Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System

A class of nonlinear Schrödinger equations involving a triad of power law terms together with a de Broglie-Bohm potential is shown to admit symmetry reduction to a hybrid Ermakov-Painlevé II equation which is linked, in turn, to the integrable Painlevé XXXIV equation. A nonlinear Schrödinger encapsu...

Повний опис

Збережено в:

Бібліографічні деталі
Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2017
Автори:	Rogers, C., Clarkson, P.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2017
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/148621
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System / C. Rogers, P.A. Clarkson // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 92 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-148621
record_format	dspace
spelling	Rogers, C. Clarkson, P.A. 2019-02-18T16:43:33Z 2019-02-18T16:43:33Z 2017 Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System / C. Rogers, P.A. Clarkson // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 92 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J15; 37K10; 76B45; 76D45 DOI:10.3842/SIGMA.2017.018 https://nasplib.isofts.kiev.ua/handle/123456789/148621 A class of nonlinear Schrödinger equations involving a triad of power law terms together with a de Broglie-Bohm potential is shown to admit symmetry reduction to a hybrid Ermakov-Painlevé II equation which is linked, in turn, to the integrable Painlevé XXXIV equation. A nonlinear Schrödinger encapsulation of a Korteweg-type capillary system is thereby used in the isolation of such a Ermakov-Painlevé II reduction valid for a multi-parameter class of free energy functions. Iterated application of a Bäcklund transformation then allows the construction of novel classes of exact solutions of the nonlinear capillarity system in terms of Yablonskii-Vorob'ev polynomials or classical Airy functions. A Painlevé XXXIV equation is derived for the density in the capillarity system and seen to correspond to the symmetry reduction of its Bernoulli integral of motion. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System
spellingShingle	Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System Rogers, C. Clarkson, P.A.
title_short	Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System
title_full	Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System
title_fullStr	Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System
title_full_unstemmed	Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System
title_sort	ermakov-painlevé ii symmetry reduction of a korteweg capillarity system
author	Rogers, C. Clarkson, P.A.
author_facet	Rogers, C. Clarkson, P.A.
publishDate	2017
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	A class of nonlinear Schrödinger equations involving a triad of power law terms together with a de Broglie-Bohm potential is shown to admit symmetry reduction to a hybrid Ermakov-Painlevé II equation which is linked, in turn, to the integrable Painlevé XXXIV equation. A nonlinear Schrödinger encapsulation of a Korteweg-type capillary system is thereby used in the isolation of such a Ermakov-Painlevé II reduction valid for a multi-parameter class of free energy functions. Iterated application of a Bäcklund transformation then allows the construction of novel classes of exact solutions of the nonlinear capillarity system in terms of Yablonskii-Vorob'ev polynomials or classical Airy functions. A Painlevé XXXIV equation is derived for the density in the capillarity system and seen to correspond to the symmetry reduction of its Bernoulli integral of motion.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/148621
citation_txt	Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System / C. Rogers, P.A. Clarkson // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 92 назв. — англ.
work_keys_str_mv	AT rogersc ermakovpainleveiisymmetryreductionofakortewegcapillaritysystem AT clarksonpa ermakovpainleveiisymmetryreductionofakortewegcapillaritysystem
first_indexed	2025-11-25T20:31:26Z
last_indexed	2025-11-25T20:31:26Z
_version_	1850524126988468224
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 018, 20 pages Ermakov–Painlevé II Symmetry Reduction of a Korteweg Capillarity System Colin ROGERS † and Peter A. CLARKSON ‡ † Australian Research Council Centre of Excellence for Mathematics & Statistics of Complex Systems, School of Mathematics, The University of New South Wales, Sydney, NSW2052, Australia E-mail: c.rogers@unsw.edu.au ‡ School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, CT2 7FS, UK E-mail: P.A.Clarkson@kent.ac.uk Received January 13, 2017, in final form March 15, 2017; Published online March 22, 2017 https://doi.org/10.3842/SIGMA.2017.018 Abstract. A class of nonlinear Schrödinger equations involving a triad of power law terms together with a de Broglie–Bohm potential is shown to admit symmetry reduction to a hybrid Ermakov–Painlevé II equation which is linked, in turn, to the integrable Painlevé XXXIV equation. A nonlinear Schrödinger encapsulation of a Korteweg-type capillary system is thereby used in the isolation of such a Ermakov–Painlevé II reduction valid for a multi- parameter class of free energy functions. Iterated application of a Bäcklund transfor- mation then allows the construction of novel classes of exact solutions of the nonlinear capillarity system in terms of Yablonskii–Vorob’ev polynomials or classical Airy functions. A Painlevé XXXIV equation is derived for the density in the capillarity system and seen to correspond to the symmetry reduction of its Bernoulli integral of motion. Key words: Ermakov–Painlevé II equation; Painlevé capillarity; Korteweg-type capillary system; Bäcklund transformation 2010 Mathematics Subject Classification: 37J15; 37K10; 76B45; 76D45 1 Introduction Giannini and Joseph [39], in a nonlinear optics context, introduced a class of symmetry reduc- tions for a cubic nonlinear Schrödinger (NLS) equation iΨt + Ψxx + ν\|Ψ\|2Ψ = 0, (1.1) with subscripts denoted partial derivatives, which resulted in the Painlevé II equation but with zero parameter α d2q dz2 = 2q3 + zq, (1.2) see also [6, 49, 61]; the reduction of the NLS equation (1.1) to equation (1.2) was derived in [37, 86]. Numerical integration led to the isolation of interesting non-stationary solutions both bounded and stable in shape for a restricted range of ratio of nonlinearily to dispersion. However, the absence of the Painlevé parameter α in that reduction does not allow the iterative construction of sequences of exact solutions via the Bäcklund transformation for the canonical Painlevé II (PII) d2q dz2 = 2q3 + zq + α, (1.3) mailto:c.rogers@unsw.edu.au mailto:P.A.Clarkson@kent.ac.uk https://doi.org/10.3842/SIGMA.2017.018 2 C. Rogers and P.A. Clarkson with α a parameter, as given by Gambier [38] and Lukashevich [59]. Here, symmetry reduction to PII (1.3) or to a hybrid Ermakov–Painlevé II equation linked to the integrable Painlevé XXXIV (PXXXIV) equation d2p dz2 = 1 2p ( dp dz )2 + 2p2 − zp− (α+ 1 2)2 2p , (1.4) with α a parameter, is obtained for a wide class of NLS equations which incorporates a triad of power law terms together with a de Broglie–Bohm potential term. It is remarked that NLS equations involving such triple power law nonlinearities arise in nonlinear optics (see [31, 91] and literature cited therein). Moreover, NLS equations containing a de Broglie–Bohm term also arise in the analysis of the propagation of optical beams [75, 90] as well as in cold plasma physics [56]. Under appropriate conditions, such ‘resonant’ NLS equations admit novel fusion or fission solitonic behaviour [55, 56, 67, 68]. The Ermakov–Painlevé II symmetry reduction is applied here to an NLS encapsulation of a nonlinear capillarity system with origin in classical work of Korteweg [54]. Iterated application of a Bäcklund transformation admitted by PII (1.3) permits the construction via the linked PXXXIV equation of novel multi-parameter wave packet solutions to the capillarity system in terms of either Yablonskii–Vorob’ev polynomials or classical Airy functions. These are shown to be valid for a multi-parameter class of model specific free energy relations. An invariance of the (1 + 1)-dimensional Korteweg capillarity system under a one-parameter class of reciprocal transformations as recently set down in [80] allows the extension of the reduction procedure to a yet wider class of capillarity systems. 