Ermakov-Painlev\'e II Symmetry Reduction of a Korteweg Capillarity System

0.46 MB·

by  Colin Rogers

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 018, 20 pages
Ermakov–Painlev´e 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: [email protected]
‡ School of Mathematics, Statistics & Actuarial Science, University of Kent,
Canterbury, CT2 7FS, UK
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1 E-mail: [email protected]
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2 Received January 13, 2017, in final form March 15, 2017; Published online March 22, 2017
r https://doi.org/10.3842/SIGMA.2017.018
a
M
Abstract. A class of nonlinear Schr¨odinger equations involving a triad of power law terms

2 togetherwithadeBroglie–Bohmpotentialisshowntoadmitsymmetryreductiontoahybrid
2 Ermakov–Painlev´e II equation which is linked, in turn, to the integrable Painlev´e XXXIV
equation. A nonlinear Schr¨odinger encapsulation of a Korteweg-type capillary system is
]
I thereby used in the isolation of such a Ermakov–Painlev´e II reduction valid for a multi-
S parameter class of free energy functions. Iterated application of a B¨acklund transfor-
.
n mation then allows the construction of novel classes of exact solutions of the nonlinear
i capillarity system in terms of Yablonskii–Vorob’ev polynomials or classical Airy functions.
l
n A Painlev´e 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.

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Key words: Ermakov–Painlev´e II equation; Painlev´e capillarity; Korteweg-type capillary
v
system; B¨acklund transformation
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3
2010 Mathematics Subject Classification: 37J15; 37K10; 76B45; 76D45
2
3
0
.
1 1 Introduction
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Giannini and Joseph [39], in a nonlinear optics context, introduced a class of symmetry reduc-
1
: tions for a cubic nonlinear Schr¨odinger (NLS) equation
v
i
X iΨ +Ψ +ν|Ψ|2Ψ = 0, (1.1)
t xx
r
a with subscripts denoted partial derivatives, which resulted in the Painlev´e II equation but with
zero parameter α
d2q
= 2q3+zq, (1.2)
dz2
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 Painlev´e parameter α in that reduction does not allow the iterative
construction of sequences of exact solutions via the B¨acklund transformation for the canonical
Painlev´e II (P )
II
d2q
= 2q3+zq+α, (1.3)
dz2
2 C. Rogers and P.A. Clarkson
withαaparameter,asgivenbyGambier[38]andLukashevich[59]. Here,symmetryreductionto
P (1.3) or to a hybrid Ermakov–Painlev´e II equation linked to the integrable Painlev´e XXXIV
II
(P ) equation
XXXIV
d2p 1 (cid:18)dp(cid:19)2 (α+ 1)2
= +2p2−zp− 2 , (1.4)
dz2 2p dz 2p
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–Painlev´e II symmetry reduction is applied here to an NLS encapsulation of
