The pulsrodon in 2+1-dimensional magneto-gasdynamics: Hamiltonian structure and integrability

Abstract

An elliptic vortex-type ansatz introduced into a 2 + 1-dimensional system governing rotating homentropic magneto-gasdynamics with a parabolic gas law is shown to lead to a finite-dimensional nonlinear dynamical system which admits exact analytical solution in terms of an elliptic function and integral representation. The dynamical system is demonstrated to be Hamiltonian and equivalent to the stationary reduction of the integrable nonlinear Schrödinger equation coupled with a Steen-Ermakov-Pinney equation. A novel magneto-gasdynamic analogue of the pulsrodon of shallow water f-plane theory is isolated thereby. Confined and time-periodic magneto-gasdynamic flows are constructed explicitly. C

On Dirichlet two-point boundary value problems for the Ermakov–Painlevé IV equation

“…On the other hand, nonlinear coupled systems of Ermakov-Ray-Reid type [3][4][5][6][7] likewise have extensive physical applications (see e.g. [8][9][10][11][12][13][14]). Just as solitonic systems and their associated classical Painlevé equations admit nonlinear superposition principles (permutability theorems) generated via invariance under Bäcklund transformations (see [1]), it is known that Ermakov-Ray-Reid systems also possess underlying nonlinear superposition laws, albeit of another kind [3,4].…”

Hybrid Ermakov-Painlevé IV Systems

“…Lyapunov stability aspects of pulsrodons and their duals has been discussed by Holm [24]. The magneto-gasdynamics pulsrodon describes a rotating, pulsating elliptic cylindrical plasma column bounded by a vacuum state [53]. Ermakov-Ray-Reid structure has also recently been isolated in [53] in connection with gas cloud evolution of a type originally investigated by Ovsiannikov [34] and Dyson [17] and subsequently by Gaffet (see [19] and work cited therein).…”

“…The magneto-gasdynamics pulsrodon describes a rotating, pulsating elliptic cylindrical plasma column bounded by a vacuum state [53]. Ermakov-Ray-Reid structure has also recently been isolated in [53] in connection with gas cloud evolution of a type originally investigated by Ovsiannikov [34] and Dyson [17] and subsequently by Gaffet (see [19] and work cited therein). Connection with integrability has been made in the 2+1-dimensional case by construction of a Lax pair [24].…”

“…The route to the Ermakov-Ray-Reid system, interestingly involves application of extensions of Ball-type moment of inertia theorems [7,8] which, in turn, may be shown to have a Lie group origin [30,40]. Ermakov-Ray-Reid systems have also been uncovered in magnetogasdynamics [43,53,60] and remarkably, have been shown to admit pulsrodon-type phenomena analogous to those observed in an elliptic warm core eddy context (Rogers [41], Rubino and Brandt [56]). Lyapunov stability aspects of pulsrodons and their duals has been discussed by Holm [24].…”

Ermakov-Modulated Nonlinear Schrödinger Models. Integrable Reduction

“…[40]. Indeed, integrable Ermakov-Ray-Reid systems have in recent years been shown to arise in a wide range of important physical contexts, including warm-core oceanographic elliptic eddy theory, rotating shallow water hydrodynamics, anisentropic gasdynamics and magnetogasdynamics [39,41,42,48]. They also arise in a spiralling elliptic soliton model of Ref.…”