2 A Ermakov–Painlevé II symmetry reduction Here, a class of (1 + 1)-dimensional nonlinear Schrödinger equations of the type iΨt + Ψxx − [ (1− C) \|Ψ\|xx \|Ψ\| − ic \|Ψ\|x \|Ψ\|2 + λ\|Ψ\|2 + µ\|Ψ\|2m + ν\|Ψ\|2n ] Ψ = 0, (2.1) which incorporates a de Broglie–Bohm potential \|Ψ\|xx/\|Ψ\| and a triad of power law terms is investigated under a symmetry reduction. Thus, constraints on the parameters in (2.1) are sought for which the class of NLS equations admits symmetry reduction either to the PII (1.3), with non-zero parameter α, or to a hybrid Ermakov–Painlevé II equation under a wave packet ansatz Ψ = [φ(ξ) + iψ(ξ)] exp(iη), (2.2a) with ξ = αt+ βt2 + γx, η = γt3 + δt2 + εγtx+ ζt+ λx. (2.2b) In the nonlinear optics context of [39] such a similarity transformation was used to reduce a standard cubic NLS equation to Painlevé II but with zero parameter α and resort was made to a numerical treatment. Asymptotic properties of PII (1.2) with α = 0 have been discussed by various authors, see, e.g., [2, 8, 23, 28, 34, 41, 62]. In the present case, on introduction of the wave packet ansatz (2.2) into (2.1), it is seen that γ2 d2φ dξ2 − dψ dξ [ 2 ( β + εγ2 ) t+ α+ 2λγ ] + cγψ \|Ψ\|3 ( φ dφ dξ + ψ dψ dξ ) −∆φ = 0, (2.3a) γ2 d2ψ dξ2 + dφ dξ [ 2 ( β + εγ2 ) t+ α+ 2λγ ] − cγφ \|Ψ\|3 ( φ dφ dξ + ψ dψ dξ ) −∆ψ = 0, (2.3b) Ermakov–Painlevé II Symmetry Reduction of a Korteweg Capillarity System 3 where ∆ = 3γt2 + 2δt+ εγx+ ζ + (εγt+ λ)2 + λ\|Ψ\|2 + µ\|Ψ\|2m + ν\|Ψ\|2n + sγ2 \|Ψ\|4 {[ φ d2φ dξ2 + ψ d2ψ dξ2 + ( dφ dξ )2 + ( dψ dξ )2 ] ( φ2 + ψ2 ) − ( φ dφ dξ + ψ dψ dξ )2 } , (2.4) with s = 1− C. The relations (2.3) together show that γ2 ( d2φ dξ2 ψ − d2ψ dξ2 φ ) − ( φ dφ dξ + ψ dψ dξ )[ 2 ( β + εγ2 ) t+ α+ 2λγ ] + cγ \|Ψ\| ( φ dφ dξ + ψ dψ dξ ) = 0, (2.5) whence it is required that β + εγ2 = 0, in which case equation (2.5) admits the integral γ2 ( dφ dξ ψ − dψ dξ φ ) − 1 2(α+ 2λγ)\|Ψ\|2 + cγ\|Ψ\| = I, (2.6) where I is an arbitrary constant of motion. On use of the relation[ φ d2φ dξ2 + ψ d2ψ dξ2 + ( dφ dξ )2 + ( dψ dξ )2 ] (φ2 + ψ2)− ( φ dφ dξ + ψ dψ dξ )2 = \|Ψ\|3d 2\|Ψ\| dξ2 , (2.7) it is seen that (2.4) yields, if β 6= 0, ∆ = γβ−1 ( ε2γ + 3 ) (ξ − αt− γx) + 2(δ + εγλ)t+ εγx+ ζ + λ2 + λ\|Ψ\|2 + µ\|Ψ\|2m + ν\|Ψ\|2n + sγ2 \|Ψ\| d2\|Ψ\| dξ2 = εξ + ζ + λ2 + λ\|Ψ\|2 + µ\|Ψ\|2m + ν\|Ψ\|2n + sγ2 \|Ψ\| d2\|Ψ\| dξ2 , (2.8) on setting βε = γ ( 3 + ε2γ ) , αε = 2(δ + εγλ). (2.9) Moreover, equations (2.3) again combine to show that γ2 ( φ d2φ dξ2 + ψ d2ψ dξ2 ) + ( dφ dξ ψ − dψ dξ φ ) (α+ 2λγ)−∆\|Ψ\|2 = 0, whence, on use of the identity (φ2 + ψ2) [( dφ dξ )2 + ( dψ dξ )2 ] − ( dφ dξ ψ − dψ dξ φ )2 ≡ ( φ dφ dξ + ψ dψ dξ )2 , together with (2.7), it is seen that γ2 [ \|Ψ\|3d 2\|Ψ\| dξ2 − ( dφ dξ ψ − dψ dξ φ )2 ] + (α+ 2λγ) ( dφ dξ ψ − dψ dξ φ ) \|Ψ\|2 −∆\|Ψ\|4 = 0. 4 C. Rogers and P.A. Clarkson The latter, by virtue of the integral of motion (2.6) and the expression (2.8) for ∆ now produces a nonlinear equation in the amplitude \|Ψ\|, namely d2\|Ψ\| dξ2 + [c1 + c2ξ]\|Ψ\|+ c3\|Ψ\|3 + c4\|Ψ\|2m+1 + c5\|Ψ\|2n+1 + c6 \|Ψ\| + c7 \|Ψ\|2 = I2 (1− s)γ4\|Ψ\|3 , (2.10) where the constants c1, c2, . . . , c7 are given by c1 = (α− δ/ε)2 − γ2(ζ + λ2) (1− s)γ4 , c2 = ε (s− 1)γ2 , c3 = λ (s− 1)γ2 , (2.11a) c4 = µ (s− 1)γ2 , c5 = ν (s− 1)γ2 , c6 = c2 (s− 1)γ2 , c7 = 2cI (1− s)γ3 , (2.11b) and it is required that s 6= 1. Below a triad of cases is set down in which the amplitude equation (2.10) reduces either directly to PII or to a hybrid Ermakov–Painlevé II equation subsequently shown to be integrable. Case (i) I = 0; m = −1 2 ; n = −1. In this case with c1 = 0, c2 = −1, c3 = −2, c4 = −α, c5 + c6 = 0, the amplitude equation reduces directly to the Painlevé II equation d2\|Ψ\| dξ2 = 2\|Ψ\|3 + ξ\|Ψ\|+ α, (2.12) corresponding to the symmetry reduction via the ansatz (2.2) of the class of NLS equations iΨt + Ψxx − [ (1− C) \|Ψ\|xx \|Ψ\| − ic \|Ψ\|x \|Ψ\|2 + Cγ2\|Ψ\|2 + Cγ2α \|Ψ\| − c2 \|Ψ\|2 ] Ψ = 0. Case (ii) I = 0; c = 0; m = −1 2 ; n = 0. Here, with c1 = −c5, c2 = −1, c3 = −2, c4 = −α, the PII equation (2.12) again results, while the associated class of NLS equations (2.1) becomes iΨt + Ψxx − [ (1− C) \|Ψ\|xx \|Ψ\| + Cγ2\|Ψ\|2 + Cγ2α \|Ψ\| + ν ] Ψ = 0. (2.13) It is remarked that in the absence of the de Broglie–Bohm term, a time-independent NLS equation of this type (2.13) incorporating a nonlinearity ∼ \|Ψ\|−1 has been derived ‘ab initio’ in [84] via a geometric model which describes stationary states of supercoiled DNA. Ermakov–Painlevé II Symmetry Reduction of a Korteweg Capillarity System 5 Case (iii) I 6= 0; c = 0; m = −1 2 ; n = 0. In this case, equation (2.10) reduces to a hybrid ‘Ermakov–Painlevé II’ equation of the type d2\|Ψ\| dξ2 + ε\|Ψ\|3 + (δξ + ζ)\|Ψ\| = σ \|Ψ\|3 , (2.14) and which will be subsequently seen to be linked to PXXXIV (1.4). It is recalled that the classical Ermakov equation with roots in [30], namely d2E dξ2 + ω(ξ)E = σ E3 , admits a nonlinear superposition principle readily derived via a Lie group approach as in [76, 81]. In the subsequent application to the Korteweg capillarity system it will be the Ermakov– Painlevé II symmetry reduction that will be exploited. With a positive solution \|Ψ\| of (2.14) to hand, the corresponding class of exact solutions for Ψ in the wave packet representation (2.2) is obtained via the integral of motion (2.6). Thus, the latter yields γ2 d dξ [ tan−1 ( φ ψ )] − α+ δ ε + cγ \|Ψ\| = I \|Ψ\|2 , whence, on integration γ2 tan−1 ( φ ψ ) = ( α− δ ε ) ξ − cγ ∫ 1 \|Ψ\| dξ + I ∫ 1 \|Ψ\|2 dξ, (2.15) where use has been made of the relation (2.9). Accordingly, with V = φ/ψ, it is seen that φ, ψ in the original wave packet representation are given by the relations φ = ± \|Ψ\|V√ 1 + V 2 , ψ = ± \|Ψ\|√ 1 + V 2 . In the sequel, the link between the Ermakov–Painlevé II equation (2.14) and PXXXIV (1.4) is used to construct novel classes of wave packet solutions of a nonlinear Korteweg capillarity system in terms of Yablonskii–Vorob’ev polynomials or classical Airy functions via the iterated application of the Bäcklund transformation for PII due to Gambier [38] and Lukashevich [59]. 