- Email: [email protected]
On modulated coupled systems. Canonical reduction via reciprocal transformations
On Modulated Coupled Systems. Canonical Reduction via Reciprocal Transformations
Journal Pre-proof
On Modulated Coupled Systems. Canonical Reduction via Reciprocal Transformations Colin Rogers, Wolfgang K Schief, Boris Malomed PII: DOI: Reference:
S1007-5704(19)30410-1 https://doi.org/10.1016/j.cnsns.2019.105091 CNSNS 105091
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
18 April 2019 23 October 2019 29 October 2019
Please cite this article as: Colin Rogers, Wolfgang K Schief, Boris Malomed, On Modulated Coupled Systems. Canonical Reduction via Reciprocal Transformations, Communications in Nonlinear Science and Numerical Simulation (2019), doi: https://doi.org/10.1016/j.cnsns.2019.105091
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Highlights • Privileged (integrable) systems admit reciprocally related modulated counterparts • Modulation may be driven by Painlev and Ermakov equations • Periodically modulated Manakov solitons may be obtained
1
On Modulated Coupled Systems. Canonical Reduction via Reciprocal Transformations Colin Rogersa , Wolfgang K Schiefa,∗, Boris Malomedb , b
a The University of New South Wales, Sydney, NSW2052, Australia The Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, Israel
Abstract It is shown how classes of modulated coupled systems of sine-Gordon, Demoulin and Manakov-type may be reduced to their unmodulated counterparts via the application of a novel reciprocal transformation. Modulations governed by recently introduced integrable Ermakov-Painlev´e II, ErmakovPainlev´e III and Ermakov-Painlev´e IV equations as well as the classical Ermakov equation are considered. In the latter case, the procedure is illustrated by the generation of a class of exact solutions to a heterogeneous Manakov system with periodic modulation. Keywords: reciprocal transformation, heterogeneous integrable system, Manakov system, modulated soliton 1. Introduction Physical systems which incorporate spatial or temporal modulation arise, notably, in both nonlinear optics and the theory of Bose-Einstein condensates [1, 2, 3, 4, 5, 6, 7]. Likewise, in classical continuum mechanics, modulated physical systems are of importance, inter alia in elastodynamics, viscoelastodynamics and in the elastostatics of inhomogeneous media (see e.g. [8, 9, 10, 11] and literature cited therein). In recent work [12], a novel reciprocal transformation was introduced which reduces non-autonomous, multi-component Ermakov-Painlev´e systems to their integrable autonomous counterparts. Here, such a reciprocal transformation is applied, in turn, to coupled systems of sine-Gordon, Demoulin and Manakov-type to reduce ∗
Corresponding author.
Preprint submitted to Commun Nonlinear Sci and Numer Simulat
October 31, 2019
them to associated unmodulated canonical forms. It is remarked that kink propagation in a particular modulated coupled sine-Gordon system has been previously investigated in [13]. Modulated single-component sine-Gordon equations have been the subject of prior work in [14, 15]. Integrability aspects of the classical two-component Demoulin system of affine geometry, such as its admittance of an auto-B¨acklund transformation are described in [16]. A modulated version of Tzitz´eica reduction of this Demoulin system was derived in a geometric context in [17]. The present work is concerned with the derivation of novel modulated coupled systems which, under a class of reciprocal transformations, retain the solitonic integrable structure of their unmodulated counterparts. In terms of physical applications, the work is motivated, in part, by the spatially modulated version of the celebrated coupled Manakov system of [18] which has been investigated in [19] in the context of rogue wave propagation. Time modulated Manakov systems have recently arisen in [20] in connection with the propagation of bright solitons in Bose-Einstein condensates. Here, coupled systems with modulation governed by integrable Ermakov-Painlev´e II, Ermakov-Painlev´e III, Ermakov-Painlev´e IV or the classical Ermakov equation are considered in turn. By way of illustration of the application of the reciprocal transformation, a novel class of exact solutions is generated for modulated Manakov systems via their integrable canonical Manakov counterparts. Privileged periodic modulated Manakov systems are recorded. In particular, a spatially modulated version of the bright soliton solution of the Manakov system is presented with the modulation being driven by the classical Ermakov equation. It is noted that reciprocal-type transformations have been previously shown, in particular, to make analytically tractable initial/boundary value problems which arise in a diversity of physical contexts. Thus, in [21] a reciprocal and Bcklund transformation were combined to solve a class of initial/boundary value problems descriptive of two-phase flow under gravity and with boundary infiltration. In [22, 23, 24], reciprocal transformations have been applied to solve nonlinear moving boundary problems of Stefantype which arise in the analysis of the melting of metals. In [25], a nonlinear model for the evolution of methacrylate in wood treatment was reduced via a reciprocal transformation to a moving boundary problem amenable to an integrable representation procedure as set down in [26]. In [27] a reciprocal transformation was used to obtained the exact solution of a nonlinear boundary value problem which arises in connection with unsaturated flow through 3
a heterogeneous porous medium. Moving boundary problems incorporating inhomogeneity occur naturally in soil mechanics. Thus, they arise notably in the analysis of the transport of liquid through soils as modelled in the homogeneous case by the classical work of Richards [28]. In most recent work [29], two distinct kinds of reciprocal transformation have been employed in conjunction to reduce a wide class of nonlinear moving boundary problems with heterogeneity to analytically solvable canonical forms. These are associated, in turn, with a classical Stefan problem and one relevant to the evolution of seapage fronts in soil mechanics. In [30], the combined action of reciprocal and integral-type transformations has been used to sequentially reduce to analytically tractable form another class of nonlinear moving boundary problems involving heterogeneity. Particular such Stefan problems arise in the description of the transport of liquids through layered porous media. The present work indicates, in particular, the potential application of reciprocal transformations to moving boundary problems for coupled systems incorporating heterogeneity. It is remarked that pointwise bounds on the solution of certain moving boundary problems for coupled solitonic systems embodied in resonant nonlinear Schr¨odinger equations have been obtained in [31]. To conclude, it is observed that the present reductions of modulated systems via reciprocal transformations may likewise be applied mutatis mutandis to corresponding temporally modulated Manakov systems. Moreover, extension may be readily made to integrable reduction of certain multi-component systems such as modulated Toda-systems. 2. A modulated coupled sine-Gordon system: canonical reduction via a reciprocal transformation Here, a class of heterogeneous coupled sine-Gordon systems is considered of the type u−v cI p , ρ utt − uxx + Φ(ρ, ρx , . . . ; x)u = q sin ρ ρ (1) cII u−v p ρ vtt − vxx + Φ(ρ, ρx , . . . ; x)v = q sin , ρ ρ
4
with modulation term ρ(x) determined by an auxiliary generalised Ermakov equation δ −ρxx + Φ(ρ, ρx , . . . ; x)ρ = 3 (2) ρ which is required to be analytically tractable. Combination of (2) with (1)1 and (2)2 in turn shows that δu cI u−v p+1 ρ utt − ρuxx + ρxx u + 3 = q−1 sin , ρ ρ ρ cII u−v δv p+1 , ρ vtt − ρvxx + ρxx v + 3 = q−1 sin ρ ρ ρ
(3)
that is, ∂ u δu cI u−v 2 ∂ ρ utt − ρ + 3 = q−1 sin , ∂x ∂x ρ ρ ρ ρ u−v v cII ∂ δv p+1 2 ∂ ρ vtt − ρ + 3 = q−1 sin . ∂x ∂x ρ ρ ρ ρ p+1
Thus, on introducing a reciprocal transformation R∗ according to ) u∗ = u/ρ, v ∗ = v/ρ R∗ , ∗ −2 ∗ dx = ρ dx, t = t
(4)
(5)
it is seen that ρp+4 u∗t∗ t∗ − u∗x∗ x∗ + δu∗ = cI ρ3−q sin(u∗ − v ∗ ),
ρp+4 vt∗∗ t∗ − vx∗∗ x∗ + δv ∗ = cII ρ3−q sin(u∗ − v ∗ ).
