Differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann hierarchy revisited

Abstract

Abstract. A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic hierarchy is revisited, its new Lax type representation is constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of the generalized Riemann type hierarchy are also discussed by means of the gradient-holonomic and geometric methods.

Dark Equations and Their Light Integrability

“…Using a combined gradient-holonomic-differential-algebraic approach [1,32,33,36,37] to test the Lax integrability of nonlinear dynamical systems, we showed that the three-component Burgers systems (1.1) and (1.2) possess 2-dimensional matrix Lax representations and corresponding symmetry recursion operators -which do not allow the usual bi-Poisoning factorization. In particular, (1.2) does not have a related countable hierarchy of, either local or non-local, conserved quantities.…”

Hidden Symmetries of Lax Integrable Nonlinear Systems

“…This system can be considered as a slight generalization of the dispersive Riemann hydrodynamic system (73), extensively studied by means of different mathematical tools in [8,9,32,35,37,41]. For the case it is well known [10,12] that the system (80) is a smooth Lax integrable bi-Hamiltonian flow on the -periodic functional manifold…”

“…A new differential-algebraic approach, elaborated in [9] for revisiting the integrability analysis of generalized Riemann type hydrodynamical Equation (73), made it possible to prove the Lax integrability of new nonlinear Hamiltonian dynamical systems representing Riemann type hydrodynamic Equations (80), (87) and (91). In particular, the integrability prerequisites of these dynamical system, such as compatible Poissonian structures, an infinite hierarchy of conservation laws and related Lax representation have been constructed by means of both the symplectic gradient-holonomic approach [10,12,48] and innovative differential-algebraic tools devised recently [8,9,35] for analyzing the integrability of a special infinite hierarchy of Riemann type hydrodynamic systems.…”

“…Among different approaches to coping with it one can mention, for instance, the classical Kowalewskaya-Painlevé method and its modifications, the Mikhaylov-Shabat [7] recursion operator method, based on analyzing the Lie-Backlund symmetries and some other techniques, which appeared to be reasonably effective in diverse applications, especially for classifying nonlinear integrable dynamical systems possessing special structure. Recently, when studying integrability properties of infinite so called Riemann type hydrodynamical hierarchies, a new direct approach to testing the Lax integrability of a priori given nonlinear dynamical systems with special structure, based on treating the related symplectic and differential-algebraic structures of differentiations, was suggested [8] and devised in [9]. By means of this technique the direct integrability problem was effectively reduced to the classical one of finding the corresponding compatible representations in suitably constructed differential rings.…”

“…for N   and extensively studied in [8,9,32,35,37,41], where it was proved that it is a Lax integrable bi-Hamiltonian flow on the manifold N M and possesses an infinite hierarchy of mutually commuting dispersive Lax integrable Hamiltonian flows. For 2 N  it is well known [10,12] that the system (92) is a smooth Lax integrable bi-Hamiltonian flow on the -periodic functional manifold 2π 2 , M whose Lax representation is given by the compatible linear system…”