3 The capillarity system In [4], Antanovskii derived the isothermal capillarity system with continuity equation ρt + div(ρv) = 0, (3.1a) augmented by the momentum equation vt + v •∇v +∇ [ δ(ρE) δρ −Π ] = 0, (3.1b) where ρ is the density, v velocity and E(ρ, \|∇ρ\|2/ρ) is the specific free energy. Herein, the standard variational derivative notation δΘ δρ = ∂Θ ∂ρ −∇ • [ ∂Θ ∂A ∇ρ ] , 6 C. Rogers and P.A. Clarkson is adopted with A = 1 2 \|∇ρ\| 2. In the above Π is an external potential, commonly taken to be that due to gravity, in which case Π = −ρg. The classical Korteweg capillarity system as set down in [54] is retrieved in the specialisation E(A, ρ) = κ(ρ)A ρ , κ(ρ) > 0, in which case, the momentum equation becomes vt + v •∇v −∇ [ κ(ρ)∇2ρ+ 1 2 \|∇ρ\| 2dκ(ρ) dρ + Π ] = 0. The classical Boussinesq capillarity system, in turn, is retrieved as the specialisation with κ constant in this Korteweg system. A system analogous to the Boussinesq model arises ‘mutatis mutandis’ in plasma physics [9]. In the case of irrotationality with v = ∇Φ, the momentum equation (3.1b) admits the Bernoulli integral Φt + 1 2 \|∇Φ\|2 + δ δρ (ρE)−Π = B(t), and on introduction of the Madelung transformation [60] Ψ = ρ1/2 exp ( 1 2 iΦ ) , (3.2) the capillarity system (3.1) may be encapsulated in the generalised NLS-type equation iΨt +∇2Ψ + [ −∇ 2\|Ψ\| \|Ψ\| − 1 2 δ(ρE) δρ + 1 2Π ] Ψ = 0, (3.3) incorporating a de Broglie–Bohm potential term. It was observed by Antanovskii et al. in [5] that if Π = 0 and E ( 1 2 \|∇ρ\| 2, ρ ) = C \|∇ρ\| 2 2ρ2 + νρ+ τ ρ , (3.4) then (3.3) reduces, if C = 1, to the cubic nonlinear Schrödinger equation iΨt +∇2Ψ− ν\|Ψ\|2Ψ = 0. If, on the other hand, C 6= 1, it is seen that reduction is obtained to a ‘resonant’ NLS-type equation [78] iΨt +∇2Ψ + [ (C − 1) ∇2\|Ψ\| \|Ψ\| − ν\|Ψ\|2 ] Ψ = 0. (3.5) Moreover, if C > 0 as in the present capillarity context, (3.5) may be transformed to a standard cubic NLS equation with the de Broglie–Bohm term removed (see, e.g., [73]). Thus, in 1 + 1 dimensions with three-parameter model energy E(12 \|∇ρ\| 2, ρ) of the type (3.4) reduction is made to a canonical integrable NLS equation. The capillarity system encapsulated in the (1 + 1)- dimensional version of (3.5) then becomes amenable to established methods of soliton theory such as inverse scattering procedures and inherits admittance of invariance under a Bäcklund transformation together with concomitant nonlinear superposition principle (see, e.g., [1, 77, 79] and literature cited therein). It is noted that a gravitational potential term Π = −ρg is readily Ermakov–Painlevé II Symmetry Reduction of a Korteweg Capillarity System 7 accommodated in the above reduction. Detailed qualitative properties of capillarity systems with model laws of the type (3.4) with Kármán–Tsien-type law κ(ρ) = C/ρ, C > 0, (3.6) have been recently set down in [16] while travelling wave propagation in (1 + 1)-dimensional capillarity theory has been investigated in [10]. Here, a more general class of model energy E(12 \|∇ρ\| 2, ρ) laws is considered, namely that with E = κ(ρ)\|∇ρ\|2 2ρ + R(ρ) ρ , so that, with Π = 0, (3.3) produces the class of NLS equations iΨt +∇2Ψ− [( 1 + dκ dρ \|Ψ\|4 ) ∇2\|Ψ\| \|Ψ\| − 1 2 ( κ(ρ) + dκ dρ \|Ψ\|2 ) (∇\|Ψ\|)2 + 1 2 dR dρ ] Ψ = 0. (3.7) Thus, capillarity systems encapsulated in (3.7) are isolated which may be aligned with NLS equations of the type (2.1) in the case c = 0, m = −2, n = 0. This occurs for the multi- parameter class of model energy laws with E ( 1 2 \|∇ρ\| 2, ρ ) = C \|∇ρ\| 2 2ρ2 + λρ− 2µ ρ2 + 2ν + τ ρ , (3.8) where λ, µ, ν and τ together with C > 0 are real constants. Importantly, this includes in the case µ = 0 the class which has been recently subject to a detailed qualitative analysis in [16]. The κ(ρ) capillarity relation is seen to be of the Kármán–Tsien type (3.6). Here, λ = −Cγ2c3, µ = −Cγ2c4, ν = −Cγ2c5, (3.9) in accordance with the relations (2.11). The associated class of NLS equations iΨt + Ψxx − [ (C − 1) \|Ψ\|xx \|Ψ\| + λ\|Ψ\|2 + µ \|Ψ\|4 + ν ] Ψ = 0, hence, admits symmetry reduction via the wave packet ansatz (2.2) to the hybrid Ermakov– Painlevé II equation (cf. (2.10)) d2\|Ψ\| dξ2 + [c1 + c5 + c2ξ] \|Ψ\|+ c3\|Ψ\|3 = σ \|Ψ\|3 , (3.10) where σ = 1 Cγ2 [( I γ )2 + µ ] . (3.11) Interestingly, this symmetry reduction to an integrable Ermakov–Painlevé II equation will be admitted by Korteweg-type capillarity systems with the particular model energy laws of the type discussed in [16]. Under the translation ζ = ξ + (c1 + c5)/c2, with c2 6= 0, (3.10) becomes d2\|Ψ\| dζ2 + c2ζ\|Ψ\|+ c3\|Ψ\|3 = σ \|Ψ\|3 , (3.12) 8 C. Rogers and P.A. Clarkson where the Madelung relation (3.2) shows that, in the present capillarity context \|Ψ\| = ρ1/2. Thus, ρ1/2 is governed by a hybrid Ermakov–Painlevé II equation while in terms of the density ρ it is seen that (3.12) produces d2ρ dζ2 = 1 2ρ ( dρ dζ )2 − 2c3ρ 2 − 2c2ζρ+ 2σ ρ , (3.13) which is equivalent to PXXXIV (1.4) (through a rescaling of the variables). This link between the Ermakov–Painlevé II equation (3.12) and PXXXIV (1.4) has been noted previously in the context of a Painlevé reduction of a classical Nernst–Planck electrodiffusion system in [3]. We remark that the special case of equation (3.13) with c3 = 0 was considered by Gambier [38, pp. 27–28], who linearised the equation. Multiplying (3.13) with c3 = 0 by ρ and differentiating gives d3ρ dζ3 = 4ζ dρ dζ + 2ρ, which has solution ρ(ζ) = C1 Ai2(z) + C2 Ai(z) Bi(z) + C3 Bi2(z), z = −c1/32 ζ, (3.14) with C1, C2 and C3 constants. The solution ρ(ζ) given by (3.14) satisfies (3.13) only if c3 = 0, σ = 0 and 4C1C2 = C2 3 . In the sequel, it is convenient to proceed with c2 = −1 2 , c3 = −1, σ = −1 4 ( α+ 1 2 )2 , whence ε = 1 2Cγ 2 > 0, λ = Cγ2 > 0, ( α+ 1 2 )2 = − 4 λ [( I γ )2 + µ ] , (3.15) where the latter requires that µ < −CI2/λ < 0. The Ermakov–Painlevé II equation (3.12) is then linked to PXXXIV (1.4) via the relation ρ = \|Ψ\|2 > 0. The well-known connection, in turn, between PII (1.3) and PXXXIV (1.4) is readily derived via the Hamiltonian system dq dz = ∂HII ∂p , dp dz = −∂HII ∂q , where the Hamiltonian HII(p, q, z;α) is given by HII(p, q, z;α) = 1 2p 2 − ( q2 + 1 2z ) p− ( α+ 1 2 ) q, leading to the coupled pair of nonlinear equations dq dz = p− q2 − 1 2z, dp dz = 2qp+ α+ 1 2 , (3.16) (see [44, 66]). Elimination of p and q successively in (3.16) duly leads to the PII (1.3) and PXXXIV (1.4). Thus, in the present capillarity context, the density distribution ρ(ζ) is given by ρ(ζ) = dw dζ + w2 + 1 2ζ, Ermakov–Painlevé II Symmetry Reduction of a Korteweg Capillarity System 9 where w(ζ) is governed by the PII equation d2w dζ2 = 2w3 + ζw + α. Here, the concern is necessarily restricted to solutions of PXXXIV (1.4) in regions in which ρ is positive. Interestingly, the importance of positive solutions of PXXXIV (1.4) also arises naturally in the setting of two-ion electro-diffusion. Thus, in the electrolytic context of [7, 13], the scaled electric field Y was shown to be governed by the PII equation d2Y dz2 = 2Y 3 + zY + α, and associated ion concentrations by p± = ±dY dz + Y 2 + 1 2z, with parameter α = 1−A−/A+ 2(1 +A−/A+) , and A± = −Φ±/D±, Φ± being the fluxes of the ion concentrations and D± diffusivity constants arising in the Einstein relation. Thus, it is seen that the ion concentrations, which are necessarily positive, are governed by PXXXIV (1.4). This positivity constraint was examined in detail in [7] for exact solutions in terms of either Yablonskii-Vorob’ev polynomials or classical Airy functions as induced by the iterated action of the Bäcklund transformation of [59] for PII (1.3). The results apply ‘mutatis mutandis’ in the present capillarity context. 