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arXiv · 1005.3942

The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations

Ziemowit Popowicz·Anatoliy K. Prykarpatsky
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Abstract

Based on the gradient-holonomic algorithm we analyze the integrability property of the generalized hydrodynamical Riemann type equation $%D_{t}^{N}u=0$ for arbitrary $N\in \mathbb{Z}_{+}.$ 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.

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Ziemowit Popowicz, Anatoliy K. Prykarpatsky. 2010-05-21. The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations. https://doi.org/10.1088/0951-7715%2F23%2F10%2F010

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