NS-Leibniz bialgebras, the Yang--Baxter equation and relative Rota--Baxter operators

We introduce NS-Leibniz bialgebras and establish their equivalence with matched pairs and Manin triples of NS-Leibniz algebras. In the coboundary case, skew-symmetric solutions of the NS-Leibniz Yang--Baxter equation yield NS-Leibniz bialgebras and correspond to $\mathcal{O}$-operators for the dual regular representation. We study quasi-triangular and factorizable structures, and relate factorizable tensors to quadratic Rota--Baxter NS-Leibniz algebras of any fixed nonzero weight. We also construct compatible bialgebra structures from skew-adjoint Nijenhuis operators commuting with the Rota--Baxter operator.