Differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann hierarchy revisited (original) (raw)
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A novel integrability analysis of a generalized Riemann type hydrodynamic hierarchy
Miskolc Mathematical Notes
The complete integrability of a generalized Riemann type hydrodynamic hierarchy is studied by means of a novel combination of symplectic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are constructed. The current investigation provides an interesting glimpse of what is apparently a far wider range of applications.
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A new Riemann type hydrodynamical hierarchy and its integrability analysis
2009
The complete integrability of a generalized Riemann type hydrodynamic system is studied by means of symplectic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, Lax type representation and related infinite hierarchy of conservation laws are constructed.
A differential–algebraic approach to studying the Lax integrability of a generalized Riemann type hydrodynamic hierarchy is revisited and a new Lax representation is constructed. The related bi-Hamiltonian integrability and compatible Poissonian structures of this hierarchy are also investigated using gradient-holonomic and geometric methods. The complete integrability of a new generalized Riemann hydrodynamic system is studied via a novel combination of symplectic and differential–algebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are obtained. In addition, the differential–algebraic approach is used to prove the complete Lax integrability of the generalized Ostrovsky–Vakhnenko and a new Burgers type system, and special cases are studied using symplectic and gradient-holonomic tools. Compatible pairs of polynomial Poissonian structures, matrix Lax representations and infinite hierarchies of conservation laws are derived.
A Multi-Component Lax Integrable Hierarchy with Hamiltonian Structure
2008
A Lax integrable multi-component hierarchy is generated from a matrix spectral problem involving two arbitrary matrices, within the the framework of zero curvature equations. Its Hamiltonian structure is established be means of the trace variational identity. An example with five components is presented, together with the first two Hamiltonian systems in the resulting soliton hierarchy.
Nonlinearity, 2010
Based on the gradient-holonomic algorithm we analyze the integrability property of the generalized hydrodynamical Riemann type equation D N t u = 0 for arbitrary N ∈ Z +. The infinite hierarchies of polynomial and non-polynomial conservation laws, both dispersive and dispersionless are constructed. Special attention is paid to the cases N = 2, 3 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.