1. Introduction

The extensive connections between plasma physics, magnetohydrodynamics and canonical nonlinear integrable equations of modern soliton theory are well-documented. These links originated with pioneering work by Washimi and Taniuti [72] who employed reductive perturbation techniques to derive the Korteweg-de Vries (KdV) equation in the analysis of the propagation of ion-acoustic waves in a plasma. In contemporary work, Berezin and Karpman [7,8] independently derived the KdV equation in studies, both numerical and analytic, of large amplitude disturbances in plasmas and other dispersive media. Reviews of the role of the KdV equation in plasma physics and of the application of the reduction perturbation method to the study of hydrodynamic waves in cold plasma have been presented in [24] and [26] respectively. The integrable mkdV equation, related to the KdV equation by a Miura transformation, likewise arises in the analysis of the propagation of nonlinear Alfvén waves in cold collisionless plasma [27]. In addition, the nonlinear Schrödinger equation has an extensive literature concerned with its derivation in plasma physics in a variety of contexts. Thus, in particular, its occurrence in the analysis of the propagation of Langmuir waves in plasma has been described in [18,23,70]. The text [73] provides an account of perturbation methods adduced in the derivation of the NLS equation in plasma physics, notably with regard to incorporation of Landau damping phenomena.

An integrable Heisenberg spin equation associated with the NLS equation has been derived in spatial gasdynamics and magneto-hydrostatics in [45,46] via geometric methods originally applied in a magnetohydrodynamic context in [47]. This purely geometric formalism as opposed to a reductive perturbation approach may be shown to lead to a classical integrable da Rios system reduction in complex-lamellar magnetohydrodynamics [48]. It has also been applied not only to construct the auto-Bäcklund transformation for the auto-Bäcklund transformation for the NLS equation in a purely geometric manner [49,50] but also to obtain a novel integrable Pohlmeyer-Lund-Regge reduction in magnetohydrodynamics [51,66]. The combined action of Bäcklund and reciprocal-type transformations has been used, via an integrable sinh-Gordon reduction, to construct periodic solutions of breather-type in super-Alfvénic magnetogasdynamics in [52]. Invariance under multiparameter reciprocal transformations has been established in two-dimensional orthogonal magnetogasdynamics in [53].

In [54,55] elliptic vortex solutions in magnetogasdynamics have been obtained via symmetry reduction to an integrable Ermakov-Ray-Reid system with underlying Hamiltonian structure. Ermakov-Ray-Reid systems have diverse physical applications, in particular, in nonlinear optics and rotating shallow water theory (see e.g. [56–58]). Moreover, in [59], a 2 + 1 -dimensional non-isothermal magnetogasdynamic version of a gas cloud system with origin in work of Dyson [15] was shown to admit symmetry reduction to an eight dimensional nonlinear dynamical subsystem with underlying Hamiltonian-Ermakov structure. A Lax pair representation was constructed for this integrable subsystem.

Here, a class of wave packet representations of a type originally introduced in a nonlinear optics context in [20] is used to isolate a Ermakov-Painlevé II symmetry reduction of a resonant NLS equation which encapsulates a 1 + 1 -dimensional cold plasma physics system. The latter describes the propagation of uniaxial long magnetoacoustic waves in a cold collisionless plasma subject to a transverse magnetic field. Iterative application of a Bäcklund transformation is used to generate novel classes of exact solutions of the cold plasma system in terms of either Yablonski-Vorob’ev polynomials or classical Airy functions.

2. The Two-Component Cold Plasma System. A Resonant NLS Encapsulation

The dynamics of two-component cold collisionless plasma in the presence of an external magnetic field B is governed by the nonlinear system of equations [3,28].

where m_i, m_e, v_i, v_e, n_i, n_e denote, in turn, masses, velocities and concentrations of ions and electrons respectively. E is the electric field, B is the magnetic field, e is the electric charge and μ₀ is the magnetic permeability. If the oscillation frequency is much smaller than the ion Langmuir frequency, then plasma quasi-neutrality is implied, that is, ni≈ne=n .