(6)
Accordingly, with p = −4, q = 3, the following result is obtained: Theorem 2.1. The heterogeneous coupled sine-Gordon system cI u−v −4 ρ utt − uxx + Φ(ρ, ρx , . . . ; x)u = 3 sin , ρ ρ cII u−v −4 ρ vtt − vxx + Φ(ρ, ρx , . . . ; x)v = 3 sin , ρ ρ 5
(7)
with modulation ρ(x) determined by the generalised Ermakov equation −ρxx + Φ(ρ, ρx , . . . ; x)ρ = under the reciprocal transformation dx∗ = ρ−2 dx, ∗
u = u/ρ,
t∗ = t ∗
v = v/ρ
)
δ ρ3
(8)
R∗
(9)
is reduced to the un-modulated coupled sine-Gordon system u∗t∗ t∗ − u∗x∗ x∗ + δu∗ = cI sin(u∗ − v ∗ ),
vt∗∗ t∗ − vx∗∗ x∗ + δv ∗ = cII sin(u∗ − v ∗ ).
(10)
It is observed that a system similar to (10) was derived, as a model of DNA, in [32]. In an analogous manner, the modulated coupled system 1 1 u u v −4 ρ uxx − 2 utt = 3 cI sin + cIII sin cos , c0 ρ ρ 2ρ 2ρ (11) 1 1 v v u ρ−4 vxx − 2 vtt = 3 cII sin + cIII sin cos c0 ρ ρ 2ρ 2ρ
under R∗ reduces to the variant sine-Gordon system u∗x∗ x∗ − vx∗∗ x∗
1 ∗ u∗ v∗ ∗ u = c sin u + c sin cos , ∗ ∗ I III c20 t t 2 2
1 v∗ u∗ − 2 vt∗∗ t∗ = cII sin v ∗ + cIII sin cos , c0 2 2
(12)
as introduced in [33] in connection with kink propagation in DNA double helices (see, also, [34]). Coupled sine-Gordon systems of the type (10) with δ = 0 arise as continuum two-component Kontorova-Frenkel dislocation models (see e.g. [35] and literature cited therein). It is noted that, if the relation ρ∗ = ρ−1
(13)
is adjoined to R∗ , then R∗2 = id. Moreover, under R∗ , (2) becomes −ρ∗x∗ x∗ + δρ∗ = where Φ∗ := Φ|R∗ . 6
Φ∗ , ρ∗3
(14)
3. Modulated Demoulin systems The reciprocal transformation R∗ as given by (9) may likewise be applied to modulated Demoulin systems 1 u/ρ e − e−(u+v)/ρ , 3 ρ 1 ρ−4 vtt − vxx + Φ(ρ, ρx , . . . ; x)v = 3 ev/ρ − e−(u+v)/ρ ρ
ρ−4 utt − uxx + Φ(ρ, ρx , . . . ; x)u =
(15)
to obtain reduction to the corresponding un-modulated system ∗
∗ +v ∗ )
v∗
−(u∗ +v ∗ )
u∗t∗ t∗ − u∗x∗ x∗ + δu∗ = eu − e−(u vt∗∗ t∗ − vx∗∗ x∗ + δv ∗ = e − e
,
.
(16)
In the case δ = 0, the latter constitutes the Demoulin system descriptive of an important subclass of surfaces in classical projective geometry. This system is integrable in the solitonic sense, and accordingly, in particular, admits invariance under a B¨acklund transformation (see e.g. [16] and cited literature). This invariance, conjugated with R∗ generates a novel invariance of the class of modulated Demoulin systems (15). In the single-component case u = v, (15) constitutes a modulated version of the classical Tzitzeica equation. It is remarked that a particular modulated version of the latter has been set down in [17]. 4. Ermakov-Painlev´ e modulation Here, modulations are considered wherein the auxiliary equation (2) in ρ constitutes a hybrid integrable Ermakov-Painlev´e II, Ermakov-Painlev´e III or Ermakov-Painlev´e IV equation [36, 37, 38, 39, 40, 41, 42]. These adopt the forms discussed in the following. 4.1. Ermakov-Painlev´e II This is the nonlinear equation ρxx − ρ3 +
(α + 12 )2 xρ =− 2 4ρ3
7
(17)
as derived in [36] via a symmetry reduction of an n+1-dimensional ‘resonant’ Manakov-type system incorporating a de Broglie-Bohm potential ∇2 |Ψ|/|Ψ|, namely ∂Ψ + ∇2 Ψ + ν(|Ψ|2 + |Ω|2 )Ψ = s(∇2 |Ψ|/|Ψ|)Ψ, i ∂t i
∂Ω + ∇2 Ω + ν(|Ψ|2 + |Ω|2 )Ω = s(∇2 |Ω|/|Ω|)Ω, ∂t
(18)
(∇2 := ∂ 2 /∂x21 + · · · + ∂ 2 /∂x2n ) .
