Whitham-Broer-Kaup Systems in Multi-dimensions: Quantum and Resonant NLS Connections

TL;DR

This paper explores multi-dimensional extensions of Whitham-Broer-Kaup systems, linking them to quantum and resonant nonlinear Schrödinger equations, and derives integrable reductions using advanced mathematical techniques.

Contribution

It introduces a novel n+1-dimensional Whitham-Broer-Kaup system and establishes its connection to multi-dimensional resonant NLS equations, along with integrable similarity reductions.

Findings

Constructed a new multi-dimensional Whitham-Broer-Kaup system.

Linked the system to multi-dimensional resonant NLS equations.

Derived integrable similarity reductions using Painlevé equations.

Abstract

An overview is presented of quantum and resonant nonlinear Schr\"odinger 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-Painlev\'e II and associated Painlev\'e XXXIV integrable similarity reductions are derived.