4 Iterated action of a Bäcklund transformation Here, the consequences of the following well-known Bäcklund transformation for PII (1.3) are applied in the present capillarity context. Theorem 4.1. If qα(z) = q(z;α) is a solution of PII (1.3) with parameter α, then qα+1(z) = −qα(z)− 2α+ 1 2q′α(z) + 2q2α(z) + z , (4.1a) qα−1(z) = −qα(z)− 2α− 1 2q′α(z)− 2q2α(z) + z , (4.1b) are solution of PII with respective parameters α+ 1 and α− 1. Proof. See Gambier [38] and Lukashevich [59]. � The iteration of the Bäcklund transformations (4.1) allows the generation of all known exact solutions of PII (1.3). We note that eliminating q′α(z) in (4.1) yields the nonlinear difference equation α+ 1 2 qα+1 + qα + α− 1 2 qα + qα−1 + 2q2α + z = 0, which is known as an alternative form of discrete Painlevé I [33]. 10 C. Rogers and P.A. Clarkson Theorem 4.2. If qα = q(z;α) and pα = p(z;α) are solutions of PII (1.3) and PXXXIV (1.4) with parameter α respectively, then qα+1 = −qα − 2α+ 1 2pα , qα−1 = −qα + 2α− 1 2pα − 4q2α + 2z , pα+1 = −pα + ( qα + 2α+ 1 2pα )2 + z, pα−1 = −pα + 2q2α + z. Proof. See Okamoto [66]; also [35]. � 4.1 Rational solutions The iterative action of the above Bäcklund transformation on the seed solution q = 0 of PII with α = 0 produces the subsequent sequence of rational solutions qn(z) = d dz ln Qn−1(z) Qn(z) , n ∈ N, (4.2) corresponding to the Painlevé parameters α = n, for n ∈ N, where the Qn(z) are the Yablonskii– Vorob’ev polynomials determined by the quadratic recurrence relations Qn+1Qn−1 = zQ2 n + 4 {( dQn dz )2 −Qn d2Qn dz2 } , (4.3) with Q−1(z) = Q0(z) = 1 [89, 92]; see also [18, 19, 22, 48, 50, 87]. The Qn(z) are monic polynomials of degree 1 2n(n+1) with each term possessing the same degree modulo 3. Moreover, on use of the invariance under q(z, α)→ −q(z;−α), it is seen that PII (1.3) also admits the associated class of rational solutions q−n(z) = d dz ln Qn(z) Qn−1(z) , n ∈ N, (4.4) corresponding to the Painlevé parameters α = −n, for n ∈ N. The rational solutions of PXXXIV (1.4) are given by pn(z) = 1 2z − 2 d2 dz2 lnQn(z) ≡ Qn+1(z)Qn−1(z) 2Q2 n(z) , n ∈ N, corresponding to the parameters α = n, with n ∈ N. It is clear from the recurrence relation (4.3) that the Qn(z) are rational functions, though it is not obvious that they are polynomials since one is dividing by Qn−1(z) at every iteration. In fact it is somewhat remarkable that the Qn(z) are polynomials. Taneda [87], used an alge- braic method to prove that the functions Qn(z) defined by (4.3) are indeed polynomials, see also [36]. The Yablonskii–Vorob’ev polynomials Qn(z) can also be expressed as determinants, see [22, 46, 47]. Clarkson and Mansfield [22] investigated the locations of the roots of the Yablonskii–Vorob’ev polynomials in the complex plane and showed that these roots have a very regular, approximately triangular structure; the term “approximate” is used since the patterns Ermakov–Painlevé II Symmetry Reduction of a Korteweg Capillarity System 11 are not exact triangles as the roots lie on arcs rather than straight lines. Recently Bertola and Bothner [11] and Buckingham and Miller [14, 15] have studied the Yablonskii–Vorob’ev polyno- mials Qn(z) in the limit as n → ∞ and shown that the roots lie in a “triangular region” with elliptic sides which meet with interior angle 2 5π, suggesting a limit to a solution of Painlevé I (PI). Indeed Buckingham and Miller [15] show that in the limit as n→∞, the rational solution qn(z) of PII tends to the tritronquée solution of PI due to Boutroux [12], which no poles (of large modulus) except in one sector of angle 2 5π (see also [45]). In the sequel, attention is restricted to the case with invariants I = In 6= 0 so that similarity reduction of the capillarity system via (2.2) leads to consideration of a Ermakov–Painlevé II equation in the amplitude \|Ψ\| and, in turn, density ρ as determined by PXXXIV (3.13). Thus, the density distribution ρ = ρ+ associated with the class of exact solutions (4.2) of PII (1.3) in terms of Yablonskii–Vorob’ev polynomials with z = ζ may be shown to be given by rational expressions derived via (3.12) to adopt the form (see [18, 19, 22]) ρ+(ζ;n) = Qn+1(ζ)Qn−1(ζ) 2Q2 n(ζ) , ρ+(ζ; 0) = 1 2ζ, (4.5) as subsequently employed in the two-ion electolytic boundary value problems investigated in [7]. Therein, the positivity of members of the class of rational solutions (4.5) of PXXXIV on appropri- ate regions has been delimited. In the present capillarity context, in such regions where ρ = ρ+ is positive, the class of solutions (4.5) is associated with wave packet representations (2.2) with φ(ξ) = ± V (ξ)√ 1 + V 2(ξ) √ Qn+1(ξ)Qn−1(ξ) 2Q2 n(ξ) , (4.6a) ψ(ξ) = ± 1√ 1 + V 2(ξ) √ Qn+1(ξ)Qn−1(ξ) 2Q2 n(ξ) , (4.6b) where V (ξ) is given by, in view of (2.15), γ2 tan−1 V (ξ) = ( α− δ ε ) ξ + 2In ∫ ξ Q2 n(s) Qn+1(s)Qn−1(s) ds = ( α− δ ε ) ξ + 2In 2n+ 1 ln Qn+1(ξ) Qn−1(ξ) , (4.7) since ∫ ξ Q2 n(s) Qn+1(s)Qn−1(s) ds = 1 2n+ 1 ln Qn+1(ξ) Qn−1(ξ) . (4.8) This result (4.8) follows since the Yablonskii–Vorob’ev polynomials Qn(ξ) satisfy the bilinear relation dQn+1 dξ Qn−1 − dQn−1 dξ Qn+1 = (2n+ 1)Q2 n, which is proved in [36, 87] (see also [50]). Likewise, there are associated classes of wave packet representations corresponding to the rational solution q−n(z) given by (4.4) which are determined by the relations (4.6), (4.7) but with the transposition n↔ n− 1. 4.2 Airy-type solutions The iterated action of the Bäcklund transformations (4.1) may, in addition, be used to ge- nerate exact solutions of PII (1.3) with parameters α = ±1 2 ,± 3 2 , . . . in terms of classical Airy 12 C. Rogers and P.A. Clarkson functions [21]. Thus, in particular, if α = 1 2 then PII (1.3) admits the exact solution q ( z; 1 2 ) = − d dz lnϕ(z), (4.9) where ϕ(z) is governed by the classical Airy function d2ϕ dz2 + 1 2zϕ = 0. (4.10) Iteration of the Bäcklund transformation (4.1a) with the Airy-type seed solution (4.9) generates as infinite sequence of exact solutions q ( z;n− 1 2 ) = d dz ln un−1(z) un(z) , n ∈ N, (4.11) where the sequence {u`(z)}, for ` ≥ 0, is determined by the recurrence relation (Toda equation) un+1un−1 = 4 {( dun dz )2 − un d2un dz2 } , (4.12a) with initial values u0(z) = 1, u1(z) = ϕ(z). (4.12b) The un(z), for n ≥ 2, are homogeneous polynomials of degree n in ϕ(z) and ϕ′(z), i.e., have the form un(z) = n∑ j=0 an,jϕ j ( dϕ dz )n−j , where an,j(z) are polynomials in z and ϕ(z) is the solution of (4.10) given by ϕ(z) = cos(ϑ) Ai ( −2−1/3z ) + sin(ϑ) Bi ( −2−1/3z ) , (4.13) with Ai(x) and Bi(x) the Airy functions and ϑ an arbitrary constant. The analogous Airy-type solutions of PXXXIV (1.4) are given by p ( z;n− 1 2 ) = −2 d2 dz2 lnun ≡ un−1un+1 2u2n , n ∈ N, for the parameter α = n− 1 2 . The Airy-type solutions of PII (1.3) and PXXXIV (1.4) can also be expressed in terms of determinants, as described in the following theorem. Theorem 4.3. Let τn(z) be the Hankel n× n determinant τn(z) = [ dj+k dzj+k ϕ(z) ]n−1 j,k=0 , n ≥ 1, with ϕ(z) given by (4.13) and τ0(z) = 1, then for n ≥ 1, q ( z;n− 1 2 ) = d dz ln τn−1(z) τn(z) , p ( z;n− 1 2 ) = −2 d2 dz2 ln τn(z), (4.14) respectively satisfy PII (1.3) and PXXXIV (1.4) with α = n− 1 2 . Ermakov–Painlevé II Symmetry Reduction of a Korteweg Capillarity System 13 Proof. See Flaschka and Newell [32], Okamoto [66]; also [21, 35]. � We remark that recently it was shown that Airy-type solutions of PII (1.3) and PXXXIV (1.4) which depend only on the Airy function Ai(x) have a