It is recalled that certain single component resonant NLS equations involving a de Broglie-Bohm potential have been shown to admit solitonic fission or fusson phenomena if s > 1 [43, 44]. The hybrid Ermakov-Painlev´e II equation (17) has also been subsequently derived in [45] as a symmetry reduction of the cold plasma system of [46] and of the classical Korteweg capillarity system in [47]. Importantly, the hybrid Ermakov-Painlev´e II equation (17) may be obtained by setting Σ = ρ2 (19) in the integrable P34 equation Σxx
2 1 1 Σ2x 2 − xΣ + 2Σ − α + , = 2Σ 2 2Σ
(20)
which, in turn, is linked to the classical PII equation Yxx = 2Y 3 + xY + α
(21)
via the Hamiltonian system Yx = −Y 2 −
x +Σ , 2
1 Σx = 2Y Σ + α + . 2
(22)
In order to implement the reciprocal transformation R∗ it is required to evaluate Z Z 1 1 ∗ x = dx = dx (23) 2 ρ (x) Σ(x)
for positive solutions Σ of P34 . A B¨acklund transformation for PII may be adduced to generate iteratively an infinite sequence of solutions Σ for which the integral (23) can be evaluated (see [38]). Thus, let Yα be a solution of 8
PII corresponding to the parameter α and Σα an associated solution of P34 . Then, another pair of solutions {Yα+1 , Σα+1 } is given by Yα+1 = −Yα −
α + 12 , Σα
and, on setting x∗α
=
2 Σα+1 = −Σα + 2Yα+1 +x
Z
1 dx Σα (x)
it may be shown that [38] . 1 3 ∗ ∗ xα+1 = α+ xα + ln (Σα+1 Σα ) α+ . 2 2
(24)
(25)
(26)
The importance of positive solutions of P34 arises naturally in a twoion electro-diffusion context wherein the scaled electric field is governed by PII and the associated ion concentrations by P34 . The physical requirement that the latter be positive has been examined in detail in [48] both for the rational solutions in terms of Yablonskii-Vorob’ev polynomials and for exact solutions involving the classical Airy function, in each case as generated by the iterative action of a B¨acklund transformation. 4.2. Ermakov-Painlev´e III This adopts the form 2 γρ4 δ ρx ρx 1 4 + 2 (αρ + β) + ρ= 3 ρxx − 2 − ρ ρx 2ρ x 2 2ρ
(27)
and is obtained by setting w = ρ2
(28)
in the classical PIII equation wxx =
wx2 wx 1 δ − + (αw2 + β) + γω 3 + . w x x w
(29)
The latter canonical nonlinear equation may be obtained, in particular, as a symmetry reduction of an integrable Ernst system as derived in [49] via the 2+1-dimensional solitonic system of [50, 51]. It was subsequently established in [52] that, remarkably, this generalised Ernst system incorporates a 9
Ermakov system reduction. It was this observation that originally motivated the study of hybrid integrable systems. Here, in view of the relation (28), in order to apply R∗ it is required to evaluate Z 1 dx (30) w(x)
where w is a positive solution of the PIII equation (29). In this connection, the latter admits rational solutions generated iteratively by a B¨acklund transformation as described in [53] whence the integral (30) may, in principle, be evaluated and regions of positivity of w(x) delimited analytically.
4.3. Ermakov-Painlev´e IV This nonlinear equation adopts the form [41] β 3 4 2 2 ρxx − ρ + 2xρ + x − α ρ = 3 4 2ρ
(31)
and may be obtained by setting σ = ρ2
(32)
in the canonical PIV equation σxx =
1 2 3 3 β σx + σ + 4xσ 2 + 2(x2 − α)σ + . 2σ 2 σ
(33)
The Ermakov-Painlev´e IV equation (31) was derived via symmetry reduction of an n-component, resonant, derivative NLS system in [39]. Here, the implementation of the reciprocal transformation R∗ requires evaluation of the integral Z 1 ∗ dx (34) x = σ(x)
for positive solutions σ(x) of the PIV equation. In this connection, it was recently recorded in [41] that a B¨acklund transformation for PIV , namely Σn+1 (x) =
[Σn,x (x) − Σ2n (x) − 2xΣn (x)]2 2Σn (x)[Σn,x (x) − Σ2n (x) − 2xΣn (x) + 4(n + 1)]
(35)
applied iteratively to the seed bound state solution ξ exp(−x2 ) , Σn (x) = √ π 1 − 21 ξ erfc(x) 10
0 < ξ < 1,
(36)
corresponding to parameters α = 1 and β = 0 generates an infinite sequence of bound state solutions corresponding to the parameters α = 2n + 1 and β = 0. The bound state solution (36) is positive for all x while the subsequent bound state solutions as generated by the iterated action of the B¨acklund transformation (35) are positive on regions of the x-axis separated by zeros. This phenomenon is illustrated graphically in [41]. Moreover, the integral (34) can be evaluated iteratively for the bound state solutions Σn (x) of PIV by application of the relation Z Z dx 2 dx = + . (37) 2 Σn+1 (x) Σn (x) Σn,x (x) − Σn (x) − 2xΣn (x) 5. An integrable modulated Manakov systems. A symmetry reduction Under the reciprocal transformation R∗ given by (9) and modulation now determined by the generalised Ermakov equation ρxx + Φ(ρ, ρx , . . . ; x)ρ =