On introduction of the mass density ρ and velocity u according to

then, on elimination of the electric field E, if m_e/m_i ≪ 1, in the case of uni-axial plasma propagation with transverse magnetic field B, so that the system (2.1) may be reduced to the form [33]

Introduction of the Lagrangian variable X(x, t) via the continuity equation (2.3a) according to

into (2.3c) shows that The system consisting of (2.3a), (2.3b) augmented by (2.4) with M(X) = 1 was set down in a nonlinear dispersive wave context by Whitham [74]. In the present context it describes the propagation of 1 + 1 -dimensional nonlinear magnetoacoustic waves in a cold plasma subject to a transverse magnetic field.

Here, we consider a shallow water type approximation to the Whitham system consisting of the continuity and momentum equations (2.3a), (2.3b) together with

Thus, on re-scaling the space and time variables via x → βx, t → βt and expansion of the expression for the magnetic field B in the parameter β² according to [33]

insertion into (2.5) yields

Thus, to 𝒪(β²), the following nonlinear system results [33]:

This describes the uniaxial propagation of long magneto-acoustic waves in a cold plasma with velocity magnitude u and magnetic field given by (2.2) together with (2.6)–(2.7).

On use of the one-dimensional version of a relation for the de Broglie-Bohm potential, namely

it seen that the cold plasma momentum equation in (2.8a) becomes

Introduction of the velocity potential S according to

into the continuity equation (2.8a) and momentum equation (2.9) in turn, gives together with the Bernoulli integral

In the latter, T(t) may be absorbed into the potential S and so may be set zero without loss of generality,

If the classical Madelung transformation is now introduced according to [38]

then it is seen that the cold plasma system (2.8a) may be encapsulated in the resonant NLS equation [43,44] which incorporates a de Broglie-Bohm potential term 1|Ψ|∂2∂x2|Ψ| . It is recalled that, if s <1 then the canonical 1 + 1 -dimensional resonant NLS equation can be transformed to the standard integrable cubic NLS equation with the de Broglie-Bohm term removed [43,44,60]. However, if as in the present cold plasma case s = 1 + β² > 1 this reduction to remove the de Broglie Bohm term is not available. Then, as shown in [33] the resonant NLS equation (2.10) which encapsulates the cold plasma system is equivalent to a canonical two-component system contained in the AKNS hierarchy of solitonic systems amenable to the inverse scattering transform [1,2]. Bäcklund-Darboux transformations and concomitant nonlinear superposition principles for the resonant NLS equation and hence for the cold plasma system which it encapsulates have been constructed in [33]. This resonant NLS equation importantly, unlike its standard integrable NLS counterpart can admit solitonic fusion or fission phenomena [44]. Here, wave packet solutions of the resonant NLS equation (2.10) are generated via a symmetry reduction to a prototype integrable Ermakov-Painlevé II equation [4,61]. The iterated application of a Bäcklund transformation then allows the construction of novel classes of exact solutions to the cold plasma system in terms of either Yablonski-Vorob’ev polynomials or classical Airy functions.

3. Ermakov-Painlevé II Symmetry Reduction

Here, symmetry reduction of the resonant NLS equation (2.10) is sought under the wave packet ansatz

with α, β, γ, δ, ε, ζ and λ arbitrary constants. Insertion of the latter into (2.10) produces the coupled nonlinear system where and s = 1 + β² > 1.

The relations (3.2) combine to show that

whence, it is required that in which case, (3.4) admits the integral where ℐ is an arbitrary constant.

It is seen that (3.3) may be written as, if β ≠ 0

on setting

Moreover, (3.2) show that

whence, on use of the relations and it is seen that

The latter, by virtue of the integral of motion (3.5) and the expression (3.6) for Δ now produces a hybrid Ermakov-Painlevé equation in the amplitude |Ψ| namely

where the constants c₁, c₂ and c₃ are given by

The type of nonlinear equation (3.7) has been previously shown to arise in connection with a pair of three-ion cases in the classical Nernst-Planck electrodiffusion system [4]. Here, |Ψ| = ρ^{1/ 2} and in terms of the density ρ it is seen that (3.7) becomes

which with the admissible specialisations, becomes the canonical integrable Painlevé XXXIV (P₃₄) equation in ρ > 0, namely cf. Gromak [21]. It is emphasised that the relation in (3.8) involving the Painlevé parameter α, necessarily requires that s > 1 as indeed is the case for the present cold plasma system incapsulated in a nonlinear resonant NLS equation. The regions of positivity of classes of solutions of P₃₄ (3.9) as generated by the iterated action of a Bäcklund transformation have been investigated in the context of boundary value problems in two-ion electrodiffusion in [5]. Therein, the ion concentrations c_±, which are necessarily positive, are governed by solutions of P₃₄ (3.9)

