On the Complete Integrability of a One Generalized Riemann Type Hydrodynamic System

The complete integrability of a generalized Riemann type hydrodynamic system is studied by means of symplectic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, Lax type representation and related infinite hierarchy of conservation laws are constructed.

💡 Research Summary

The paper investigates the complete integrability of a generalized Riemann‑type hydrodynamic system in one spatial dimension. The authors begin by formulating the system as a three‑component first‑order evolution equation
(u_{t}=v,; v_{t}=w,; w_{t}=F(u,v,w))
where the nonlinear term (F) is taken to be a polynomial in the fields, allowing a broad class of physical models (e.g., non‑Newtonian fluids, anisotropic media). The central aim is to demonstrate that this system possesses an infinite hierarchy of commuting flows and conserved quantities, thereby qualifying as a completely integrable system in the sense of the Magri‑Lenard scheme.

To this end, two compatible Poisson operators (\vartheta) and (\eta) are constructed explicitly. (\vartheta) is a first‑order, non‑symmetric differential operator, while (\eta) is a second‑order, symmetric operator containing polynomial weight functions. Their compatibility (