On symmetry reduction and some classes of invariant solutions of the (1+3)-dimensional homogeneous Monge-Ampère equation

We study the relationship between structural properties of the two-dimensional nonconjugate subalgebras of the same rank of the Lie algebra of the Poincaré group P(1,4) and the properties of reduced equations for the (1+3)-dimensional homogeneous Monge-Ampère equation. In this paper, we present some of the results obtained concerning symmetry reduction of the equation under investigation to identities. Some classes of the invariant solutions (with arbitrary smooth functions) are presented.

A nonlocal Boussinesq equation: multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N = 2, 3, 4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.

Revisiting cosmologies in teleparallelism

We discuss the most general field equations for cosmological spacetimes for theories of gravity based on non-linear extensions of the non-metricity scalar and the torsion scalar. Our approach is based on a systematic symmetry-reduction of the metric-affine geometry which underlies these theories. While for the simplest conceivable case the connection disappears from the field equations and one obtains the Friedmann equations of General Relativity, we show that in $f(\mathbb{Q})$ cosmology the connection generically modifies the metric field equations and that some of the connection components become dynamical. We show that $f(\mathbb{Q})$ cosmology contains the exact General Relativity solutions and also exact solutions which go beyond. In $f(\mathbb{T})$~cosmology, however, the connection is completely fixed and not dynamical.

Symmetry reduction in a hyperpolarization-activated homotetrameric ion channel

Plants obtain nutrients from the soil via transmembrane transporters and channels in their root hairs, from which ions radially transport in towards the xylem for distribution across the plant body. We determined structures of the hyperpolarization-activated channel AKT1 from Arabidopsis thaliana, which mediates K+ uptake from the soil into plant roots. These structures of AtAKT1 embedded in lipid nanodiscs show that the channel undergoes a reduction of C4 to C2 symmetry, possibly to regulate its electrical activation.