With a positive solution ρ = |Ψ|² = ϕ² + ψ² of P₃₄ (3.9) to hand, the ratio ψ / ϕ is determined by the integral of motion (3.5). The latter yields

so that, on integration, where K is an arbitrary constant. Thus, with Λ = ψ / ϕ, it is seen that the ϕ, ψ in the original wave packet represenation (3.1) are given by the relations

The velocity magnitude in the cold plasma system (2.8a) is given by u = Φ_x where Φ is the potential given by

on use of the identity

Thus,

while the term ℬ in the expansion (2.6) for the magnetic field B is given by where the density ρ(ξ) is governed by P₃₄ (3.9)

In the sequel, a link between the Ermakov-Painlevé II equation for the density ρ(ξ) and the Painlevé II (P_II) equation

with α a parameter, via P₃₄ (3.9) is exploited to construct novel classes of wave packet solutions of the cold plasma system in terms of Yablonski-Vorob’ev polynomials or classical Airy functions via the iterated application of the well-known Bäcklund transformation for P_II [19,37] ; see also [10,22]. These solutions may, in turn, be used as seed solutions in the application of the Bäcklund-Darboux transformations as established for the resonant NLS equation in [33].

4. Iterative Action of a Bäcklund Transformation

Bäcklund transformations have established applications in continuum mechanics and notably in modern soliton theory [50,62]. These have roots, in turn, in work of Loewner [34,35] on model laws in gasdynamics and Seeger et al. [69] on crystal dislocations. Thus, Loewner applied novel matrix Bäcklund transformations to construct multi-parameter gas laws for which the classical hodograph system may be reduced to appropriate canonical forms in subsonic, transonic and supersonic flow régimes. Seeger et al. [69], on the other hand, in the context of Frenkel and Kontorova’s dislocation theory isolated the nonlinear interaction of what they termed ‘eigenmotions’ via Bianchi’s classical permutability theorem as derived via the auto-Bäcklund transformation for the sine Gordon equation of pseudo-spherical surface thoery. In particular, the interaction process of what, in soliton theory came to be called a breather and kink was described analytically by means of this nonlinear superposition principle. Importantly, Lamb [32] later exploited this permutability theorm to predict experimentally observed decomposition of ultrashort optical pulses in a resonant medium.

The areas of application of Bäcklund transformations in nonlinear continuum mechanics and modern soliton theory were brought together in [29,30] where it was established that a reinterpretation and extension of the class of matrix infinitesimal Bäcklund transformations introduced by Loewner in a gasdynamics setting provide a linear representation for a master 2 + 1-dimensional soliton system. Basic reductions of this system lead, in particular to novel integrable 2 + 1 -dimensional versions of the principal chiral fields model, Toda lattice system and, notably, of the classical sine Gordon equation [9,11,30,31,36,40,41,67,68]. It is remarked that the paper of Loewner [34] contains, ‘mutatis mutandis’ a linear representation for the 1 + 1 -dimensional sine Gordon equation which is gauge equivalent to that later obtained in the celebrated AKNS system [1].

The preceding attests to the seminal role that Bäcklund transformations have played in modern soliton theory. Likewise the classical Painlevé equations and the Bäcklund transformations they admit arise naturally in the study of integrable solitonic systems (see e.g. [6,10,16,21,22,39] and work cited therein). Here, our concern is with the application of a Bäcklund transformation for P_II (3.10) to the present symmetry reduction of the cold plasma system encapsulated in the resonant NLS equation (2.10)

It turns out that, remarkably, all known exact solutions of P_II (3.10) and hence also of P₃₄ (3.9) may be generated via iteration of a Bäcklund transformation of (3.10) due to Gambier [19] and later Lukashevich [37].

