Separable trajectory

This article defines a property that can be evaluated for a trajectory on the space of functions on a manifold

Definition

Let $M$ be a manifold and $u=u(t,x)$ be a function $\mathbb{R}\times M\to \mathbb{R}$, where:

• $t$ denotes the time parameter, and varies in $\mathbb{R}$
• $x$ denotes the spatial parameter, and varies in $M$

In other words, $u$ is a trajectory (or path) in the space of all functions from $M$ to $\mathbb{R}$.

Then $u$ is said to be separable if $u(t,x)=f(t)g(x)$ for suitable functions $f:\mathbb{R}\to \mathbb{R}$ and $g:M\to \mathbb{R}$. Equivalently, $u$ is separable if and only if it satisfies the property:

$u(t_1,x_1)u(t_2,x_2)=u(t_{!},x_2)u(t_2,x_1)$

for all times $t_1,t_2\in \mathbb{R}$ and points $x_1,x_2\in M$.

Note that for a separable trajectory, we get the following differential equations for $f$ and $g$:

$f^{\prime }(t)=\lambda f(t)$

and

$(F(g))(x)=\lambda g(x)$

where $\lambda \in \mathbb{R}$ is a real number. In fact the entire family of separable trajectories is parametrized by $\lambda$.

The solution to the first equation is $f(t)=C{e}^{\lambda t}$.

When $F$ is a linear differential operator, the solution to the second equation is any eigenvector $g$ for $F$ having eigenvalue $\lambda$ (we can use the term eigenvector only when the operator is linear though an equivalent notion makes sense for arbitrary differnetial operators).