Integrability of and differential-algebraic structures for spatially 1D hydrodynamical systems of Riemann type

Denis Blackmore, Yarema A. Prykarpatsky, Nikolai N. Bogolubov, Anatolij K. Prykarpatski

Research output: Contribution to journalArticlepeer-review

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Abstract

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.

Original languageEnglish (US)
Pages (from-to)59-81
Number of pages23
JournalChaos, Solitons and Fractals
Volume59
DOIs
StatePublished - Feb 2014

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematics(all)
  • Physics and Astronomy(all)
  • Applied Mathematics

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