δ , ρ3
(38)
it is readily seen that the class of heterogeneous Manakov systems i
∂u + ρ4 uxx + ρ−2 (|u|2 + |v|2 )u + ρ4 Φ(ρ, ρx , . . . ; x)u = 0, ∂t
∂v i + ρ4 vxx + ρ−2 (|u|2 + |v|2 )v + ρ4 Φ(ρ, ρx , . . . ; x)v = 0 ∂t
(39)
is reduced to the canonical integrable Manakov system i
∂u∗ + u∗x∗ x∗ + (|u∗ |2 + |v ∗ |2 )u∗ + δu∗ = 0, ∂t∗
∂v ∗ i ∗ + vx∗∗ x∗ + (|u∗ |2 + |v ∗ |2 )v ∗ + δv ∗ = 0. ∂t
(40)
It is interesting to note that in [19], in a rogue wave propagation context, a Manakov-type system with certain modulated dispersion and nonlinearity terms together with an external potential was set down which admits reduction to the canonical integrable two-component Manakov system. The latter 11
has an extensive literature both in nonlinear optics and the theory of BoseEinstein condensates. Moreover, it was recorded in [54] (see, also, [55]) that the Manakov system (40) remains integrable if one introduces an additional linear coupling, namely, i
∂u∗ + u∗x∗ x∗ + (|u∗ |2 + |v ∗ |2 )u∗ + cv ∗ + δu∗ = 0, ∂t∗
(41) ∂v ∗ ∗ ∗ 2 ∗ 2 ∗ ∗ ∗ i ∗ + vx∗ x∗ + (|u | + |v | )v + cu + δv = 0, ∂t where c is a constant. This corresponds to a heterogeneous Manakov-type system with additional coupling terms cv and cu in (39)1 and (39)2 respectively. In the case when the auxiliary equation (38) is taken as the classical Ermakov equation (see e.g. [12] and [56, 57]) ρxx + ω(x)ρ =
δ , ρ3
(42)
the integration of the reciprocal relation in (5) to determine x∗ may be achieved by use of the well-known nonlinear superposition principle for (42). This is readily derived by Lie group methods as in [58, 59] and states that p ρ = c1 α2 (x) + 2c2 α(x)β(x) + c3 β 2 (x) (43)
where α(x), β(x) are two linearly independent solutions of (42)|δ=0 with corresponding constant Wronskian W = αβx − βαx and with the constants ci such that c1 c3 − c22 = δ/W 2 . (44) Thus,
∗
x =
Z
Z
dx c1 α2 (x) + 2c2 α(x)β(x) + c3 β 2 (x) Z β(x)/α(x) 1 dζ = = S(β/α). W c1 + 2c2 ζ + c3 ζ 2
dx = ρ2 (x)
(45)
5.1. A symmetry reduction. Application of the Ermakov invariant Here, a symmetry reduction of the canonical Manakov system (40) is sought with u∗ = q1∗ (x∗ ) exp[−iσt∗ + ir1∗ (x∗ )],
v ∗ = q2∗ (x∗ ) exp[−iσt∗ + ir2∗ (x∗ )] 12
(46)
whence, on substitution, it is seen that ∗ ∗2 ∗2 ∗ ∗ ∗ ∗2 q1x ∗ x∗ + (q1 + q2 )q1 + (δ + σ)q1 − q1 r1x∗ = 0,
∗ ∗2 ∗ ∗2 ∗ ∗2 ∗ q2x ∗ x∗ + (q1 + q2 )q2 + (δ + σ)q2 − q2 r2x∗ = 0,
∗ ∗ ∗ ∗ r1x ∗ x∗ /r1x∗ = −2q1x∗ /q1 ,
(47)
∗ ∗ ∗ ∗ r2x ∗ x∗ /r2x∗ = −2q2x∗ /q2 .
The latter pair of relations shows that ∗ ∗2 r1x ∗ = λ/q1 ,
∗ ∗2 r2x ∗ = µ/q2
(48)
whence, the nonlinear coupled system ∗ ∗2 ∗2 ∗ ∗ q1x ∗ x∗ + (q1 + q2 )q1 + (δ + σ)q1 = ∗ q2x ∗ x∗
+
(q1∗2
+
q2∗2 )q2∗
+ (δ +
σ)q2∗
λ2 , q1∗3
µ2 = ∗3 q2
(49)
results. It is remarked that a similarity ansatz of the type (46) has previously been applied to a Manakov system in [60] and [61]. In the latter paper, it was noted that the nonlinear coupled system derived is descriptive of the motion of particles interacting with a quartic potential and an inverse square potential. Moreover, for a particular choice of parameters, a Lax representation was obtained. Here, it is shown that the system (49) is amenable to exact solution via application of its Hamiltonian together with its admitted Ermakov invariant. The latter adopts the form " ∗ 2 ∗ 2 # 1 q q 2 2 ∗ ∗ ∗ 2 (q1∗ q2x + µ2 1∗ , (50) E= ∗ − q2 q1x∗ ) + λ ∗ 2 q1 q2 while the Hamiltonian is given by (q1∗2 + q2∗2 )2 1 ∗2 λ2 µ2 ∗2 ∗2 ∗2 H= q ∗ + q2x∗ + + (δ + σ)(q1 + q2 ) + ∗2 + ∗2 . (51) 2 1x 2 q1 q2 On application of the identity 2 2 ∗ ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗2 (q1∗2 + q2∗2 )(q1x ∗ + q2x∗ ) − (q1 q2x∗ − q2 q1x∗ ) ≡ (q1 q1x∗ + q2 q2x∗ )
13
(52)
and use of the pair of invariants E and H, it is seen that Σ∗2 λ2 µ2 ∗ ∗ − (δ + σ)Σ − ∗2 − ∗2 Σ 2H − 2 q1 q2 " ∗ 2 ∗ 2 # q q 1 = Σ∗2 − 2E − λ2 2∗ − µ2 1∗ ∗, q1 q2 4 x
(53)
where Σ∗ = q1∗2 + q2∗2 .
(54)
On reduction, (53) yields 1 Σ∗2 ∗ ∗ − (δ + σ)Σ − 2E + λ2 + µ2 = Σ∗2 Σ 2H − ∗ 2 4 x
(55)
which serves to determine Σ∗ in terms of elliptic functions. Moreover, the Ermakov invariant relation (50) shows that ∗ 2 ∗ 2 ∗ 2 q1 q2 q1 −1 2 2 ∗2 ∗2 d tan = 2E − λ −µ , (56) (q1 + q2 ) ∗ ∗ dx q2 q1 q2∗ whence, on introduction of the new independent variable x according to dx = Σ∗−1 dx,
(57)
it follows that q1∗ /q2∗ is determined via the tractable integral Z dΛ p =x 2 2E − λ cot2 Λ − µ2 tan2 Λ where
−1
Λ = tan
q1∗ q2∗
.
(58)
(59)
With Σ∗ and Λ determined in turn by (55) and (58), q1∗ and q2∗ in the symmetry reduction are now given by q1∗ = ±Σ∗1/2 sin Λ,
q2∗ = ±Σ∗1/2 cos Λ
(60)
and r1∗ , r2∗ by integration of the relations ∗ 2 ∗ r1x ∗ = λ/(Σ sin Λ),
∗ ∗ 2 r2x ∗ = µ/(Σ cos Λ).
