Ermakov–Ray–Reid systems in nonlinear optics

Abstract

A hydrodynamics-type system incorporating a Madelung-Bohm-type quantum potential, as derived by Wagner et al via Maxwell's equations and the paraxial approximation in nonlinear optics, is reduced to a nonlinear Schrödinger canonical form. A two-parameter nonlinear Ermakov-Ray-Reid system that arises from this model, and which governs the evolution of beam radii in an elliptically polarised medium is shown to be reducible to a classical Pöschl-Teller equation. A class of exact solutions to the Ermakov-type system is constructed in terms of elliptic dn functions. It is established that integrable twocomponent Ermakov-Ray-Reid subsystems likewise arise in a coupled (2+1)dimensional nonlinear optics model descriptive of the two-pulse interaction in a Kerr medium. The Hamiltonian structure of these subsystems allows their complete integration.

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].…”

Universal and Integrable Aspects of an Elliptic Vortex Representation in 2+1‐Dimensional Magneto‐Gasdynamics

“…Hence, as in , we have established the Hamiltonian character of the dynamical system (), () when and w are functionally dependent. In particular, if we demand that () be of Ermakov–Ray–Reid type [19, 14, 15], that is then and the existence of a second integral of motion, namely the Ray–Reid invariant [14, 15] with renders the Hamiltonian system integrable [21]. This case is now analyzed in detail with regard to the compatibility of the constraints () and () which play a key role in the classification of equations of state presented in .…”

“…Accordingly, the elliptic vortex solutions (7) of the gasdynamic system (14), (21) are essentially encapsulated in the Hamiltonian system (8), (9), (11). The pressure p and entropy S are determined by (16) and (21) 1 , respectively, with the functions F and being arbitrary.…”

“…It is not difficult to show that the ideal gas law (23) with the entropy given by (27) 2 is not compatible with the class of constitutive equations (21) unless the gas is perfect. However, if we assume that the temperature T is a function of time t only then the ideal gas law (23) shows that ρ, p, and t are no longer independent and the assumption under which (21) has been derived is violated. Thus, comparison of the two expressions (16) and (23) for p leads to σ = 0 and…”

Hybrid Ermakov-Painlevé IV Systems

“…Conte [13] and Clarkson [12] together with literature cited therein). On the other hand, Ermakov-type systems and their generalisations likewise have extensive applications, notably in nonlinear optics [15,20,21,23,48,49,62] where, inportantly, they have been used to model the evolution of the size and shape of the light spot and wave front in elliptical Gaussian beams [15,23].…”