# Integrability, exact reductions and special solutions of the KP-Whitham equations

## TL;DR

This paper investigates reductions of the KP-Whitham system, demonstrating their integrability and deriving exact solutions relevant to physical scenarios involving solitons and harmonic waves.

## Contribution

It introduces new integrable reductions of the KP-Whitham system and analyzes their structure and solutions using the Haantjes tensor and hydrodynamic reductions.

## Key findings

- The KP-Whitham system can be split into decoupled and coupled parts.
- Both parts are shown to be integrable via the Haantjes tensor and hydrodynamic reductions.
- Exact solutions for physical scenarios are derived from these reductions.

## Abstract

Reductions of the KP-Whitham system, namely the (2+1)-dimensional hydrodynamic system of five equations that describes the slow modulations of periodic solutions of the Kadomtsev-Petviashvili (KP) equation, are studied. Specifically, the soliton and harmonic wave limits of the KP-Whitham system are considered, which give rise in each case to a four-component (2+1)-dimensional hydrodynamic system. It is shown that a suitable change of dependent variables splits the resulting four-component systems into two parts: (i) a decoupled, independent two-component system comprised of the dispersionless KP equation, (ii) an auxiliary, two-component system coupled to the mean flow equations, which describes either the evolution of a linear wave or a soliton propagating on top of the mean flow. The integrability of both four-component systems is then demonstrated by applying the Haantjes tensor test as well as the method of hydrodynamic reductions. Various exact reductions of these systems are then presented that correspond to concrete physical scenarios.