Whitham–Broer–Kaup systems in multi-dimensions: Quantum and resonant NLS connections

An overview is presented of quantum and resonant nonlinear Schrödinger equation links to Whitham–Broer–Kaup type systems. A novel [Formula: see text] dimensional extension of the Whitham–Broer–Kaup hydrodynamic system is constructed with connection to an equivalent multi-dimensional resonant NLS equation. Hybrid Ermakov–Painlevé II and associated Painlevé XXXIV integrable similarity reductions are derived.