Differential-algebraic approach to constructing representations of commuting differentiations in functional spaces and its application to nonlinear integrable dynamical systems

摘要

There is developed a differential-algebraic approach to studying therepresentations of commuting differentiations in functional differential ringsunder nonlinear differential constraints. An example of the differential idealwith the only one conserved quantity is analyzed in detail, the correspondingLax type representations of differentiations are constructed for an infinitehierarchy of nonlinear dynamical systems of the Burgers and Korteweg-de Vriestype. A related infinite bi-Hamiltonian hierarchy of Lax type dynamical systemsis constructed.