anonlinearcapillaritysystemwithorigininclassicalworkofKorteweg[54]. Iteratedapplication
of a B¨acklund transformation admitted by P (1.3) permits the construction via the linked
II
P equation of novel multi-parameter wave packet solutions to the capillarity system in
XXXIV
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–Painlev´e II symmetry reduction
Here, a class of (1+1)-dimensional nonlinear Schr¨odinger equations of the type
(cid:20) (cid:21)
|Ψ| |Ψ|
iΨ +Ψ − (1−C) xx −ic x +λ|Ψ|2+µ|Ψ|2m+ν|Ψ|2n Ψ = 0, (2.1)
t xx |Ψ| |Ψ|2
which incorporates a de Broglie–Bohm potential |Ψ| /|Ψ| and a triad of power law terms is
xx
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 P (1.3),
II
with non-zero parameter α, or to a hybrid Ermakov–Painlev´e 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 Painlev´e II but with zero parameter α and resort was made
to a numerical treatment. Asymptotic properties of P (1.2) with α = 0 have been discussed
II
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
γ2d2φ − dψ(cid:2)2(cid:0)β+εγ2(cid:1)t+α+2λγ(cid:3)+ cγψ (cid:18)φdφ +ψdψ(cid:19)−∆φ = 0, (2.3a)
dξ2 dξ |Ψ|3 dξ dξ
γ2d2ψ + dφ(cid:2)2(cid:0)β+εγ2(cid:1)t+α+2λγ(cid:3)− cγφ (cid:18)φdφ +ψdψ(cid:19)−∆ψ = 0, (2.3b)
dξ2 dξ |Ψ|3 dξ dξ
Ermakov–Painlev´e 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
(cid:40)(cid:34) (cid:35) (cid:41)
+ sγ2 φd2φ +ψd2ψ +(cid:18)dφ(cid:19)2+(cid:18)dψ(cid:19)2 (cid:0)φ2+ψ2(cid:1)−(cid:18)φdφ +ψdψ(cid:19)2 , (2.4)
|Ψ|4 dξ2 dξ2 dξ dξ dξ dξ
with s = 1−C. The relations (2.3) together show that
γ2(cid:18)d2φψ− d2ψφ(cid:19)−(cid:18)φdφ +ψdψ(cid:19)(cid:2)2(cid:0)β+εγ2(cid:1)t+α+2λγ(cid:3)
dξ2 dξ2 dξ dξ
(cid:18) (cid:19)
cγ dφ dψ
+ φ +ψ = 0, (2.5)
|Ψ| dξ dξ
whence it is required that
β+εγ2 = 0,
in which case equation (2.5) admits the integral
(cid:18) (cid:19)
dφ dψ
γ2 ψ− φ − 1(α+2λγ)|Ψ|2+cγ|Ψ| = I, (2.6)
dξ dξ 2
where I is an arbitrary constant of motion.
On use of the relation
(cid:34) (cid:35)
d2φ d2ψ (cid:18)dφ(cid:19)2 (cid:18)dψ(cid:19)2 (cid:18) dφ dψ(cid:19)2 d2|Ψ|
φ +ψ + + (φ2+ψ2)− φ +ψ = |Ψ|3 , (2.7)
dξ2 dξ2 dξ dξ dξ dξ dξ2
it is seen that (2.4) yields, if β (cid:54)= 0,
∆ = γβ−1(cid:0)ε2γ +3(cid:1)(ξ−αt−γx)+2(δ+εγλ)t+εγx+ζ +λ2
sγ2d2|Ψ|
+λ|Ψ|2+µ|Ψ|2m+ν|Ψ|2n+
|Ψ| dξ2
sγ2d2|Ψ|
= εξ+ζ +λ2+λ|Ψ|2+µ|Ψ|2m+ν|Ψ|2n+ , (2.8)
|Ψ| dξ2
on setting
βε = γ(cid:0)3+ε2γ(cid:1), αε = 2(δ+εγλ). (2.9)
Moreover, equations (2.3) again combine to show that
(cid:18) d2φ d2ψ(cid:19) (cid:18)dφ dψ (cid:19)
γ2 φ +ψ + ψ− φ (α+2λγ)−∆|Ψ|2 = 0,
dξ2 dξ2 dξ dξ
whence, on use of the identity
(cid:34) (cid:35)
(cid:18)dφ(cid:19)2 (cid:18)dψ(cid:19)2 (cid:18)dφ dψ (cid:19)2 (cid:18) dφ dψ(cid:19)2
(φ2+ψ2) + − ψ− φ ≡ φ +ψ ,
dξ dξ dξ dξ dξ dξ
together with (2.7), it is seen that
(cid:34) (cid:35)
d2|Ψ| (cid:18)dφ dψ (cid:19)2 (cid:18)dφ dψ (cid:19)
γ2 |Ψ|3 − ψ− φ +(α+2λγ) ψ− φ |Ψ|2−∆|Ψ|4 = 0.
