On The Complete Integrability Of The Ostrovsky-Vakhnenko Equation

TL;DR

This paper proves the complete integrability of the Ostrovsky-Vakhnenko equation using advanced mathematical tools, constructing new hierarchies of integrable systems and revealing connections with known equations.

Contribution

It introduces a new bi-infinite hierarchy of Lax type integrable systems and links the Ostrovsky-Vakhnenko equation to the Degasperis-Processi equation.

Findings

Constructed polynomial Poissonian structures and Lax representations.

Established a hierarchy of conservation laws.

Connected the equation to the Degasperis-Processi equation.

Abstract

The complete integrability of the Ostrovsky-Vakhnenko equation is studied by means of symplectic gradient-holonomic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, Lax type representation and related infinite hierarchies of conservation laws are constructed. A new bi-infinite hierarchy of completely Lax type integrable Riemann type hydrodynamical systems is proposed. It is demonstrated that at s=3 the corresponding Riemann type hydrodynamical equation is related with the Degasperis-Processi equation, whose reduction gives rise to the Ostrovsky-Vakhnenko equation.