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A new class of elastic materials with orthorhombic symmetry is introduced. For such materials, group velocity surfaces of quasi-longitudinal waves have ellipsoidal shapes. Relations between elastic constants that guaranty ellipsoidal shapes of the group velocity surfaces are indicated. The model materials are used for approximate determination of effective elastic constants of heterogeneous materials by acoustical methods. Examples of reconstruction of elastic constants from acoustical data are presented, and possible errors of the reconstructions are discussed. [ABSTRACT FROM AUTHOR]

This paper presents a brief study of (2-dimensional, spacelike) wave surfaces to a null direction l on a space-time (M , g) and studies how certain imposed symmetries on the set of such wave surfaces can be used to describe other geometrical features of l and (M , g) . It is mainly a review of known material but contains some novelties. For example, the brief discussion of the nature of wave surfaces (when viewed geometrically as wave fronts to a null ray direction) in Wave Surfaces Section is new in the sense that although it appeared in the author's work by the present author, it has not, to the best of his knowledge, appeared in this form anywhere else. Further, the work on conical symmetry and plane waves are, to the best of the author's knowledge, original with him from earlier papers and are reviewed here while the work on complete wave surface (sectional curvature-) symmetry is believed to be entirely new. Geometrical use of the sectional curvature function is employed in many places. The consequences of the various symmetry conditions imposed on the collection of all wave surfaces to a null direction spanned by a null vector l are described in terms of l spanning a principal null direction of the Weyl tensor (if non-zero) at the point concerned (in the sense of Petrov and Bel). [ABSTRACT FROM AUTHOR]

In this paper, we investigated the Gilson-Pickering (GP) equation and many new solutions are obtained with the aid of two different approaches, namely Jacobi elliptic functions and exponential rational function approach. Different choices of the parameters in obtained results lead to the solutions of some well known models, which are Camassa-Holm equation, the Fornberg-Whitham equation and the Rosenau-Hyman equation. The methods considered here can also help to have a panoply of new wave surfaces concerning other related partial differential equations. Further more, 2D and 3D graphical presentations of these surfaces are presented for the various parameters. Moreover, bifurcation behavior of nonlinear travelling waves of GP equation is discussed. Bifurcation theory of planer dynamical system is utilized to observe that considered model contains nonlinear periodic wave, bell shaped solitary wave and shock wave.

• To study wave surfaces and bifurcation of nonlinear periodic waves forn Gilson-Pickering equation. • To use bifurcation theory of planer dynamical system. • To implement Jacobi elliptic functions and exponential rational function approaches. • Solutions of Camassa-Holm, the Fornberg-Whitham and the Rosenau-Hyman equations. • New methodology for obtaining the optical soliton solutions. In this paper, we investigated the Gilson-Pickering (GP) equation and many new solutions are obtained with the aid of two different approaches, namely Jacobi elliptic functions and exponential rational function approach. Different choices of the parameters in obtained results lead to the solutions of some well known models, which are Camassa-Holm equation, the Fornberg-Whitham equation and the Rosenau-Hyman equation. The methods considered here can also help to have a panoply of new wave surfaces concerning other related partial differential equations. Further more, 2D and 3D graphical presentations of these surfaces are presented for the various parameters. Moreover, bifurcation behavior of nonlinear travelling waves of GP equation is discussed. Bifurcation theory of planer dynamical system is utilized to observe that considered model contains nonlinear periodic wave, bell shaped solitary wave and shock wave. [ABSTRACT FROM AUTHOR]