Abstract. Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm. The bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated and an infinite hierarchy of commuting to each other conservation laws of dispersive type are found. The well defined regularization of the model is constructed and its Lax type integrability is discussed. A generalized hydrodynamical Riemann type system is considered, infinite hierarchies of conservation laws, related compatible Poisson structures and a Lax type representation for the special case N = 3 are constructed.

Symplectic and symmetry analysis for studying MHD superfluid flows is devised, a new version of the Z. Peradzyński (Int. J. Theor. Phys. 29(11):1277-1284, 1990 helicity theorem based on differential-geometric and group-theoretical methods is derived. Having reanalyzed the Peradzyński helicity theorem within the modern symplectic theory of differential-geometric structures on manifolds, a new unified proof and a new generalization of this theorem for the case of compressible MHD superfluid flow are proposed. As a by-product, a sequence of nontrivial helicity type local and global conservation laws for the case of incompressible superfluid flow, playing a crucial role for studying the stability problem under suitable boundary conditions, is constructed. KeywordsThe MHD superfluid equations · The Peradzynski helicity theorem · Symplectic and symmetry analysis · Conservation laws · Vorticity invariants · Hamiltonian and Poissonian structures

The authors dedicate this article to their friend and teacher academician Prof. Anatoliy M. Samoilenko on the occasion of his 70th Birthday with great compliments and gratitude to his brilliant talent and impressive impact to modern theory of nonlinear dynamical systems of mathematical physics and nonlinear analysis.

UDC 517.9The backgrounds of quantum mathematics, a new discipline in mathematical physics, are discussed and analyzed from both historical and analytical points of view. The magic properties of the second quantization method, invented by Fock in 1934, are demonstrated, and an impressive application to the theory of nonlinear dynamical systems is considered.

Abstract. Invariant ergodic measures for generalized Boole type transformations are studied using an invariant quasi-measure generating function approach based on special solutions to the Frobenius-Perron operator. New two-dimensional Boole type transformations are introduced, and their invariant measures and ergodicity properties are analyzed.