Modulated Kepler-Ermakov triads. Integrable Hamiltonian structure and parametrisation

A hybrid Kepler-Ermakov system modulated by means of a classical nonlinear superposition principle is reduced via a class of involutory transformations to its unmodulated counterpart. In the case of certain underlying Hamiltonian-type and dual associated structure, parametric representations are applied which allow systematic integration on application of admitted invariants.


Introduction
Here it is shown how admitted Ermakov and Hamiltonian invariants may be conjugated to provide a systematic procedure for the integration of the Kepler-Ermakov extension of the Ermakov-Ray-Reid system. Application is made of a class of involutory transformations. Nonlinear coupled systems which represent an extension of classical work of Ermakov [1] have proved to possess diverse physical applications. Thus, in [2,3] what are now termed Ermakov-Ray-Reid systems, namely wherein a dot denotes a derivative with respect to the independent variable t where x = x(t), y = y(t). In subsequent work, 2 + 1 dimensional Ermakov-Ray-Reid systems were introduced in [4], while n + 1dimensional extensions were constructed in [5,6]. Multi-component Ermakov-Ray-Reid systems were derived in a hydrodynamic context [7] via symmetry reduction of a multi-layer fluid model. Therein, such n-component systems were shown to admit iterative reduction to n − 2 linear equations augmented by the canonical Ermakov-Ray-Reid system (1). Darboux-type transformations were constructed which link distinct nonlinear systems of the latter type.
The notion of hybrid Kepler-Ermakov systems was originally introduced by Athorne [8]. Here, modulated versions of this system are introduced which are reducible via a class of involutory transformations to their unmodulated counterparts. It is established how parametric representations for certain Kepler-Ermakov systems which admit pairs of invariants allow their systematic integration.

A Kepler-Ermakov triad: modulation
Here, a hybrid modulated three-component Kepler-Ermakov system is introduced according to with Ω(t) determined by the classical Ermakov equation The above triad yields If the relations in I * are augmented by the relation Ω * = 1/Ω then the involutory property I ** = I results. The class of involutory transformations I * is of a type originally applied in [9] to autonomise the standard Ermakov-Ray-Reid system corresponding to H = 0 in (5). Application has been subsequently been made to reduce Ermakov-modulated nonlinear Schrödinger models [10][11][12][13] and coupled solitonic sine-Gordon, Demoulin and Manakov systems [14] to their integrable counterparts. In [15,16] modulated systems have been recently arisen in the analysis of heterogeneous moving boundary problems such as model the transport of liquids through a porous medium such as soil.
The canonical unmodulated Kepler-Ermakov system (7) is seen to admit the integral of motion (7) aligns with the classical invariant (2) of the Ermakov-Ray-Reid system.

A Hamiltonian Kepler-Ermakov system parametrisation
The autonomous Hamiltonian Ermakov-Ray-Reid system was shown in [17] to admit the parametrisation admitted by the nonlinear system (10).
The Hamiltonian system parametrisation (10) was originally derived in [17] in the context of rotating shallow water hydrodynamics. Therein, in an analysis of motion over a circular paraboloidal bottom topography, a Hamiltonian Ermakov-Ray-Reid system such as (10) above was obtained which describes the temporal evolution of the semi-axes of the moving elliptic shoreline on the underlying rigid basin.
Ermakov-Ray-Reid systems have important application in nonlinear optics (see [18,19] and literature cited therein). In particular, they can describe the evolution of the size and shape of the light spot and wave front in elliptic Gaussian beams [20]. Such systems with underlying Hamiltonian structure not only arise in nonlinear optics but also in the magnetogasdynamics of bounded plasma columns [21] and the analysis of non-isothermal gas cloud evolution [22]. In [23] a Hamiltonian Ermakov-Ray-Reid system was derived via a 2 + 1-dimensional Madelung system incorporating both a Bohm quantum and logarithmic potentials. This involved an 8-dimensional nonlinear dynamical system of a type with origin in the analysis of the evolution of elliptic warmcore oceanographic eddies [24]. The wide-ranging physical applications of Ermakov-Ray-Reid systems in both physics and nonlinear continuum mechanics have recently been surveyed in [25].
The Kepler-Ermakov type system Application is now made of the invariants  and  in the identity x y x y xy yx xx yy 15 whence, The original variables x(t), y(t) in the Hamiltonian Kepler-Ermakov system (13) are now determined via the latter relation together with Σ 2 = x 2 + y 2 , whence

A dual integrable parametrisation
is determined by an elliptic integral. On introduction of U according to

Ermakov modulation
Nonlinear superposition principles derived via invariance under Bäcklund transformations are admitted generically by solitonic systems and have their genesis in classical geometry (qv [27][28][29] and literature cited therein). Ermakov-Ray-Reid system (1) were shown in [2,3] to likewise generically possess nonlinear superposition principles albeit not of the type admitted by solitonic systems. In particular, the classical onecomponent Ermakov reduction (4) is the constant Wronskian of Ω 1 Ω 2 . This now classical result may be systematically retrieved and extended by Lie group methods with application made thereby to the discretization of the classical Ermakov equation [30].
The classical Ermakov equation (4) arises in the modelling of the dynamics of cosmologies [31]. In terms of application in continuum mechanics, the nonlinear superposition principle (43) allows the exact solution of a range of initial value problems which describe the large amplitude radial oscillation of thin-shelled tubes of hyperelastic Mooney-Rivlin material subject to various boundary loadings [32][33][34][35]. The derivation of the Ermakov equation (4) in the context of nonlinear elastodynamics with application to the analysis of the propagation of transverse waves in both neo-Hookean and Mooney-Rivlin solids has been detailed in [36].
Here, on application of the nonlinear superposition principle (43) in the present context, the class of Kepler-Ermakov systems (3) modulated via the Ermakov equation (4) and which is reducible to the canonical system (7) by means of the involutory transformations I * is given explicitly by This class of modulated Kepler-Ermakov systems includes, importantly, the specialisations corresponding to the subcases associated with the Hamiltonian and dual parametrisations in (13) and (32) respectively.

Conclusion
Here, a modulated generalization of the Kepler-Ermakov system has been presented. It is remarked that there continues to be research interest in Ermakov systems and their extensions. Thus, subsequent to the extensive review of the diverse physical applications of such nonlinear systems in [25] recent developments in [37][38][39][40] may be cited.

Data availability statement
No new data were created or analysed in this study.