14
(61)
The reciprocal transformation R∗ now delivers the associated class of solutions of the heterogeneous Manakov system (39) with u = ρu∗ (x∗ ), v = ρv ∗ (x∗ ), Z ∗ x = ρ−2 dx,
(62)
where, if the auxiliary equation (42) is the classical Ermakov equation (42) then x∗ is given by (45). In particular, if ω = const = 2 in (42) then, with α = cos x, β = sin x, p ρ = c1 cos2 x + 2c2 sin x cos x + c3 sin2 x, (63) c1 c3 − c22 = δ/2 . Thus, in this case, an integrable periodically-modulated Manakov system (39) results. 5.2. A spatially Ermakov-modulated Manakov soliton We now illustrate the periodic spatial modulation (62), (63) governed by the classical Ermakov equation by applying it to the standard bright onesoliton solution of the Manakov system (40) [18]. Thus, it may be verified directly that i(k3 x∗ +(k2 −k2 +δ)t) ∗ 2 3 k1 eik4 e u , 2k22 = k12 + kˆ12 , (64) = ˆ ikˆ4 ∗ ∗ v k1 e cosh[k2 (x − 2k3 t)] wherein the remaining constants are arbitrary but real, constitutes a solution of the Manakov system (40). Accordingly, the canonical quantity Σ∗ = |u∗ |2 + |v ∗ |2 =
2k22 cosh2 [k2 (x∗ − 2k3 t)]
(65)
exhibits the typical sech2 -profile of a soliton. The propagation of this soliton is depicted in Figure 1 for k2 = 1 and k3 = 1/2. The modulated analogue of the soliton solution (64) which satisfies the periodically-modulated Manakov system (39), (42), (63) is now generated via the relations (62). In particular, one obtains the modulated bright soliton Σ = |u|2 + |v|2 =
2k22 [ρ(x)]2 . cosh2 [k2 (x∗ (x) − 2k3 t)] 15
(66)
Figure 1: The bright one-soliton solution (65) of the Manakov system (40) for k2 = 1 and k3 = 1/2.
*
Figure 2: The relation x∗ = x∗ (x) for c1 = 1/3, c2 = 0, c3 = 3. Here, = 1 (red) and = 6 (blue) respectively.
Integration of the relation (62)3 yields, up to an additive constant, ! c tan x + c x 1 1 π 3 2 ∗ arctan p + , (67) x = p + p 2 c1 c3 − c22 c1 c3 − c22 c1 c3 − c22 π
where b · c denotes the standard floor function. In the above, we have assumed that c3 > 0. In the case, c1 = 1/3, c2 = 0 and c3 = 3, the dependence of x∗ on x is depicted in Figure 2 for = 1 and = 6. It is evident that the modulated soliton Σ∗ travels “on average” at the same speed as the Manakov soliton Σ. However, the shape of the soliton Σ∗ is modulated in two different ways. On the one hand, its amplitude is modulated due to the factor ρ2 . 16
On the other hand, it is deformed “horizontally” due to the dependence of sech2 on x∗ (x) − 2k3 t which slightly deviates form x − 2k3 t. The larger , the less pronounced this distortion becomes. Figure 3 depicts the modulated Manakov soliton Σ∗ for the choice of parameters adopted in the preceding and = 1, 2, 3, 4, 5, 6. 6. Conclusion Here, a novel kind of reciprocal transformation has been used to reduce certain spatially-modulated two-component systems of partial differential equations to their unmodulated counterparts. The reciprocal reduction method has its genesis in an autonomisation procedure as originally presented in [62] for the physically important coupled Ermakov-Ray-Reid system (see, e.g., [12]). The method may be applied, mutatis mutandis, to the temporallymodulated analogues of these systems. In addition, such reduction may be readily extended to multi-component systems of physical interest. Thus, in particular under the reciprocal transformation ψi∗ = ψi /ρ dx∗ = ρ−2 dx,
(68)
t∗ = t
with −ρxx + Φ(ρ, ρx , . . . ; x)ρ =
δ ρ3
(69)
it is seen that the modulated Toda lattice system ρ−4 ψn,tt − ψn,xx + Φ(ρ, ρx , . . . ; x)ψn =
1 [−eψn+1 /ρ + 2eψn /ρ − eψn−1 /ρ ] (70) ρ3
is reduced to the corresponding unmodulated system ∗
∗
∗
∗ ψn+1 ∗ ∗ + 2eψn − eψn−1 . ψn,t ∗ t∗ − ψn,x∗ x∗ + δψn = −e
(71)
Thus, if δ = 0 the standard integrable Toda lattice system results [63]. Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 17
Figure 3: The modulated Manakov soliton (66), (67) for = 1, 2, 3, 4, 5, 6.
18
References [1] J. Belmonte-Beita, V.M. P´erez-Garcia and V. Vekslechnik, Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities, Physical Review Letters 98, 064102-1–064102-4 (2007). [2] J. Belmonte-Beita, V.M. P´erez-Garcia, V. Vekslechik and V.V. Konotop, Localised nonlinear waves in systems with time and space-modulated nonlinearities, Physical Review Letters 100, 064102-1–064102–4 (2008). [3] X.Y. Tang and P.K. Shukla, Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schr¨odinger equation with external potential, Phys. Rev. A 76, 01312 (2007). [4] J.F. Zhang, Yi S. Li, J. Meng, L. Wo and B.A. Malomed, Matter-wave solitons and finite-amplitude Bloch waves in optical lattices with a spatially modulated nonlinearity, Phys. Rev. A 82, 033614 (2010). [5] W.P. Zhong, M.R. Belic, B.A. Malomed and T. Huang, Solitary waves in the nonlinear Schr¨odinger equation with Hermite-Gaussian modulation of the local linearity, Phys. Rev. E 84, 046611 (2011). [6] W.P. Zhong, M.R. Belic and T. Huang, Solitary waves in the nonlinear Schr¨odinger equation with spatially modulated Bessel nonlinearity, J. Opt. Soc. An. B30, 1276–1283 (2013). [7] C. Rogers, G. Saccomandi and L. Vergori, Ermakov-modulated nonlinear Schr¨odinger models. Integrable reduction, J. Nonlinear Mathematical Physics 23, 108–126 (2016). [8] F.C. Karal and J.B. Keller, Elastic wave propagation in homogeneous and inhomogeneous media, J. Acoust. Soc. America 31, 694–705 (1959). [9] D.W. Barclay, T.B. Moodie and C. Rogers, Cylindrical impact waves in inhomogeneous Maxwellian visco-elastic media, Acta Mechanica 29, 93–117 (1978). [10] D.L. Clements, C. Atkinson and C. Rogers, Antiplane crack problems for an inhomogeneous elastic material, Acta Mechanica 29, 199–211 (1978).
