED

The study of isolated system is of little interest to physicists since it is not applicable to practical situations. Analytical mechanics should be brought closer to control theory. Behavior of a system coupled to external controlling devices should be examined. There is a class of controlling devices representable by external forces. We proposed a framework for describing the dynamics of mechanical devices subject to control by external forces. The original version of the theory was presented in G. Marmo, W. Tulczyjew and P. Urbański, Dynamics of autonomous systems with external forces, Acta Phys. Polon. B33 (2002) 1181–1240. The geometric constructions used in original paper are clarified. The Legendre–Fenchel transformation is described and the concept of external force is introduced. Liouville structures are defined.