On integrability of a new dynamical system associated with the BBM-type hydrodynamic flow

TL;DR

This paper demonstrates the complete integrability of a new BBM-type hydrodynamic system by constructing its hierarchy of conservation laws and compatible Poisson structures, revealing deep links with Riemann-type equations.

Contribution

It introduces a novel integrability analysis of a higher-order BBM-type equation using the gradient-holonomic scheme and constructs its infinite hierarchy of conservation laws.

Findings

Proved the complete integrability of the new BBM-type system.

Constructed an infinite hierarchy of conservation laws.

Identified two compatible Poisson structures.

Abstract

This article explores the exceptional integrability property of a family of higher-order Benjamin-Bona-Mahony (BBM)-type nonlinear dispersive equations. Here, we highlight its deep relationship with a generalized infinite hierarchy of the integrable Riemann-type hydrodynamic equations. A previous Lie symmetry analysis revealed a particular case which was conjectured to be integrable. Namely, a Lie-Baecklund symmetry exists, thus highlighting another associated dynamical system. Here, we investigate these two equations using the gradient-holonomic integrability scheme. Moreover, we construct their infinite hierarchy of conservation laws analytically, using three compatible Poisson structures to prove the complete integrability of both dynamical systems. We investigate these two equations using the so-called gradient-holonomic integrability scheme. Based on this scheme, applied to the equation under consideration, we have analytically constructed its infinite hierarchy of conservation laws, derived two compatible Poisson structures and proved its complete integrability.