19
[11] D.L. Clements and C. Rogers, On the Bergman operator method and anti-plane contact problems involving an inhomogeneous half-space, SIAM J. Appl. Math. 34, 764–773 (1978). [12] C. Rogers and W.K. Schief, Ermakov-type systems in nonlinear physics and continuum mechanics, in Nonlinear Systems and Their Remarkable Mathematical Structures (Ed. N. Euler) CRC Press (2018) 541–576. [13] B.A. Malomed, Motion of a kink in a spatially modulated sine-Gordon system Phys. Lett. A 144, 351–356 (1990). [14] B.A. Malomed and M.I. Tribelsky, Emission of waves by moving kinks in a spatially modulated sine-Gordon system, Physical Review B41, 11271– 11281 (1990). [15] A. S´anchez, A.R. Bishop and F. Dominguez-Adame, Kink stability, propagation and length scale competition in the periodically modulated sine-Gordon equation, Phys. Rev. E 49, 4603–4615 (1994). [16] E.V. Ferapontov and W.K. Schief, Surfaces of Demoulin: differential geometry, B¨acklund transformation and integrability, J. Geom. Phys. 30, 343–363 (1999). [17] M. Dunajski and P. Plansangkate, Strominger-Yau-Zaslow geometry, affine spheres and Painlev´e III, Comm. Math. Phys. 290, 997–1024 (2009). [18] S.V. Manakov, On the theory of two-dimensional stationary focussing of electromagnetic waves, Sov. Phys. JETP 38, 248–253 (1974). [19] W.P. Zhong, M.R. Belic and B.A. Malomed, Rogue waves in a twocomponent Manakov system with variable coefficients and an external potential, Phys. Rev. E 92, 053201 (2015). [20] T. Kanna, R.B. Mareesaran and F. Mertens, Non-autonomous bright solitons and their stability in Rabi coupled binary Bose-Einstein condensates, J. Phys. Commun 1, 045005 (2017). [21] C. Rogers, M.P. Stallybrass and D.L. Clements, On two phase filtration under gravity and with boundary infiltration: application of a B¨acklund transformation, J. Nonlinear Analysis, Theory, Methods and Applications, 7, 785–799 (1983). 20
[22] C. Rogers, Application of a reciprocal transformation to a two phase Stefan problem, J. Phys. A: Math. Gen. 18, L105–L109 (1985). [23] C. Rogers, On a class of moving boundary problems in nonlinear heat conduction: application of a B¨acklund transformation, Int. J. Nonlinear Mechanics 21, 249–256 (1986). [24] C. Rogers and B. Yu Guo, A note on the onset of melting in a class of simple metals. Condition on the applied boundary flux, Acta Mathematica Scientia 8, 425–430 (1988). [25] A.S. Fokas, C. Rogers and W.K. Schief, Evolution of methacrylate distribution during wood saturation. A nonlinear moving boundary problem, Appl. Math. Lett. 18, 321–328 (2005). [26] A.S. Fokas, A unifed transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London A 453, 1411–1443 (1997). [27] C. Rogers and P. Broadbridge, On a nonlinear moving boundary problem with heterogeneity: application of a B¨acklund transformation, Zeit. angew. Math. Phys. 39, 122–128 (1988). [28] L.A. Richards, Capillary conduction of liquids through porous mediums, J. Applied Physics 1, 318–333 (1931). [29] C. Rogers, On Stefan-type moving boundary problems with heterogeneity: canonical reduction via conjugation of reciprocal transformations, Acta Mechanica 230, 839–850 (2019). [30] C. Rogers, Moving boundary problems for heterogeneous media. Integrability via conjugation of reciprocal and integral transformations, J. Nonlinear Math. Phys., 26, 313–325 (2019). [31] J. Flavin and C. Rogers, Upper estimates for a moving boundary problem for a resonant nonlinear Schr¨odinger equation, Stud. Appl. Math. 121, 189–198 (2008). [32] S. Homma and S. Takeno, Topological solitons and modulated structure of bases in DNA double helices: A dynamic plane base-rotator model, Prog. Theor. Phys. 70, 308–311 (1983); A coupled base-rotator model for structure and dynamics of DNA: Local fluctuations in helical twist angles and topological solitons, ibid. 72, 679–693 (1984). 21
[33] C.T. Zhang, Soliton excitations in deoxyribonucleic acid (DNA) double helices, Physical Rev. A 35, 886–891 (1987). [34] S. Yomosa, Solitary excitations in deoxyribonuclei acid (DNA) double helices, Phys. Rev. A 30, 474–480 (1984). [35] O.M. Braun and Y.S. Kivshar, The Frenkel-Kontorova Model: Concepts, Methods and Applications, Springer (2004). [36] C. Rogers, A novel Ermakov-Painlev´e II system: N+1-dimensional coupled NLS and elastodynamic reductions, Stud. Appl. Math. 133, 214–231 (2014). [37] P. Amster and C. Rogers, On a Ermakov-Painlev´e II reduction in threeion electrodiffusion. A Dirichlet boundary value problem, Discrete and Continuous Dynamical Systems 35, 3277–3292 (2015). [38] C. Rogers and W.K. Schief, On Ermakov-Painlev´e II systems. Integrable reduction, Meccanica 51, 2967–2974 (2016). [39] C. Rogers, Hybrid Ermakov-Painlev´e IV systems, J. Nonlinear Math. Phys. 21, 628–642 (2014). [40] P. Amster and C. Rogers, On Dirichlet two-point boundary value problems for the Ermakov-Painlev´e IV equation, J. Applied Math. & Computing 48, 71–81 (2015). [41] C. Rogers, A.P. Bassom and P.A. Clarkson, Ermakov-Painlev´e IV systems. Canonical reduction, J. Math. Anal. Appl. 462, 1225–1241 (2018). [42] C. Rogers, On hybrid Ermakov-Painlev´e systems. Integrable reduction, J. Nonlinear Math. Phys. 24, 239–249 (2017). [43] O.K. Pashaev and J.H. Lee, Resonance solitons as black holes in a Madelung fluid, Mod. Phys. Lett. A 17, 1601–1619 (2002). [44] O.K. Pashaev, J.H. Lee and C. Rogers, Soliton resonances in a generalised nonlinear Schr¨odinger equation, J. Phys. A: Math. & Theor. 41, 452001–452009 (2008).