completely different structure to those which involve a linear combination of the Airy functions Ai(x) and Bi(x), see [21]. In particular, for n ∈ 2N the solution (4.14) of PXXXIV (1.4) has no poles on the real axis when ϕ(z) = Ai(−2−1/3z) and decays algebraically as z → ±∞. This special solution arose in a study of the double scaling limit of unitary random matrix ensembles by Its, Kuijlaars, and Östensson [42, 43], who identify the solution as a tronquée solution of PXXXIV, i.e., has no poles in a sector of the complex plane (see also [21]). The iterative application of the Bäcklund transformations (4.1) to generate Airy-type solu- tions of PII (1.3) has been used to solve boundary value problems associated with the classical Nernst–Planck system for two-ion electro-diffusion [7, 74]. The repeated action of the Bäcklund transformations in this electrolytic setting has been recently associated with quantised fluxes of ionic species in [13]. It is remarked that the Painlevé structure underlying a multi-ion elec- trodiffusion model has been systematically investigated in [26]. In the present capillarity context, the density distributions associated with the class of exact solutions of PII (1.3) determined by the relations (4.11), (4.12) are given by [7] ρ ( ξ;n− 1 2 ) = un−1(ξ)un+1(ξ) 2u2n(ξ) , ρ ( ξ;−1 2 ) = 0, where un(z) is given by (4.12). The physical requirement of positivity of these solutions corre- sponding to the specialisation ϕ(z) = Ai(−2−1/3z) in (4.13), i.e., when ϑ = 0, has been examined in detail in the electrolytic context of [7]. The implications of the results carry over to the present capillarity study. In general, with regard to the velocity magnitude v = \|v\|, alignment of the Madelung trans- formation (3.2) with (2.2) produces the velocity potential relation Φ = 2 tan−1 ( ψ + φ tan η φ− ψ tan η ) = 2 [ tan−1 ( ψ φ ) + η ] , (4.15) on use of the identity tan−1 ( x+ y 1− xy ) ≡ tan−1 x+ tan−1 y. Integration of the invariant relation (2.6) where in the present context c = 0, yields −γ2 tan−1 ( ψ φ ) = ( α− δ ε ) ξ + 2I ∫ ξ u2n(s) un+1(s)un−1(s) ds = ( α− δ ε ) ξ + I n ln un+1(ξ) un−1(ξ) , since ∫ ξ u2n(s) un+1(s)un−1(s) ds = 1 2n ln un+1(ξ) un−1(ξ) . (4.16) The result (4.16) holds as the un(ξ) satisfy the bilinear relation dun+1 dξ un−1 − dun−1 dξ un+1 = 2nu2n, which follows from Theorem 4.2. Consequently (4.15) yields \|v\| = − 2 γ2 [ γ ∂ ∂ξ + (εγt+ λ) ∂ ∂η ] [( α− δ ε ) ξ + I ∫ 1 \|Ψ\|2 dξ + η ] , 14 C. Rogers and P.A. Clarkson whence, v = − 2I γρ(ξ) + 2εγt− α γ , on use of the relation (2.9). Insertion of the latter expression into the momentum equation of the capillarity system, namely vt + vvx + [ δ δρ (ρE) ] x = 0, on integration, produces the Bernoulli integral 2I2 γ2ρ2 + 2εγx+ δ δρ (ρE) = B(t). Here (3.8) shows that ρE = C γ 2 2ρ ρ2x + λρ2 + 2µ ρ + 2νρ+ τ, whence, δ δρ (ρE) = ∂ ∂ρ (ρE)− C ∂ ∂x ( ρx ρ ) = 2λρ+ 2µ ρ2 + 2ν + C ( ρ2x 2ρ2 − ρxx ρ ) , in which ρ(x, t) = R(ξ) with ξ = αt+ βt2 + γx so that ρx = γR′(ξ). Thus, 2I2 γ2R2 + 2εγx+ 2λR+ 2µ R2 + 2ν + Cγ2 [ 1 2R2 ( dR dξ )2 − 1 R d2R dξ2 ] = B(t), and with B(t) = −2ε(αt + βt2), an equation equivalent to PXXXIV (1.4) for the density ρ(ζ) is retrieved, namely d2ρ dζ2 = 1 2ρ ( dρ dζ )2 + 2λρ2 Cγ2 + 2εζρ Cγ2 + 2(I2 + γ2µ) Cγ4ρ , ζ = ξ + c1 + c5 c2 , which aligns with (3.13) in view of the relations (3.9), (3.11) and (3.15). Thus, it is seen that, remarkably, in the present context PXXXIV is associated with the density ρ corresponding to a symmetry reduction of the Bernoulli integral of motion of the capillarity system. 5 Invariance under a reciprocal transformation The application of reciprocal-type transformations to (1 + 1)-dimensional nonlinear physical systems has its origin in the isolation of novel invariance properties in gasdynamics and magne- togasdynamics [70, 71]. They have been subsequently applied to both obtain analytic solution to moving boundary problems of Stefan-type [72] and to link integrable systems of modern soliton theory (see, e.g., [27, 65] together with [79] and work cited therein). In 3 + 1 dimen- sions, reciprocal-type transformations have been shown to have application in discontinuity wave propagation theory [29]. In the present capillarity context, the (1 + 1)-dimensional version of the system (3.1) with Π = 0, namely ρt + (ρv)x = 0, vt + vvx + [ δ δρ (ρE) ] x = 0, Ermakov–Painlevé II Symmetry Reduction of a Korteweg Capillarity System 15 was recently shown in [80] to be invariant under the one-parameter (χ) class of reciprocal-type transformations ρ∗ = ρ 1 + χρ , q∗ = q, E∗(A∗, ρ∗) = E(A, ρ), A∗ = A (1 + χρ)6 , dx∗ = (1 + χρ) dx− χρq dt, dt∗ = dt, 0 < \|1 + χρ\| <∞, where A = 1 2ρ 2 x. A direct corollary of this result is that the (1 + 1)-dimensional Korteweg-type capillarity system ρt + (ρv)x = 0, (5.1a) vt + vvx + ( −κ(ρ)ρxx − 1 2 dκ dρ ρ2x + dR dρ ) x = 0, (5.1b) is invariant under the one-parameter class of reciprocal transformations dx∗ = (1 + χρ) dx− χρq dt, dt∗ = dt, ρ∗ = ρ 1 + χρ , q∗ = q, augmented by the relations κ∗ = (1 + χρ)5κ, R∗ = R 1 + χρ . This invariant transformation may be applied to seed solutions of the capillarity system (5.1) as previously determined in terms of Yablonskii–Vorob’ev polynomials or classical Airy functions to construct extended χ-dependent classes of exact solutions valid for model energy laws E∗ with χ-deformed Kármán–Tsien capillarity relation κ∗ = C ρ∗(1− χρ∗)4 , together with R∗ = λρ∗2 1− χρ∗ − 2µ (1− χρ∗)2 ρ∗ + 2νρ∗ + τ(1− χρ∗). The original κ(ρ), R(ρ) associated with the PXXXIV reduction are retrieved in the limit χ→ 0. 6 General perspectives on model laws in continuum mechanics and solitonic connections: conclusion Here, model energy laws have been isolated for which a Korteweg-type capillarity system ad- mits symmetry reduction to a integrable hybrid Ermakov–Painlevé equation. A Bäcklund and reciprocal transformation have been used in turn to generate novel classes of exact solutions and to extend the range of the reduction. The derivation of multi-parameter model constitu- tive laws for which systems in nonlinear continuum mechanics become analytically tractable via the application of Bäcklund or reciprocal transformations has an extensive literature. Thus, in gasdynamics, Loewner [57, 58] applied matrix Bäcklund transformations to construct model constitutive laws for which the classical hodograph equations may be systematically reduced to appropriate tractable canonical forms in subsonic, transonic and supersonic flow régimes. The celebrated Kármán–Tsien two-parameter pressure-density model law of [88] as extensively applied in subsonic gasdynamics, arises as a particular reduction. The Bäcklund transforma- tions as introduced in the model gas law context of [58], suitably interpreted and extended, 16 C. Rogers and P.A. Clarkson remarkably, turn out to have application in (2 + 1)-dimensional soliton theory [52, 53]. In non- linear elastodynamics, model multi-parameter stress-strain laws were constructed in [17] which allow the analytic treatment of aspects of shock-less pulse propagation in bounded nonlinear elastic media. Comparison of experimental stress-strain relations with such model laws was investigated, in particular, for the dynamic compression of saturated soil, dry sand and clay silt. A Bäcklund transformation may be introduced at the level of the stress-strain laws for the uniaxial Lagrangian elastodynamic system treated in [17]. The