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Special relativity (SR) with a privileged frame is a framework, which, like the standard relativity theory, is based on the relativity principle and the universality of the (two-way) speed of light but includes a privileged frame as an essential element. It is developed using the following first principles: (1) Anisotropy of the one-way speed of light in an inertial frame is due to its motion with respect to the privileged frame; (2) Space-time transformations between inertial frames leave the equation of anisotropic light propagation invariant; (3) A set of the transformations possesses a group structure. The Lie group theory apparatus is applied to define groups of transformations. The correspondingly modified general relativity (GR), like the standard GR, is based on the equivalence principle but with the properly modified space-time local symmetry in which an invariant combination differs from the Minkowski interval of the standard SR. Applying the modified GR to cosmology yields the luminosity distance -- redshift relation corrected such that the observed deceleration parameter can be negative as it was derived from the data for type Ia supernovae. Thus, the observed negative values of the deceleration parameter can be explained without introducing the dark energy. A number of other observations, such as Baryon Acoustic Oscillations (BAO) and Cosmic Microwave Background (CMB) also can be well fit to the cosmological model arising from the GR based on the SR with a privileged frame., Comment: arXiv admin note: text overlap with arXiv:1610.08771

The modern view, that there exists a preferred frame of reference related to the cosmic microwave background (CMB), is in apparent contradiction with the principles of special relativity. The purpose of the present study is to develop a counterpart of the special relativity theory that is consistent with the existence of a preferred frame but, like the standard relativity theory, is based on the relativity principle and universality of the (\textit{two-way}) speed of light. In the framework developed, a degree of anisotropy of the one-way velocity acquires meaning of a characteristic of the really existing anisotropy caused by motion of an inertial frame relative to the preferred frame. The anisotropic special relativity kinematics is developed based on the first principles: (1) Space-time transformations between inertial frames leave the equation of anisotropic light propagation invariant and (2) A set of the transformations possesses a group structure. The Lie group theory apparatus is applied to define groups of space-time transformations between inertial frames. Applying the consequences of the transformations to the problem of calculating the CMB temperature distribution yields an equation in which the angular dependence coincides with that obtained on the basis of the standard relativity theory but the mean temperature is corrected by the terms second order in the observer velocity. From conceptual point of view, it eliminates the inconsistency of the usual approach when formulas of the standard special relativity are applied to define effects caused by motion with respect to the preferred frame.

The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for surface elevation for a prescribed relation between the orders of the two expansion parameters, the amplitude parameter $\alpha$ and the long wavelength (or shallowness) parameter $\beta$, is developed. Unlike the heuristic approaches found in the literature, when modifications are made in the equation for surface elevation itself, the procedure starts from the consistently truncated asymptotic expansions for unidirectional waves, a counterpart of the Boussinesq system of equations for the surface elevation and the bottom velocity, from which the leading order and higher order equations for the surface elevation can be obtained by iterations. The relations between the orders of the two small parameters are taken in the form $\beta=O(\alpha^n)$ and $\alpha=O(\beta^m)$ with $n$ and $m$ specified to some important particular cases. The analysis shows, in particular, that some evolution equations, proposed before as model equations in other physical contexts (like the Gardner equation, the modified KdV equation, and the so-called 5th-order KdV equation), can emerge as the leading order equations in the asymptotic expansion for the unidirectional water waves, on equal footing with the KdV equation. The results related to the higher orders of approximation provide a set of consistent higher order model equations for unidirectional water waves which replace the KdV equation with higher-order corrections in the case of non-standard ordering when the parameters $\alpha$ and $\beta$ are not of the same order of magnitude. (See the paper for the complete abstract.), Comment: 17 pages, LaTeX, uses IOP style files, accepted for publication in J. Phys. A: Math. Theor. (see also the copyright notice in the paper)