# Popowicz Z. The conserved quantities for the generalized Riemann equation, talk at The Workshop on GDEq and Integrability, 11-15 October 2010, Hradec nad Moravici, Czech Republic (abstract)

Speaker: Ziemowit Popowicz

Title: The conserved quantities for the generalized Riemann equation

Abstract:
We analyze the integrability property of the generalized hydrodynamical Riemann type equation ${\displaystyle D_{t}^{N}u=0}$ for arbitrary ${\displaystyle N}$. The infinite hierarchies of polynomial and non-polynomial conservation laws, both dispersive and dispersionless, are presented. Special attention is paid to the cases ${\displaystyle N=2}$, ${\displaystyle 3}$ and ${\displaystyle N=4}$, for which the conservation laws, Lax type representations and Hamiltonian structures are analyzed in detail. We also show that the case ${\displaystyle N=2}$ 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.

Slides: Prikarpatsky A.K., Golenia J., Artemovych O.D., Pavlov M., Popowicz Z. The conserved quantities for the generalized Riemann equation (presentation at Workshop Geom. Diff. Eq. and Integrability, 11-15 October 2010, Hradec nad Moravici, Czech Republic).pdf

References:
Anatoliy K. Prykarpatsky, Orest D. Artemovych, Ziemowit Popowicz, Maxim V. Pavlov Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg-de Vries hydrodynamical equations, J. Phys. A: Math. Theor. 43 (2010) 295205, arXiv:1005.2660

Ziemowit Popowicz, Anatoliy K. Prykarpatsky The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations, Nonlinearity 23 (2010) 2517-2537, arXiv:1005.3942