dξ2 dξ dξ dξ dξ
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| c c
+[c +c ξ]|Ψ|+c |Ψ|3+c |Ψ|2m+1+c |Ψ|2n+1+ 6 + 7
dξ2 1 2 3 4 5 |Ψ| |Ψ|2
I2
= , (2.10)
(1−s)γ4|Ψ|3
where the constants c ,c ,...,c are given by
1 2 7
(α−δ/ε)2−γ2(ζ +λ2) ε λ
c = , c = , c = , (2.11a)
1 (1−s)γ4 2 (s−1)γ2 3 (s−1)γ2
µ ν c2 2cI
c = , c = , c = , c = , (2.11b)
4 (s−1)γ2 5 (s−1)γ2 6 (s−1)γ2 7 (1−s)γ3
and it is required that s (cid:54)= 1.
Below a triad of cases is set down in which the amplitude equation (2.10) reduces either
directlytoP ortoahybridErmakov–Painlev´eIIequationsubsequentlyshowntobeintegrable.
II
Case (i) I = 0; m = −1; n = −1.
2
In this case with
c = 0, c = −1, c = −2, c = −α, c +c = 0,
1 2 3 4 5 6
the amplitude equation reduces directly to the Painlev´e II equation
d2|Ψ|
= 2|Ψ|3+ξ|Ψ|+α, (2.12)
dξ2
corresponding to the symmetry reduction via the ansatz (2.2) of the class of NLS equations
(cid:20) |Ψ| |Ψ| Cγ2α c2 (cid:21)
iΨ +Ψ − (1−C) xx −ic x +Cγ2|Ψ|2+ − Ψ = 0.
t xx |Ψ| |Ψ|2 |Ψ| |Ψ|2
Case (ii) I = 0; c = 0; m = −1; n = 0.
2
Here, with
c = −c , c = −1, c = −2, c = −α,
1 5 2 3 4
the P equation (2.12) again results, while the associated class of NLS equations (2.1) becomes
II
(cid:20) |Ψ| Cγ2α (cid:21)
iΨ +Ψ − (1−C) xx +Cγ2|Ψ|2+ +ν Ψ = 0. (2.13)
t xx
|Ψ| |Ψ|
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–Painlev´e II Symmetry Reduction of a Korteweg Capillarity System 5
Case (iii) I =(cid:54) 0; c = 0; m = −1; n = 0.
2
In this case, equation (2.10) reduces to a hybrid ‘Ermakov–Painlev´e II’ equation of the type
d2|Ψ| σ
+ε|Ψ|3+(δξ+ζ)|Ψ| = , (2.14)
dξ2 |Ψ|3
and which will be subsequently seen to be linked to P (1.4). It is recalled that the classical
XXXIV
Ermakov equation with roots in [30], namely
d2E σ
+ω(ξ)E = ,
dξ2 E3
admitsanonlinearsuperpositionprinciplereadilyderivedviaaLiegroupapproachasin[76,81].
In the subsequent application to the Korteweg capillarity system it will be the Ermakov–
Painlev´e 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
(cid:20) (cid:18) (cid:19)(cid:21)
d φ δ cγ I
γ2 tan−1 −α+ + = ,
dξ ψ ε |Ψ| |Ψ|2
whence, on integration
(cid:18) (cid:19) (cid:18) (cid:19) (cid:90) (cid:90)
φ δ 1 1
γ2tan−1 = α− ξ−cγ dξ+I dξ, (2.15)
ψ ε |Ψ| |Ψ|2
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
|Ψ|V |Ψ|
φ = ±√ , ψ = ±√ .
1+V2 1+V2
In the sequel, the link between the Ermakov–Painlev´e II equation (2.14) and P (1.4)
XXXIV
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 B¨acklund transformation for P due to Gambier [38] and Lukashevich [59].
