The Gurevich–Zybin system

Abstract

Abstract. We present three different linearizable extensions of the Gurevich-Zybin system. Their general solutions are found by reciprocal transformations.In this paper we rewrite the Gurevich-Zybin system as a Monge-Ampere equation. By application of reciprocal transformation this equation is linearized. Infinitely many local Hamiltonian structures, local Lagrangian representations, local conservation laws and local commuting flows are found. Moreover, all commuting flows can be written as Monge-Ampere equations similar to the Gurevich-Zybin system.The Gurevich-Zybin system describes the formation of a large scale structures in the Universe. The second harmonic wave generation is known in nonlinear optics. In this paper we prove that the Gurevich-Zybin system is equivalent to a degenerate case of the second harmonic generation. Thus, the Gurevich-Zybin system is recognized as a degenerate first negative flow of two-component Harry Dym hierarchy up to two Miura type transformations. A reciprocal transformation between the Gurevich-Zybin system and degenerate case of the second harmonic generation system is found. A new solution for the second harmonic generation is presented in implicit form.

Generic regularity and Lipschitz metric for the Hunter–Saxton type equations

“…It is also a special case of the Gurevich-Zybin system modelling the dynamics of nondissipative dark matter [16]. Its local well-posedness, global existence and blow-up phenomena were discussed recently in [18].…”

A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability

“…We also notice that dynamical system (16), as it was shown before in [21,22], can be transformed via the substitution…”

“…Quite different conservation laws have been obtained in [21][22][23] using the recursion operator technique. The corresponding recursion operator proves to generate no new conservation law, if one applies it to the non-polynomial conservations laws (24).…”

Isospectral integrability analysis of dynamical systems on discrete manifolds

“…And like their counterparts analyzed above, the integrability properties of (5.2) are important for several practical applications. Naturally, it would be interesting to apply our direct gradient-holonomic integrability approach to the hierarchy (5.2) and find its differential-difference analog using the known [31,32,34] corresponding Lax representations. As one can easily check, one of the discrete analogs of the corresponding linear Lax "spectral" problem for (5.1) for s = 2 has the form ∆f n = l n [u, z; λ]f n , l n [u, z; λ] := 1 − λD n u n −D n z n 2λ 2 1 + λD n u n ,…”