22
[45] C. Rogers and P.A. Clarkson, Ermakov-Painlev´e II reduction in cold plasma physics. Application of a B¨acklund transformation, J. Nonlinear Mathematical Physics 25, 247–261 (2018). [46] J.H. Lee, O.K. Pashaev, C. Rogers and W.K. Schief, The resonant nonlinear Schr¨odinger equation in cold plasma physics. Application of B¨acklund-Darboux transformations and superposition principles, J. Plasma Phys. 73, 257–272 (2007). [47] C. Rogers and P.A. Clarkson, Ermakov-Painlev´e II symmetry reduction of a Korteweg capillarity system, Symmetry, Integrability and Geometry: Methods and Applications 13, 018 (2017). [48] L.K. Bass, J.J.C. Nimmo, C. Rogers and W.K. Schief, Electrical structures of interfaces. A Painlev´e II model, Proc. Roy. Soc. London Ser. A 466, 2117–2136 (2010). [49] W.K. Schief, On a 2+1-dimensional integrable Ernst-type equation, Proc. Roy. Soc. London A 446, 381–398 (1994). [50] B.G. Konopelchenko and C. Rogers, On 2+1-dimensional nonlinear systems of Loewner-type, Phys. Lett. A158, 391–397 (1991). [51] B.G. Konopelchenko and C. Rogers, On generalised Loewner systems: novel integrable equations in 2+1-dimensions, J. Math. Phys. A34, 214– 242 (1993). [52] W.K. Schief, C. Rogers and A. Bassom, Ermakov systems with arbitrary order and dimension. Structure and linearisation, J. Phys. A: Math. Gen. 29, 903–911 (1996). [53] A.E. Milne and P.A. Clarkson, Rational solutions and B¨acklund transformations for the third Painlev´e equation, in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations (Ed. P.A. Clarkson), NATO ASI Series: Mathematical and Physical Sciences, Kluwer Academic Publishers 413, 341–352 (1993). [54] M.V. Tratnik and J.E. Sipe, Bound solitary waves in a birefringent optical fiber, Phys. Rev. A 38, 2011–2017 (1988)
23
[55] B. Deconinck, P.G. Kevrekidis, H.E. Nistazakis and D.J. Frantzeskakis, Linearly coupled Bose-Einstein condensates: From Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A 70, 063605 (2004). [56] K. Andriopoulos and P. Leach, The Ermakov equation: A commentary, Appl. Anal. Discrete Math. 2, 146–157 (2008). [57] Ya. Prykarpatskyy, Steen-Ermakov-Pinney equation and integrable nonlinear deformation of the one-dimensional Dirac equation, J. Mathematical Sciences 231, 820–826 (2018). [58] C. Rogers and U. Ramgulam, A nonlinear superposition principle and Lie group invariance: application in rotating shallow water theory, Int. J. Nonlinear Mech. 24, 229–236 (1989). [59] C. Rogers, W.K. Schief and P. Winternitz, Lie theoretical generalisation and discretisation of the Pinney equation, J. Math. Anal. Appl. 216, 246–264 (1997). [60] P.L. Christiansen, J.C. Eilbeck, V.Z. Enolski and N.A. Kostov, Quasiperiodic solutions of the coupled nonlinear Schr¨odinger equations, Proc. R. Soc. London A 451, 685–700 (1995). [61] P.L. Christiansen, J.C. Eilbeck, V.Z. Enolski and N.A. Kostov, Quasiperiodic and periodic solutions for coupled nonlinear Schr¨odinger equations of Manakov type, Proc. R. Soc. London A 456, 2263–2281(2000). [62] C. Athorne, C. Rogers, U. Ramgulam and A.Osbaldestin, On linearisation of the Ermakov system, Phys. Lett. A 143, 207–212 (1990). [63] A.V. Mikhailov, Integrability of the two-dimensional generalization of Toda chain, JETP Lett. 30, 414–418 (1979).
24
Journal Pre-proof
On Modulated Coupled Systems. Canonical Reduction via Reciprocal Transformations Colin Rogers, Wolfgang K Schief, Boris Malomed PII: DOI: Reference:
S1007-5704(19)30410-1 https://doi.org/10.1016/j.cnsns.2019.105091 CNSNS 105091
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
18 April 2019 23 October 2019 29 October 2019
Please cite this article as: Colin Rogers, Wolfgang K Schief, Boris Malomed, On Modulated Coupled Systems. Canonical Reduction via Reciprocal Transformations, Communications in Nonlinear Science and Numerical Simulation (2019), doi: https://doi.org/10.1016/j.cnsns.2019.105091
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Highlights • Privileged (integrable) systems admit reciprocally related modulated counterparts • Modulation may be driven by Painlev and Ermakov equations • Periodically modulated Manakov solitons may be obtained
1
On Modulated Coupled Systems. Canonical Reduction via Reciprocal Transformations Colin Rogersa , Wolfgang K Schiefa,∗, Boris Malomedb , b
a The University of New South Wales, Sydney, NSW2052, Australia The Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, Israel
Abstract It is shown how classes of modulated coupled systems of sine-Gordon, Demoulin and Manakov-type may be reduced to their unmodulated counterparts via the application of a novel reciprocal transformation. Modulations governed by recently introduced integrable Ermakov-Painlev´e II, ErmakovPainlev´e III and Ermakov-Painlev´e IV equations as well as the classical Ermakov equation are considered. In the latter case, the procedure is illustrated by the generation of a class of exact solutions to a heterogeneous Manakov system with periodic modulation. Keywords: reciprocal transformation, heterogeneous integrable system, Manakov system, modulated soliton 1. Introduction Physical systems which incorporate spatial or temporal modulation arise, notably, in both nonlinear optics and the theory of Bose-Einstein condensates [1, 2, 3, 4, 5, 6, 7]. Likewise, in classical continuum mechanics, modulated physical systems are of importance, inter alia in elastodynamics, viscoelastodynamics and in the elastostatics of inhomogeneous media (see e.g. [8, 9, 10, 11] and literature cited therein). In recent work [12], a novel reciprocal transformation was introduced which reduces non-autonomous, multi-component Ermakov-Painlev´e systems to their integrable autonomous counterparts. Here, such a reciprocal transformation is applied, in turn, to coupled systems of sine-Gordon, Demoulin and Manakov-type to reduce ∗
Corresponding author.