single action of this Bäcklund transformation to the classical Hooke’s law generates the multi-parameter class of (T, e) laws applied extensively therein. Moreover application of a nonlinear superposition principle associ- ated with the Bäcklund transformation permits the construction of more general model nonlinear (T, e)-laws for which the (1+1)-dimensional elastodynamic system may be iteratively reduced to that associated with the canonical Hooke’s law. There is again a remarkable solitonic connection in that the nonlinear superposition principle acting on the (T, e)-laws turns out to be nothing but the permutability theorem for the potential Korteweg–de Vries hierarchy (see, e.g., [79]). In nonlinear elastostatics, model stress-deformation laws have been introduced by Neuber in [63, 64] in connection with the problem of determining the stress-distribution in shear-strained isotropic prismatical bodies. Loewner-type Bäcklund transformations were applied in [24] to the Neuber elastostatic system to solve a class of indentation boundary value problems for both Neuber–Sokolovsky and power law model stress-deformation relations. The application of model B-H and D-E constitutive laws in the analysis of the propagation of plane polarised electromagnetic waves through nonlinear dielectric media has been described in [51]. In general terms, it was shown in [82] that model constitutive laws as constructed via the Bäcklund ap- proach introduced by Loewner, corresponds to solitonic solutions generated by a Darboux-type transformation. With regard to reciprocal transformations and their role in the construction of model con- stitutive laws one may cite, in particular, the investigation of Storm in [85] concerning heat conduction in simple monatomic metals. Therein, a class of model (cp(T ), k(T )) temperature T -dependent laws was introduced for which a (1 + 1)-dimensional nonlinear heat conduction equation may be reduced via a reciprocal transformation to the classical linear heat equation. The applicability of these model laws was justified in [85] for appropriate specific heat cp(T ) and thermal conductivity k(T ) over wide temperature ranges for such materials as aluminium, silver, sodium, cadium, zinc, copper and lead. This kind of reduction via a reciprocal transformation may be extended to hyperbolic systems that correspond to multi-parameter model laws that arise in Cattaneo-type conduction and nonlinear visco-elasticity (see, e.g., [83]). The preceding attests to the importance and wide range of physical applications of model constitutive laws in nonlinear continuum mechanics together with intriguing solitonic connec- tions. The six classical Painlevé equations arise in a wide range of physical applications and play a fundamental role in modern soliton theory (see, e.g., [20, 25, 34, 40]). In the present work, model multi-parameter specific energy laws have been isolated which allow symmetry reduction of a Korteweg capillarity system to consideration of a hybrid Ermakov–Painlevé II equation and thereby to the linked integrable PXXXIV equation. In conclusion, it is remarked that the Ermakov–Painlevé II symmetry reduction presented here is also valid for a superfluidity model system involving a de Broglie Bohm potential as set down in [69]. References [1] Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, London Mathe- matical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge, 1991. [2] Ablowitz M.J., Segur H., Solitons and the inverse scattering transform, SIAM Studies in Applied Mathe- matics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. https://doi.org/10.1017/CBO9780511623998 https://doi.org/10.1017/CBO9780511623998 Ermakov–Painlevé II Symmetry Reduction of a Korteweg Capillarity System 17 [3] Amster P., Rogers C., On a Ermakov–Painlevé II reduction in three-ion electrodiffusion. A Dirichlet bound- ary value problem, Discrete Contin. Dyn. Syst. 35 (2015), 3277–3292. [4] Antanovskii L.K., Microscale theory of surface tension, Phys. Rev. E 54 (1996), 6285–6290. [5] Antanovskii L.K., Rogers C., Schief W.K., A note on a capillarity model and the nonlinear Schrödinger equation, J. Phys. A: Math. Gen. 30 (1997), L555–L557. [6] Assanto G., Minzoni A.A., Smyth N.F., On optical Airy beams in integrable and non-integrable systems, Wave Motion 52 (2015), 183–193. [7] Bass L., Nimmo J.J.C., Rogers C., Schief W.K., Electrical structures of interfaces: a Painlevé II model, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010), 2117–2136. [8] Bassom A.P., Clarkson P.A., Law C.K., McLeod J.B., Application of uniform asymptotics to the second Painlevé transcendent, Arch. Rational Mech. Anal. 143 (1998), 241–271, solv-int/9609005. [9] Belashov V.Yu., Vladimirov S.V., Solitary waves in dispersive complex media. Theory, simulation, applica- tions, Springer Series in Solid-State Sciences, Vol. 149, Springer-Verlag, Berlin, 2005. [10] Benzoni-Gavage S., Planar traveling waves in capillary fluids, Differential Integral Equations 26 (2013), 439–485. [11] Bertola M., Bothner T., Zeros of large degree Vorob’ev–Yablonski polynomials via a Hankel determinant identity, Int. Math. Res. Not. 2015 (2015), 9330–9399, arXiv:1401.1408. [12] Boutroux P., Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre, Ann. Sci. École Norm. Sup. (3) 30 (1913), 255–375. [13] Bracken A.J., Bass L., Rogers C., Bäcklund flux quantization in a model of electrodiffusion based on Painlevé II, J. Phys. A: Math. Theor. 45 (2012), 105204, 20 pages, arXiv:1201.0673. [14] Buckingham R.J., Miller P.D., Large-degree asymptotics of rational Painlevé-II functions: noncritical be- haviour, Nonlinearity 27 (2014), 2489–2578, arXiv:1310.2276. [15] Buckingham R.J., Miller P.D., Large-degree asymptotics of rational Painlevé-II functions: critical behaviour, Nonlinearity 28 (2015), 1539–1596, arXiv:1406.0826. [16] Carles R., Danchin R., Saut J.-C., Madelung, Gross–Pitaevskii and Korteweg, Nonlinearity 25 (2012), 2843–2873, arXiv:1111.4670. [17] Cekirge H.M., Varley E., Large amplitude waves in bounded media I. Reflexion and transmission of large amplitude shockless pulses at an interface, Philos. Trans. Roy. Soc. London Ser. A 273 (1973), 261–313. [18] Clarkson P.A., Painlevé equations – nonlinear special functions, J. Comput. Appl. Math. 153 (2003), 127– 140. [19] Clarkson P.A., Remarks on the Yablonskii–Vorob’ev polynomials, Phys. Lett. A 319 (2003), 137–144. [20] Clarkson P.A., Painlevé equations – nonlinear special functions, in Orthogonal polynomials and special functions, Lecture Notes in Math., Vol. 1883, Springer, Berlin, 2006, 331–411. [21] Clarkson P.A., On Airy solutions of the second Painlevé equation, Stud. Appl. Math. 137 (2016), 93–109, arXiv:1510.08326. [22] Clarkson P.A., Mansfield E.L., The second Painlevé equation, its hierarchy and associated special polyno- mials, Nonlinearity 16 (2003), R1–R26. [23] Clarkson P.A., McLeod J.B., A connection formula for the second Painlevé transcendent, Arch. Rational Mech. Anal. 103 (1988), 97–138. [24] Clements D.L., Rogers C., On the theory of stress concentration for shear-strained prismatical bodies with a non-linear stress-strain law, Mathematika 22 (1975), 34–42. [25] Conte R. (Editor), The Painlevé property. One century later, CRM Series in Mathematical Physics, Springer- Verlag, New York, 1999. [26] Conte R., Rogers C., Schief W.K., Painlevé structure of a multi-ion electrodiffusion system, J. Phys. A: Math. Theor. 