Rational Factorization of Hamiltonian Flows in the Space Dual to the Lie Algebra of Fractional Integrodifferential Operators and Benney-Type Integrable Hydrodynamic Systems

For the Lax-type Hamiltonian flows in the space dual to the Lie algebra of fractional integrodifferential operators, we develop a rational factorization method, which enables one to get new integrable hierarchies of nonlinear fractional-differential dynamical systems on the Lie algebra of ordinary integrodifferential operators and obtain infinite sequences of their conservation laws. By using the Bäcklund transformation, we show that the system of two flows of this kind for a pair of fractional integrodifferential operators related by a gauge transformation is equivalent to a system of two evolutionary equations for fractional differential operators, which determine the corresponding rational factorization, and find the Hamiltonian representation for this system of evolutionary equations. We show that its quasiclassical approximation is a system of two evolutionary equations for polynomials of fractional degree in a certain complex parameter. The proposed method is used to construct a new integrable hierarchy of nonlinear fractionaldifferential systems on the Lie algebra of ordinary integrodifferential operators and an infinite sequence of its conservation laws, as well as a new integrable hierarchy of Benney-type hydrodynamic systems, which is its quasiclassical approximation.