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Chaos Solitons Fractals, 2014,  59,  59–81 (Mi chsf1)  

Integrability of and differentialalgebraic 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 differentialalgebraic 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 differentialalgebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are obtained.
In addition, the differentialalgebraic approach is used to prove the complete Lax integrability of the generalized OstrovskyVakhnenko 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.

National Science Foundation CMMI-1029809
Scientific and Technological Research Council of Turkey (TÜBITAK) TUBITAK/NASU-110T558 Project
D.B. acknowledges the National Science Foundation (Grant CMMI-1029809), A.P. and Y.P. acknowledge the Scientific and Technological Research Council of Turkey (TUBITAK/NASU-110T558 Project) for partial support of their research.


DOI: https://doi.org/10.1016/j.chaos.2013.11.012


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: 09.02.2013
:21.11.2013
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