40 (2007), F1031–F1040, arXiv:0711.0615. [27] Degasperis A., Holm D.D., Hone A.N.W., A new integrable equation with peakon solutions, Theoret. and Math. Phys. 133 (2002), 1463–1474, nlin.SI/0205023. [28] Deift P.A., Zhou X., Asymptotics for the Painlevé II equation, Comm. Pure Appl. Math. 48 (1995), 277–337. [29] Donato A., Ramgulam U., Rogers C., The 3 + 1-dimensional Monge–Ampère equation in discontinuity wave theory: application of a reciprocal transformation, Meccanica 27 (1992), 257–262. https://doi.org/10.3934/dcds.2015.35.3277 https://doi.org/10.1103/PhysRevE.54.6285 https://doi.org/10.1088/0305-4470/30/16/001 https://doi.org/10.1016/j.wavemoti.2014.10.003 https://doi.org/10.1098/rspa.2009.0620 https://doi.org/10.1007/s002050050105 http://arxiv.org/abs/solv-int/9609005 https://doi.org/10.1007/b138237 https://doi.org/10.1093/imrn/rnu239 http://arxiv.org/abs/1401.1408 https://doi.org/10.1088/1751-8113/45/10/105204 http://arxiv.org/abs/1201.0673 https://doi.org/10.1088/0951-7715/27/10/2489 http://arxiv.org/abs/1310.2276 https://doi.org/10.1088/0951-7715/28/6/1539 http://arxiv.org/abs/1406.0826 https://doi.org/10.1088/0951-7715/25/10/2843 http://arxiv.org/abs/1111.4670 https://doi.org/10.1098/rsta.1973.0001 https://doi.org/10.1016/S0377-0427(02)00589-7 https://doi.org/10.1016/j.physleta.2003.10.016 https://doi.org/10.1007/978-3-540-36716-1_7 https://doi.org/10.1111/sapm.12123 http://arxiv.org/abs/1510.08326 https://doi.org/10.1088/0951-7715/16/3/201 https://doi.org/10.1007/BF00251504 https://doi.org/10.1007/BF00251504 https://doi.org/10.1112/S0025579300004472 https://doi.org/10.1007/978-1-4612-1532-5 https://doi.org/10.1088/1751-8113/40/48/F01 https://doi.org/10.1088/1751-8113/40/48/F01 http://arxiv.org/abs/0711.0615 https://doi.org/10.1023/A:1021186408422 https://doi.org/10.1023/A:1021186408422 http://arxiv.org/abs/nlin.SI/0205023 https://doi.org/10.1002/cpa.3160480304 https://doi.org/10.1007/BF00424364 18 C. Rogers and P.A. Clarkson [30] Ermakov V.P., Second-order differential equations: conditions for complete integrability, Univ. Izv. Kiev 20 (1880), no. 9, 1–25, see Appl. Anal. Discrete Math. 2 (2008), 123–145. [31] Eslami M., Mirzazadeh M., Optical solitons with Biswas–Milovic equation for power law and dual-power law nonlinearities, Nonlinear Dynam. 83 (2016), 731–738. [32] Flaschka H., Newell A.C., Monodromy- and spectrum-preserving deformations. I, Comm. Math. Phys. 76 (1980), 65–116. [33] Fokas A.S., Grammaticos B., Ramani A., From continuous to discrete Painlevé equations, J. Math. Anal. Appl. 180 (1993), 342–360. [34] Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents. The Riemann–Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, Amer. Math. Soc., Providence, RI, 2006. [35] Forrester P.J., Witte N.S., Application of the τ -function theory of Painlevé equations to random matrices: PIV, PII and the GUE, Comm. Math. Phys. 219 (2001), 357–398, math-ph/0103025. [36] Fukutani S., Okamoto K., Umemura H., Special polynomials and the Hirota bilinear relations of the second and the fourth Painlevé equations, Nagoya Math. J. 159 (2000), 179–200. [37] Gagnon L., Winternitz P., Lie symmetries of a generalised nonlinear Schrödinger equation. II. Exact solu- tions, J. Phys. A: Math. Gen. 22 (1989), 469–497. [38] Gambier B., Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes, Acta Math. 33 (1910), 1–55. [39] Giannini J.A., Joseph R.I., The role of the second Painlevé transcendent in nonlinear optics, Phys. Lett. A 141 (1989), 417–419. [40] Gromak V.I., Laine I., Shimomura S., Painlevé differential equations in the complex plane, De Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2002. [41] Hastings S.P., McLeod J.B., A boundary value problem associated with the second Painlevé transcendent and the Korteweg–de Vries equation, Arch. Rational Mech. Anal. 73 (1980), 31–51. [42] Its A.R., Kuijlaars A.B.J., Östensson J., Critical edge behavior in unitary random matrix ensembles and the thirty-fourth Painlevé transcendent, Int. Math. Res. Not. 2008 (2008), no. 9, rnn017, 67 pages, arXiv:0704.1972. [43] Its A.R., Kuijlaars A.B.J., Östensson J., Asymptotics for a special solution of the thirty fourth Painlevé equation, Nonlinearity 22 (2009), 1523–1558, arXiv:0811.3847. [44] Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with ra- tional coefficients. II, Phys. D 2 (1981), 407–448. [45] Joshi N., Kitaev A.V., On Boutroux’s tritronquée solutions of the first Painlevé equation, Stud. Appl. Math. 107 (2001), 253–291. [46] Kajiwara K., Masuda T., A generalization of determinant formulae for the solutions of Painlevé II and XXXIV equations, J. Phys. A: Math. Gen. 32 (1999), 3763–3778, solv-int/9903014. [47] Kajiwara K., Ohta Y., Determinant structure of the rational solutions for the Painlevé II equation, J. Math. Phys. 37 (1996), 4693–4704, solv-int/9607002. [48] Kametaka Y., Noda M., Fukui Y., Hirano S., A numerical approach to Toda equation and Painlevé II equation, Mem. Fac. Eng. Ehime Univ. 9 (1986), 1–24. [49] Kaminer I., Segev M., Christodoulides D.N., Self-accelerating self-trapped optical beams, Phys. Rev. Lett. 106 (2011), 213903, 4 pages. [50] Kaneko M., Ochiai H., On coefficients of Yablonskii–Vorob’ev polynomials, J. Math. Soc. Japan 55 (2003), 985–993, math.CA/0205178. [51] Kazakia J.Y., Venkataraman R., Propagation of electromagnetic waves in a nonlinear dielectric slab, Z. Angew. Math. Phys. 26 (1975), 61–76. [52] Konopelchenko B., Rogers C., On (2+1)-dimensional nonlinear systems of Loewner-type, Phys. Lett. A 158 (1991), 391–397. [53] Konopelchenko B., Rogers C., On generalized Loewner systems: novel integrable equations in 2 + 1 dimen- sions, J. Math. Phys. 34 (1993), 214–242. [54] Korteweg D.K., Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité, Arch. Néerl. 6 (1901), 1–24. https://doi.org/10.2298/AADM0802123E https://doi.org/10.1007/s11071-015-2361-1 https://doi.org/10.1007/BF01197110 https://doi.org/10.1006/jmaa.1993.1405 https://doi.org/10.1006/jmaa.1993.1405 https://doi.org/10.1090/surv/128 https://doi.org/10.1007/s002200100422 http://arxiv.org/abs/math-ph/0103025 https://doi.org/10.1088/0305-4470/22/5/013 https://doi.org/10.1007/BF02393211 https://doi.org/10.1016/0375-9601(89)90860-8 https://doi.org/10.1515/9783110198096 https://doi.org/10.1515/9783110198096 https://doi.org/10.1007/BF00283254 https://doi.org/10.1093/imrn/rnn017 http://arxiv.org/abs/0704.1972 https://doi.org/10.1088/0951-7715/22/7/002 http://arxiv.org/abs/0811.3847 https://doi.org/10.1016/0167-2789(81)90021-X https://doi.org/10.1111/1467-9590.00187 https://doi.org/10.1088/0305-4470/32/20/309 http://arxiv.org/abs/solv-int/9903014 https://doi.org/10.1063/1.531648 https://doi.org/10.1063/1.531648 http://arxiv.org/abs/solv-int/9607002 https://doi.org/10.1103/PhysRevLett.106.213903 https://doi.org/10.2969/jmsj/1191418760 http://arxiv.org/abs/math.CA/0205178 https://doi.org/10.1007/BF01596279 https://doi.org/10.1016/0375-9601(91)90680-7 https://doi.org/10.1063/1.530377 Ermakov–Painlevé II Symmetry Reduction of a Korteweg Capillarity System 19 [55] Lee J.-H., Pashaev O.K., Solitons of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions: Hirota bilinear method, Theoret. and Math. Phys. 152 (2007), 991–1003, nlin.SI/0611003. [56] Lee J.