In Noether’s original presentation of her celebrated theorem of 1918, allowance was made for the dependence of the coefficient functions of the differential operator, which generated the infinitesimal transformation of the action integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades, this diminution of the power of Noether’s theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether’s theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence on the independent variables.

We explore the phase-space of homogeneous and anisotropic spacetimes within symmetric teleparallel $f(Q)$ gravity. Specifically, we consider the Kantowski-Sachs and locally rotational Bianchi III geometries to describe the physical space. By analyzing the phase-space, we reconstruct the cosmological history dictated by $f(Q)$ gravity and comment about the theory’s viability. Our findings suggest that the free parameters of the connection must be constrained to eliminate nonlinear terms in the field equations. Consequently, new stationary points emerge, rendering the theory cosmologically viable. We identify the existence of anisotropic accelerated universes, which may correspond to the preinflationary epoch.

We study the Jimbo – Miwa equation and two of its extended forms, as proposed by Wazwaz et al., using Lie’s group approach. Interestingly, the travelling – wave solutions for all the three equations are similar. Moreover, we obtain certain new reductions which are completely different for each of the three equations. For example, for one of the extended forms of the Jimbo – Miwa equation, the subsequent reductions leads to a second – order equation with Hypergeometric solutions. In certain reductions, we obtain simpler first – order and linearisable second – order equations, which helps us to construct the analytic solution as a closed – form function. The variation in the nonzero Lie brackets for each of the different forms of the Jimbo – Miwa also presents a different perspective. Finally, singularity analysis is applied in order to determine the integrability of the reduced equations and of the different forms of the Jimbo – Miwa equation.

The 1 + 2 dimensional Bogoyavlensky-Konopelchenko equation is investigated for its solution and conservation laws using the Lie point symmetry analysis. In the recent past, certain work has been do...