The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations

Abstract

Based on the gradient-holonomic algorithm we analyze the integrability property of the generalized hydrodynamical Riemann type equation for arbitrary $N\in {\mathbb{Z}}_{+}.$ The infinite hierarchies of polynomial and non-polynomial conservation laws, both dispersive and dispersionless are constructed. Special attention is paid to the cases and N=4 for which the conservation laws, Lax type representations and bi-Hamiltonian structures are analyzed in detail. We also show that the case N=2 is equivalent to a generalized Hunter-Saxton dynamical system, whose integrability follows from the results obtained. As a byproduct of our analysis we demonstrate a new set of non-polynomial conservation laws for the related Hunter-Saxton equation.

17 pages

References in corpus (1)

• Differential-Algebraic Integrability Analysis of the Generalized Riemann Type and Korteweg-de Vries Hydrodynamical Equations

Cited by in corpus (5)

• The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited
• The matrix Lax representation of the generalized Riemann equations and its conservation Laws
• Conservation laws and symmetries of Hunter-Saxton equation: revisited
• Conservation laws of the generalized Riemann equations at $N=2,3,4$
• On the Complete Integrability of Nonlinear Dynamical Systems on Discrete Manifolds within the Gradient-Holonomic Approach