-H., Pashaev O.K., Rogers C., Schief W.K., The resonant nonlinear Schrödinger equation in cold plasma physics. Application of Bäcklund–Darboux transformations and superposition principles, J. Plasma Phys. 73 (2007), 257–272. [57] Loewner C., A transformation theory of the partial differential equations of gas dynamics, Tech. Notes Nat. Adv. Comm. Aeronaut. 1950 (1950), 1–56. [58] Loewner C., Generation of solutions of systems of partial differential equations by composition of infinitesimal Baecklund transformations, J. Anal. Math. 2 (1953), 219–242. [59] Lukashevich N.A., On the theory of Painlevé’s second equation, Differ. Equ. 7 (1971), 853–854. [60] Madelung E., Quantentheorie in hydrodynamischen Form, Z. Phys. 40 (1926), 322–326. [61] Mahalov A., Suslov S.K., An “Airy gun”: self-accelerating solutions of the time-dependent Schrödinger equation in vacuum, Phys. Lett. A 377 (2012), 33–38. [62] Miles J.W., On the second Painlevé transcendent, Proc. Roy. Soc. London Ser. A 361 (1978), 277–291. [63] Neuber H., Kerbspannungslehre. Grundlagen für genaue Festigkeitsberechnung mit Berücksichtigung von Konstruktionsform und Werkstoff, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1958. [64] Neuber H., Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law, J. Appl. Mech. 28 (1961), 544–550. [65] Oevel W., Rogers C., Gauge transformations and reciprocal links in 2 + 1 dimensions, Rev. Math. Phys. 5 (1993), 299–330. [66] Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, PII and PIV, Math. Ann. 275 (1986), 221–255. [67] Pashaev O.K., Lee J.-H., Resonance solitons as black holes in Madelung fluid, Modern Phys. Lett. A 17 (2002), 1601–1619, hep-th/9810139. [68] Pashaev O.K., Lee J.-H., Rogers C., Soliton resonances in a generalized nonlinear Schrödinger equation, J. Phys. A: Math. Theor. 41 (2008), 452001, 9 pages. [69] Roberts P.H., Berloff N.G., The nonlinear Schrödinger equation as a model of superfluidity, in Quantum Vortex Dynamics and Superfluid Turbulence, Lecture Notes in Phys., Vol. 571, Editors C.F. Barenghi, R.J. Donnelly, W.F. Vinen, Springer, Berlin, 2001, 235–257. [70] Rogers C., Reciprocal relations in non-steady one-dimensional gasdynamics, Z. Angew. Math. Phys. 19 (1968), 58–63. [71] Rogers C., Invariant transformations in non-steady gasdynamics and magnetogasdynamics, Z. Angew. Math. Phys. 20 (1969), 370–382. [72] Rogers C., On a class of moving boundary problems in nonlinear heat conduction: application of a Bäcklund transformation, Internat. J. Non-Linear Mech. 21 (1986), 249–256. [73] Rogers C., Integrable substructure in a Korteweg capillarity model. A Kármán–Tsien type constitutive relation, J. Nonlinear Math. Phys. 21 (2014), 74–88. [74] Rogers C., Bassom A.P., Schief W.K., On a Painlevé II model in steady electrolysis: application of a Bäcklund transformation, J. Math. Anal. Appl. 240 (1999), 367–381. [75] Rogers C., Malomed B., Chow K., An H., Ermakov–Ray–Reid systems in nonlinear optics, J. Phys. A: Math. Theor. 43 (2010), 455214, 15 pages. [76] Rogers C., Ramgulam U., A nonlinear superposition principle and Lie group invariance: application in rotating shallow water theory, Internat. J. Non-Linear Mech. 24 (1989), 229–236. [77] Rogers C., Schief W.K., Intrinsic geometry of the NLS equation and its auto-Bäcklund transformation, Stud. Appl. Math. 101 (1998), 267–287. [78] Rogers C., Schief W.K., Geodesic motion in multidimensional unified gauge theories, Nuovo Cimento B 114 (1999), 1409–1412. [79] Rogers C., Schief W.K., Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. [80] Rogers C., Schief W.K., The classical Korteweg capillarity system: geometry and invariant transformations, J. Phys. A: Math. Theor. 47 (2014), 345201, 20 pages. https://doi.org/10.1007/s11232-007-0083-3 http://arxiv.org/abs/nlin.SI/0611003 https://doi.org/10.1017/S0022377806004648 https://doi.org/10.1017/S0022377806004648 https://doi.org/10.1007/BF02825638 https://doi.org/10.1007/BF01400372 https://doi.org/10.1016/j.physleta.2012.10.041 https://doi.org/10.1098/rspa.1978.0103 https://doi.org/10.1007/978-3-642-53069-2 https://doi.org/10.1115/1.3641780 https://doi.org/10.1142/S0129055X93000073 https://doi.org/10.1007/BF01458459 https://doi.org/10.1142/S0217732302007995 http://arxiv.org/abs/hep-th/9810139 https://doi.org/10.1088/1751-8113/41/45/452001 https://doi.org/10.1007/3-540-45542-6_23 https://doi.org/10.1007/BF01603278 https://doi.org/10.1007/BF01590430 https://doi.org/10.1007/BF01590430 https://doi.org/10.1016/0020-7462(86)90032-6 https://doi.org/10.1080/14029251.2014.894721 https://doi.org/10.1006/jmaa.1999.6589 https://doi.org/10.1088/1751-8113/43/45/455214 https://doi.org/10.1088/1751-8113/43/45/455214 https://doi.org/10.1016/0020-7462(89)90042-5 https://doi.org/10.1111/1467-9590.00093 https://doi.org/10.1111/1467-9590.00093 https://doi.org/10.1017/CBO9780511606359 https://doi.org/10.1088/1751-8113/47/34/345201 20 C. Rogers and P.A. Clarkson [81] Rogers C., Schief W.K., Winternitz P., Lie-theoretical generalization and discretization of the Pinney equa- tion, J. Math. Anal. Appl. 216 (1997), 246–264. [82] Schief W.K., Rogers C., Loewner transformations: adjoint and binary Darboux connections, Stud. Appl. Math. 100 (1998), 391–422. [83] Seymour B., Varley E., A Bäcklund transformation for a nonlinear telegraph equation, in Wave Phenomena: Modern Theory and Applications, North-Holland Mathematics Studies, Vol. 97, Editors C. Rogers and T.M. Moodie, North-Holland, Amsterdam, 1984, 299–306. [84] Shi Y., Hearst J.E., The Kirchoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling, J. Chem. Phys. 101 (1994), 5186–5200. [85] Storm M.L., Heat conduction in simple metals, J. Appl. Phys. 22 (1951), 940–951. [86] Tajiri M., Similarity reductions of the one- and two-dimensional nonlinear Schrödinger equations, J. Phys. Soc. Japan 52 (1983), 1908–1917. [87] Taneda M., Remarks on the Yablonskii–Vorob’ev polynomials, Nagoya Math. J. 159 (2000), 87–111. [88] Tsien H.-S., Two-dimensional subsonic flow of compressible fluids, J. Aeronaut. Sci. 6 (1939), 399–407. [89] Vorob’ev A.P., On the rational solutions of the second Painlevé equation, Differ. Equ. 1 (1965), 79–81. [90] Wagner W.G., Haus H.A., Marburger J.H., Large-scale self-trapping of optical beams in the paraxial ray approximation, Phys. Rev. 175 (1968), 256–266. [91] Xu Y., Suarez P., Milovic D., Khan K.R., Mahmood M.F., Biswas A., Belic M., Raman solitons in nanoscale optical waveguides, with metamaterials, having polynomial law non-linearity, J. Modern Opt. 63 (2016), S32–S37. [92] Yablonskii A.I., On rational solutions of the second Painlevé equation, Vesti AN BSSR, Ser. Fiz.-Tech. Nauk (1959), no. 3, 30–35. https://doi.org/10.1006/jmaa.1997.5674 https://doi.org/10.1111/1467-9590.00082 https://doi.org/10.1111/1467-9590.00082 https://doi.org/10.1016/S0304-0208(08)71273-8 http://doi.org/10.1063/1.468506 https://doi.org/10.1063/1.1700076 https://doi.org/10.1143/JPSJ.52.1908 https://doi.org/10.1143/JPSJ.52.1908 https://doi.org/10.1017/S0027763000007431 https://doi.org/10.2514/8.916 https://doi.org/10.1103/PhysRev.175.256 https://doi.org/10.1080/09500340.2016.1193240 1 Introduction 2 A Ermakov–Painlevé II symmetry reduction 3 The capillarity system 4 Iterated action of a Bäcklund transformation 4.1 Rational solutions 4.2 Airy-type solutions 5 Invariance under a reciprocal transformation 6 General perspectives on model laws in continuum mechanics and solitonic connections: conclusion References

Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System

Репозитарії

Схожі ресурси