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In this paper, we consider two nonlinear models arising in mathematical physics, namely, the Landau–Ginzburg–Higgs (LGH) equation and the nonlinear dispersive modified Benjamin–Bona (DMBBM) equation. The LGH model describes the exchange between mid-latitude Rossby and equatorial waves, as well as nonlinear waves with long-range and weak scattering interactions between tropical tropospheres and mid-latitude. The DMBBM model explains surface wave propagation estimates in a nonlinear dispersive medium. We employed He's semi-inverse approach in order to solve these models. Specifically, we present hyperbolic wave solutions. The proposed method is straightforward, reliable, and effective, and its potential for use in solving additional partial differential equations in applied research is encouraging. Appropriate values for the parameters are taken into consideration when simulating certain 2D and 3D graphs that correspond to select solutions using Matlab software. [ABSTRACT FROM AUTHOR]

In this paper we propose four kinds of two-dimensional square beam-like zigzag lattice structures and study their bandgaps and directional propagation of elastic waves. The band structures are calculated by using the finite element method. Both the in-plane and out-of-plane waves are investigated simultaneously via the three-dimensional Euler beam elements. The mechanism of the bandgap generation is analyzed by studying the vibration modes at the bandgap edges. The effects of the geometry parameters of the xy- and z-zigzag lattices on the bandgaps are investigated and discussed. Multiple complete bandgaps are found owing to the separation of the degeneracy by introducing bending arms. The bandgaps are sensitive to the geometry parameters of the periodic systems. The deformed displacement fields of the harmonic responses of a finite lattice structure subjected to harmonic loads at different positions are illustrated to show the directional wave propagation. An extension of the proposed concept to the hexagonal lattices is also presented. The research work in this paper is relevant to the practical design of cellular structures with enhanced vibro-acoustics performance. [ABSTRACT FROM AUTHOR]

This paper presents an elastodynamic analysis of two-dimensional time-harmonic elastic wave propagation in periodically multilayered elastic composites, which are also frequently referred to as one-dimensional phononic crystals, with a periodic array of strip-like interior or interface cracks. The transfer matrix method and the boundary integral equation method in conjunction with the Bloch-Floquet theorem are applied to compute the elastic wave fields in the layered periodic composites. The effects of the crack size, spacing, and location, as well as the incidence angle and the type of incident elastic waves on the wave propagation characteristics in the composite structure are investigated in details. In particular, the band-gaps, the localization and the resonances of elastic waves are revealed by numerical examples. In order to understand better the wave propagation phenomena in layered phononic crystals with distributed cracks, the energy flow vector of Umov and the corresponding energy streamlines are visualized and analyzed. The numerical results demonstrate that large energy vortices obstruct elastic wave propagation in layered phononic crystals at resonance frequencies. They occur before the cracks reflecting most of the energy transmitted by the incoming wave and disappear when the problem parameters are shifted from the resonant ones. [ABSTRACT FROM AUTHOR]