The $N$-wave equations with $\mathcal{PT}$ symmetry

We study extensions of $N$-wave systems with $\mathcal{PT}$ symmetry and describe the types of (nonlocal) reductions leading to integrable equations invariant under the $\mathcal P$ (spatial reflection) and $\mathcal T$ (time reversal) symmetries. We derive the corresponding constraints on the fundamental analytic solutions and the scattering data. Based on examples of three-wave and four-wave systems (related to the respective algebras $sl(3,\mathbb C)$) and $so(5,\mathbb C)$), we discuss the properties of different types of one- and two-soliton solutions. We show that the $\mathcal{PT}$-symmetric three-wave equations can have regular multisoliton solutions for some specific choices of their parameters.