Chaos Solitons Fractals, 2014, Volume 59, Pages 59–81
Integrability of and differential–algebraic structures for spatially 1D hydrodynamical systems of Riemann type
D. Blackmorea, Ya. A. Prikarpatskybc, N. N. Bogolyubov (Jr.)de, A. K. Prikarpatskif
a Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102-1982, United States
b Department of Applied Mathematics, Agrarian University of Krakow, Poland
c Institute of Mathematics of NAS, Kyiv, Ukraine
d Abdus Salam International Centre of Theoretical Physics, Trieste, Italy
e V.A. Steklov Mathematical Institute of RAS, Moscow, Russian Federation
f AGH University of Science and Technology, Craców 30059, Poland
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.
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