II
3 The capillarity system
In [4], Antanovskii derived the isothermal capillarity system with continuity equation
ρ +div(ρv) = 0, (3.1a)
t
augmented by the momentum equation
(cid:20) (cid:21)
δ(ρE)
vt+v•∇v+∇ −Π = 0, (3.1b)
δρ
where ρ is the density, v velocity and E(ρ,|∇ρ|2/ρ) is the specific free energy. Herein, the
standard variational derivative notation
(cid:20) (cid:21)
δΘ ∂Θ ∂Θ
= −∇• ∇ρ ,
δρ ∂ρ ∂A
6 C. Rogers and P.A. Clarkson
is adopted with A = 1|∇ρ|2. In the above Π is an external potential, commonly taken to be
2
that due to gravity, in which case Π = −ρg. The classical Korteweg capillarity system as set
down in [54] is retrieved in the specialisation
κ(ρ)A
E(A,ρ) = , κ(ρ) > 0,
ρ
in which case, the momentum equation becomes
(cid:20) (cid:21)
dκ(ρ)
vt+v•∇v−∇ κ(ρ)∇2ρ+ 12|∇ρ|2 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
δ
Φ + 1|∇Φ|2+ (ρE)−Π = B(t),
t 2 δρ
and on introduction of the Madelung transformation [60]
Ψ = ρ1/2exp(cid:0)1iΦ(cid:1), (3.2)
2
the capillarity system (3.1) may be encapsulated in the generalised NLS-type equation
(cid:20) ∇2|Ψ| δ(ρE) (cid:21)
iΨ +∇2Ψ+ − − 1 + 1Π Ψ = 0, (3.3)
t |Ψ| 2 δρ 2
incorporating a de Broglie–Bohm potential term.
It was observed by Antanovskii et al. in [5] that if Π = 0 and
E(cid:0)1|∇ρ|2,ρ(cid:1) = C|∇ρ|2 +νρ+ τ, (3.4)
2 2ρ2 ρ
then (3.3) reduces, if C = 1, to the cubic nonlinear Schr¨odinger equation
iΨ +∇2Ψ−ν|Ψ|2Ψ = 0.
t
If, on the other hand, C =(cid:54) 1, it is seen that reduction is obtained to a ‘resonant’ NLS-type
equation [78]
(cid:20) ∇2|Ψ| (cid:21)
iΨ +∇2Ψ+ (C −1) −ν|Ψ|2 Ψ = 0. (3.5)
t
|Ψ|
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(1|∇ρ|2,ρ) of the type (3.4) reduction is made
2
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 B¨acklund
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–Painlev´e 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 K´arm´an–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(1|∇ρ|2,ρ) laws is considered, namely that with
2
κ(ρ)|∇ρ|2 R(ρ)
E = + ,
2ρ ρ
so that, with Π = 0, (3.3) produces the class of NLS equations
(cid:20)(cid:18) dκ (cid:19) ∇2|Ψ| (cid:18) dκ (cid:19) dR(cid:21)
iΨ +∇2Ψ− 1+ |Ψ|4 − 1 κ(ρ)+ |Ψ|2 (∇|Ψ|)2+ 1 Ψ = 0. (3.7)
t dρ |Ψ| 2 dρ 2 dρ
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(cid:0)1|∇ρ|2,ρ(cid:1) = C|∇ρ|2 +λρ− 2µ +2ν + τ, (3.8)
2 2ρ2 ρ2 ρ
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 K´arm´an–Tsien type (3.6). Here,
λ = −Cγ2c , µ = −Cγ2c , ν = −Cγ2c , (3.9)
3 4 5
in accordance with the relations (2.11). The associated class of NLS equations
(cid:20) (cid:21)
|Ψ| µ
iΨ +Ψ − (C −1) xx +λ|Ψ|2+ +ν Ψ = 0,
t xx |Ψ| |Ψ|4
hence, admits symmetry reduction via the wave packet ansatz (2.2) to the hybrid Ermakov–
Painlev´e II equation (cf. (2.10))
d2|Ψ| σ
+[c +c +c ξ]|Ψ|+c |Ψ|3 = , (3.10)
dξ2 1 5 2 3 |Ψ|3
where
(cid:34) (cid:35)
1 (cid:18)I(cid:19)2
σ = +µ . (3.11)
Cγ2 γ
Interestingly, this symmetry reduction to an integrable Ermakov–Painlev´e 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+(c +c )/c , with c (cid:54)= 0, (3.10) becomes
