What are some examples of (semi-)explicit solutions of the incompressible Euler equations which satisfy the following

1. they are pressureless
2. they are periodic in space
3. they have nontrivial time dependence (i.e. not steady state)?

In other words, I am looking for explicit or semi explicit solutions $u:\mathbb{R}\times \mathbb{T}^d\rightarrow \mathbb{R}$, where $\Bbb T^d$, $d\ge 2$ is the $d$-dimensional torus, to the Euler system of PDEs $$ \begin{cases} \partial_t u+u\cdot \nabla u=0,\\ \operatorname{div} u =0, \end{cases} $$ with $\partial_t u$ not everywhere zero.