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Certain solutions of autonomous PDEs without any boundary conditions describing the spatiotemporal evolution of a dependent variable in an unbounded spatial domain can be characterised as a travelling wave moving with constant speed. In the simplest case, such PDEs can be reduced to a single autonomous second order ODE with one dependent variable. For certain parameter values it has been shown using perturbations methods in combination with ans\"{a}tze that numerous such second order ODEs have analytical travelling wave solutions described by a simple sigmoid function. However, this methodology provides no leverage on the problem of finding a generalised class of models possessing such analytical travelling wave solutions. The most efficient methods for both finding analytical solutions and constructing classes of ODEs are based on Lie symmetries which are transformations known as one parameter $\mathcal{C}^{\infty}$ diffeomorphisms mapping solutions to other solutions. Recently, analytical solutions of a second order ODE encapsulating numerous oscillatory models as well as some of the previously mentioned travelling wave models with simple analytical solutions have been found by means of a two dimensional Lie algebra. Based on this Lie algebra, we construct the most general class of integrable autonomous second order ODEs for which these symmetries are manifest. Moreover, we show that a sub-class of second order ODEs has simple analytical travelling wave solutions described by a sigmoid function. Lastly, we characterise the action of the two symmetries in this Lie algebra on these simple analytical travelling wave solutions and we relate our sub-class of ODEs to previously known integrable travelling wave models., Comment: 21 pages, 1 figure

In this paper, we derive general monotone quantities and geometric inequalities associated with $p$-capacitary functions in asymptotically flat $3$-manifolds with simple topology and nonnegative scalar curvature. The inequalities become equalities on the spatial Schwarzschild manifolds outside rotationally symmetric spheres. This generalizes Miao's result \cite{M} from $p=2$ to $p\in (1, 3)$. As applications, we recover mass-to-$p$-capacity and $p$-capacity-to-area inequalities due to Bray-Miao \cite{BM} and Xiao \cite{Xiao}., Comment: In this version, we extended the range of $k$ from $[0, 1]$ to $(-1, 1]$ in Theorem 1.1. As a consequence, we removed the assumption of nonnegative Hawking mass in Theorem 1.3