1 5 2 2
d2|Ψ| σ
+c ζ|Ψ|+c |Ψ|3 = , (3.12)
dζ2 2 3 |Ψ|3
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–Painlev´e II equation while in terms of the density ρ
it is seen that (3.12) produces
d2ρ 1 (cid:18)dρ(cid:19)2 2σ
= −2c ρ2−2c ζρ+ , (3.13)
dζ2 2ρ dζ 3 2 ρ
which is equivalent to P (1.4) (through a rescaling of the variables). This link between the
XXXIV
Ermakov–Painlev´e II equation (3.12) and P (1.4) has been noted previously in the context
XXXIV
of a Painlev´e reduction of a classical Nernst–Planck electrodiffusion system in [3]. We remark
that the special case of equation (3.13) with c = 0 was considered by Gambier [38, pp. 27–28],
3
who linearised the equation. Multiplying (3.13) with c = 0 by ρ and differentiating gives
3
d3ρ dρ
= 4ζ +2ρ,
dζ3 dζ
which has solution
ρ(ζ) = C Ai2(z)+C Ai(z)Bi(z)+C Bi2(z), z = −c1/3ζ, (3.14)
1 2 3 2
with C , C and C constants. The solution ρ(ζ) given by (3.14) satisfies (3.13) only if c = 0,
1 2 3 3
σ = 0 and 4C C = C2.
1 2 3
In the sequel, it is convenient to proceed with
c = −1, c = −1, σ = −1 (cid:0)α+ 1(cid:1)2,
2 2 3 4 2
whence
(cid:34) (cid:35)
ε = 1Cγ2 > 0, λ = Cγ2 > 0, (cid:0)α+ 1(cid:1)2 = −4 (cid:18)I(cid:19)2+µ , (3.15)
2 2 λ γ
where the latter requires that µ < −CI2/λ < 0. The Ermakov–Painlev´e II equation (3.12) is
then linked to P (1.4) via the relation ρ = |Ψ|2 > 0.
XXXIV
The well-known connection, in turn, between P (1.3) and P (1.4) is readily derived
II XXXIV
via the Hamiltonian system
dq ∂H dp ∂H
II II
= , = − ,
dz ∂p dz ∂q
where the Hamiltonian H (p,q,z;α) is given by
II
H (p,q,z;α) = 1p2−(cid:0)q2+ 1z(cid:1)p−(cid:0)α+ 1(cid:1)q,
II 2 2 2
leading to the coupled pair of nonlinear equations
dq dp
= p−q2− 1z, = 2qp+α+ 1, (3.16)
dz 2 dz 2
(see [44, 66]). Elimination of p and q successively in (3.16) duly leads to the P (1.3) and
II
P (1.4). Thus, in the present capillarity context, the density distribution ρ(ζ) is given by
XXXIV
dw
ρ(ζ) = +w2+ 1ζ,
dζ 2
Ermakov–Painlev´e II Symmetry Reduction of a Korteweg Capillarity System 9
where w(ζ) is governed by the P equation
II
d2w
= 2w3+ζw+α.
dζ2
Here, the concern is necessarily restricted tosolutions of P (1.4) in regions in whichρis
XXXIV
positive. Interestingly, the importance of positive solutions of P (1.4) also arises naturally
XXXIV
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 P equation
II
d2Y
= 2Y3+zY +α,
dz2
and associated ion concentrations by
dY
p = ± +Y2+ 1z,
± dz 2
with parameter
1−A /A
− +
α = ,
2(1+A /A )
− +
and A = −Φ /D , Φ being the fluxes of the ion concentrations and D diffusivity constants
± ± ± ± ±
arisingintheEinsteinrelation. Thus,itisseenthattheionconcentrations,whicharenecessarily
positive, are governed by P (1.4). This positivity constraint was examined in detail in [7]
XXXIV
for exact solutions in terms of either Yablonskii-Vorob’ev polynomials or classical Airy functions
asinducedbytheiteratedactionoftheB¨acklundtransformationof[59]forP (1.3). Theresults
II
apply ‘mutatis mutandis’ in the present capillarity context.
