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 n+1-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.