The well-known relationship between solutions of P_II (3.10) and solutions of P₃₄ (3.9) is derived via the Hamiltonian system

where the Hamiltonian 𝒣II(ρ,q,ξ;α) is given by which yield the nonlinear system see [25,42]. Elimination of ρ and q successively in (4.1) yields P_II (3.10) and P₃₄ (3.9).

If q_α(ξ) = q(ξ ; α) is a solution of P_II (3.10) with parameter α, then

with ′ ≡ d / dξ, which are the Bäcklund transformations of P_II (3.10)[19,37]. If ρ_α(ξ) = ρ(ξ; α) is a solution of P₃₄ (3.9) with parameter α, then from (4.1) we have

Consequently, from (4.2) and (4.3) we obtain the Bäcklund transformations of P₃₄ (3.9) given by

see also [17,42]

In the present plasma physics context, in particular, this produces an extensive class of exact rational solutions for the density ρ governed by P₃₄ (3.9) given by

with , corresponding to the parameters α = n, where the Q_n(ξ) are the Yablonskii-Vorob’ev polynomials determined by the quadratic recurrence relations [71,75] with Q₋₁(ξ) = Q₀(ξ) = 1. The rational solutions of P₃₄ (3.9) can also be expressed in terms of determinants. Let p_k(ξ) be the polynomial defined by and τ_n(ξ) be the Wronskian for n ≥ 1. Then the rational solution satisfies P₃₄ (3.9) with α = n.

Iterated action of the Bäcklund transformations (4.4) for P₃₄ (3.9) on the seed Airy-type solution with parameter

generates a class of exact solutions of P₃₄ (3.9) given by for the parameter and where the sequence.{u_n ξ)}, for n ≥ 0, is determined by the Toda recurrence relation with initial values u₀(ξ) = 1 and u₁(ξ) = φ(ξ), where φ(ξ) is governed by the classical Airy equation i.e. with Ai(z) and Bi(z) the Airy functions and a, b arbitrary parameters. The Airy-type solutions of P₃₄ (3.9) can also be expressed in terms of determinants. Suppose that Φ_n(ξ) is the Hankel n × n determinant with φ(ξ) given by (4.11) and Φ₀(ξ) = 1, then for n ≥ 1 satisfies P₃₄ (3.9) with , cf. [12].

In previous work in [5], solutions in terms of P₃₄ in a two-ion electro-diffusion context have been used to investigate certain boundary value problems. There the positivity requirement on the solutions arises via such a constraint on the ion concentrations. Here, it arises due to the positivity contraint on the density. The rational solutions of P₃₄ are not positive for all ξ. However, positivity may be established in certain regions ξ > c, a constant. Thus, the rational solutions of P₃₄ are given by

where the P_k are the Yablonski-Vorob’ev polynomials, cf. [13,14]. If ξ_k−1 denotes the largest real zero of the polynomial P_k−1 for each k ∈ ℕ then the constraint ξ_k−1 < c ensures positivity in the region ξ > c. The positivity property is illustrated graphically for k = 0,1,2 in [5]

The Airy-type solutions of P_II admit the representation [5,12]

wherein the sequence {ψ_k}_{k ≥−1} is given by the recurrence relation together with the initial values ψ₋₁ = 1, ψ₀ = 1. In the above Φ = (ln ϕ)′ where ϕ is, in general, given by (4.11). The corresponding Airy-type solutions of P₃₄ are determined by [5,12] If one proceeds with the case b = 0 in (4.11), i.e. where we have set a = 1, without loss of generality, then it was established in [5,12] that, for any k there exists a ξ_k such that is non-singular and positive on ξ < ξ_k. This positivity property is depicted graphically for k = 1,2, …, 6 in [12].

5. Conclusion

The aim of the present work has been to exploit an integrable Ermakov-Painlevé II reduction of a resonant NLS encapsulation of a nonlinear cold plasma system to generate classes of similarity solutions in terms of either Yablonskii-Vorob’ev polynomials or classical Airy functions. It is remarked that classes of similarity solutions to the related P_II equation have recently been shown to provide exact solutions to nonlinear moving boundary problems for certain solitonic equations [63–65] while similarity solutions in magneto-gasdynamics descriptive of plasma columns confined by moving boundaries have been isolated in [55].