4 Iterated action of a B¨acklund transformation
Here, the consequences of the following well-known B¨acklund transformation for P (1.3) are
II
applied in the present capillarity context.
Theorem 4.1. If q (z) = q(z;α) is a solution of P (1.3) with parameter α, then
α II
2α+1
q (z) = −q (z)− , (4.1a)
α+1 α 2q(cid:48) (z)+2q2(z)+z
α α
2α−1
q (z) = −q (z)− , (4.1b)
α−1 α 2q(cid:48) (z)−2q2(z)+z
α α
are solution of P with respective parameters α+1 and α−1.
II
Proof. See Gambier [38] and Lukashevich [59]. (cid:4)
The iteration of the B¨acklund transformations (4.1) allows the generation of all known exact
solutions of P (1.3).
II
We note that eliminating q(cid:48) (z) in (4.1) yields the nonlinear difference equation
α
α+ 1 α− 1
2 + 2 +2q2 +z = 0,
q +q q +q α
α+1 α α α−1
which is known as an alternative form of discrete Painlev´e I [33].
10 C. Rogers and P.A. Clarkson
Theorem 4.2. If q = q(z;α) and p = p(z;α) are solutions of P (1.3) and P (1.4)
α α II XXXIV
with parameter α respectively, then
2α+1
q = −q − ,
α+1 α
2p
α
2α−1
q = −q + ,
α−1 α 2p −4q2 +2z
α α
(cid:18) 2α+1(cid:19)2
p = −p + q + +z,
α+1 α α
2p
α
p = −p +2q2 +z.
α−1 α α
Proof. See Okamoto [66]; also [35]. (cid:4)
4.1 Rational solutions
The iterative action of the above B¨acklund transformation on the seed solution q = 0 of P with
II
α = 0 produces the subsequent sequence of rational solutions
d Q (z)
q (z) = ln n−1 , n ∈ N, (4.2)
n
dz Q (z)
n
correspondingtothePainlev´eparametersα = n, forn ∈ N, wheretheQ (z)aretheYablonskii–
n
Vorob’ev polynomials determined by the quadratic recurrence relations
(cid:40) (cid:41)
(cid:18)dQ (cid:19)2 d2Q
Q Q = zQ2 +4 n −Q n , (4.3)
n+1 n−1 n dz n dz2
with Q (z) = Q (z) = 1 [89, 92]; see also [18, 19, 22, 48, 50, 87]. The Q (z) are monic
−1 0 n
polynomialsofdegree 1n(n+1)witheachtermpossessingthesamedegreemodulo3. Moreover,
2
on use of the invariance under
q(z,α) → −q(z;−α),
it is seen that P (1.3) also admits the associated class of rational solutions
II
d Q (z)
q (z) = ln n , n ∈ N, (4.4)
−n
dz Q (z)
n−1
corresponding to the Painlev´e parameters α = −n, for n ∈ N. The rational solutions of P
XXXIV
(1.4) are given by
d2 Q (z)Q (z)
p (z) = 1z−2 lnQ (z) ≡ n+1 n−1 , n ∈ N,
n 2 dz2 n 2Q2(z)
n
corresponding to the parameters α = n, with n ∈ N.
It is clear from the recurrence relation (4.3) that the Q (z) are rational functions, though
n
it is not obvious that they are polynomials since one is dividing by Q (z) at every iteration.
n−1
In fact it is somewhat remarkable that the Q (z) are polynomials. Taneda [87], used an alge-
n
braic method to prove that the functions Q (z) defined by (4.3) are indeed polynomials, see
n
also [36]. The Yablonskii–Vorob’ev polynomials Q (z) can also be expressed as determinants,
n
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