On a Nonlocal Ostrovsky-Whitham Type Dynamical System, Its Riemann Type Inhomogeneous Regularizations and Their Integrability

Abstract

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.

A vertex operator representation of solutions to the Gurevich-Zybin hydrodynamical equation

“…It is worth also mentioning that the scheme devised above for finding the corresponding vertex operator representations for the Gurevich-Zybin (2.1) can be similarly generalized for treating other equations of the infinite hierarchy (1.1) when N ≥ 3, having taking into account the existence of their suitable Lax type representations found before in recent works [4,5,25].…”

“…These methods appeared to be very effective [1] in investigating many types of nonlinear spatially one-dimensional systems of hydrodynamical type and, in particular, the characteristics method in the form of a "reciprocal" transformation of variables has been used recently in studying a so called Gurevich-Zybin system [2,3] in [9] and a Whitham type system in [5,6]. Moreover, this method was further effectively applied to studying solutions to a generalized [5] where N ∈ Z + , u ∈ M 1 ⊂ C ∞ (R/2πZ; R) is a smooth function on a periodic functional manifold M 1 and t ∈ R is the evolution parameter. Making use of novel methods, devised in [8,18,25] and based both on the spectral theory [10,17,19,20] and differential algebra techniques, the Lax type representations for the cases N = 1, 4 were constructed in explicit form.…”

Isospectral integrability analysis of dynamical systems on discrete manifolds

“…In the same way one can find the second compatible with (4.28) Poissonian operator 32) where the Hamiltonian function is…”

“…For example, in [31][32][33] these types of differential-algebraic tools were used to study the integrability of a generalized (owing to D. Holm and M. Pavlov) Riemann hydrodynamical hierarchy of dynamical systems of the form…”

“…Note that these Poissonian structures are also Nötherian for the whole hierarchy of dynamical systems 32) where t j ∈ R, j ∈ Z + , are the corresponding evolution parameters, and which, owing to (2.11), commute with each other on the manifold M (N ) . Therefore, it is possible to apply Lemma 1.4 to any one of the dynamical systems (2.32) if the related vector fields commuting with (1.1) are assumed known.…”