Preprint submitted to Commun Nonlinear Sci and Numer Simulat
October 31, 2019
them to associated unmodulated canonical forms. It is remarked that kink propagation in a particular modulated coupled sine-Gordon system has been previously investigated in [13]. Modulated single-component sine-Gordon equations have been the subject of prior work in [14, 15]. Integrability aspects of the classical two-component Demoulin system of affine geometry, such as its admittance of an auto-B¨acklund transformation are described in [16]. A modulated version of Tzitz´eica reduction of this Demoulin system was derived in a geometric context in [17]. The present work is concerned with the derivation of novel modulated coupled systems which, under a class of reciprocal transformations, retain the solitonic integrable structure of their unmodulated counterparts. In terms of physical applications, the work is motivated, in part, by the spatially modulated version of the celebrated coupled Manakov system of [18] which has been investigated in [19] in the context of rogue wave propagation. Time modulated Manakov systems have recently arisen in [20] in connection with the propagation of bright solitons in Bose-Einstein condensates. Here, coupled systems with modulation governed by integrable Ermakov-Painlev´e II, Ermakov-Painlev´e III, Ermakov-Painlev´e IV or the classical Ermakov equation are considered in turn. By way of illustration of the application of the reciprocal transformation, a novel class of exact solutions is generated for modulated Manakov systems via their integrable canonical Manakov counterparts. Privileged periodic modulated Manakov systems are recorded. In particular, a spatially modulated version of the bright soliton solution of the Manakov system is presented with the modulation being driven by the classical Ermakov equation. It is noted that reciprocal-type transformations have been previously shown, in particular, to make analytically tractable initial/boundary value problems which arise in a diversity of physical contexts. Thus, in [21] a reciprocal and Bcklund transformation were combined to solve a class of initial/boundary value problems descriptive of two-phase flow under gravity and with boundary infiltration. In [22, 23, 24], reciprocal transformations have been applied to solve nonlinear moving boundary problems of Stefantype which arise in the analysis of the melting of metals. In [25], a nonlinear model for the evolution of methacrylate in wood treatment was reduced via a reciprocal transformation to a moving boundary problem amenable to an integrable representation procedure as set down in [26]. In [27] a reciprocal transformation was used to obtained the exact solution of a nonlinear boundary value problem which arises in connection with unsaturated flow through 3
a heterogeneous porous medium. Moving boundary problems incorporating inhomogeneity occur naturally in soil mechanics. Thus, they arise notably in the analysis of the transport of liquid through soils as modelled in the homogeneous case by the classical work of Richards [28]. In most recent work [29], two distinct kinds of reciprocal transformation have been employed in conjunction to reduce a wide class of nonlinear moving boundary problems with heterogeneity to analytically solvable canonical forms. These are associated, in turn, with a classical Stefan problem and one relevant to the evolution of seapage fronts in soil mechanics. In [30], the combined action of reciprocal and integral-type transformations has been used to sequentially reduce to analytically tractable form another class of nonlinear moving boundary problems involving heterogeneity. Particular such Stefan problems arise in the description of the transport of liquids through layered porous media. The present work indicates, in particular, the potential application of reciprocal transformations to moving boundary problems for coupled systems incorporating heterogeneity. It is remarked that pointwise bounds on the solution of certain moving boundary problems for coupled solitonic systems embodied in resonant nonlinear Schr¨odinger equations have been obtained in [31]. To conclude, it is observed that the present reductions of modulated systems via reciprocal transformations may likewise be applied mutatis mutandis to corresponding temporally modulated Manakov systems. Moreover, extension may be readily made to integrable reduction of certain multi-component systems such as modulated Toda-systems. 2. A modulated coupled sine-Gordon system: canonical reduction via a reciprocal transformation Here, a class of heterogeneous coupled sine-Gordon systems is considered of the type u−v cI p , ρ utt − uxx + Φ(ρ, ρx , . . . ; x)u = q sin ρ ρ (1) cII u−v p ρ vtt − vxx + Φ(ρ, ρx , . . . ; x)v = q sin , ρ ρ
4
with modulation term ρ(x) determined by an auxiliary generalised Ermakov equation δ −ρxx + Φ(ρ, ρx , . . . ; x)ρ = 3 (2) ρ which is required to be analytically tractable. Combination of (2) with (1)1 and (2)2 in turn shows that δu cI u−v p+1 ρ utt − ρuxx + ρxx u + 3 = q−1 sin , ρ ρ ρ cII u−v δv p+1 , ρ vtt − ρvxx + ρxx v + 3 = q−1 sin ρ ρ ρ
(3)
that is, ∂ u δu cI u−v 2 ∂ ρ utt − ρ + 3 = q−1 sin , ∂x ∂x ρ ρ ρ ρ u−v v cII ∂ δv p+1 2 ∂ ρ vtt − ρ + 3 = q−1 sin . ∂x ∂x ρ ρ ρ ρ p+1
Thus, on introducing a reciprocal transformation R∗ according to ) u∗ = u/ρ, v ∗ = v/ρ R∗ , ∗ −2 ∗ dx = ρ dx, t = t
(4)
(5)
it is seen that ρp+4 u∗t∗ t∗ − u∗x∗ x∗ + δu∗ = cI ρ3−q sin(u∗ − v ∗ ),
ρp+4 vt∗∗ t∗ − vx∗∗ x∗ + δv ∗ = cII ρ3−q sin(u∗ − v ∗ ).
(6)
Accordingly, with p = −4, q = 3, the following result is obtained: Theorem 2.1. The heterogeneous coupled sine-Gordon system cI u−v −4 ρ utt − uxx + Φ(ρ, ρx , . . . ; x)u = 3 sin , ρ ρ cII u−v −4 ρ vtt − vxx + Φ(ρ, ρx , . . . ; x)v = 3 sin , ρ ρ 5
(7)
with modulation ρ(x) determined by the generalised Ermakov equation −ρxx + Φ(ρ, ρx , . . . ; x)ρ = under the reciprocal transformation dx∗ = ρ−2 dx, ∗
u = u/ρ,
t∗ = t ∗
v = v/ρ
)
δ ρ3
(8)
R∗
(9)
is reduced to the un-modulated coupled sine-Gordon system u∗t∗ t∗ − u∗x∗ x∗ + δu∗ = cI sin(u∗ − v ∗ ),
vt∗∗ t∗ − vx∗∗ x∗ + δv ∗ = cII sin(u∗ − v ∗ ).
(10)
It is observed that a system similar to (10) was derived, as a model of DNA, in [32]. In an analogous manner, the modulated coupled system 1 1 u u v −4 ρ uxx − 2 utt = 3 cI sin + cIII sin cos , c0 ρ ρ 2ρ 2ρ (11) 1 1 v v u ρ−4 vxx − 2 vtt = 3 cII sin + cIII sin cos c0 ρ ρ 2ρ 2ρ
under R∗ reduces to the variant sine-Gordon system u∗x∗ x∗ − vx∗∗ x∗
1 ∗ u∗ v∗ ∗ u = c sin u + c sin cos , ∗ ∗ I III c20 t t 2 2
1 v∗ u∗ − 2 vt∗∗ t∗ = cII sin v ∗ + cIII sin cos , c0 2 2
(12)
as introduced in [33] in connection with kink propagation in DNA double helices (see, also, [34]). Coupled sine-Gordon systems of the type (10) with δ = 0 arise as continuum two-component Kontorova-Frenkel dislocation models (see e.g. [35] and literature cited therein). It is noted that, if the relation ρ∗ = ρ−1
(13)
is adjoined to R∗ , then R∗2 = id. Moreover, under R∗ , (2) becomes −ρ∗x∗ x∗ + δρ∗ = where Φ∗ := Φ|R∗ . 6
Φ∗ , ρ∗3
(14)
3. Modulated Demoulin systems The reciprocal transformation R∗ as given by (9) may likewise be applied to modulated Demoulin systems 1 u/ρ e − e−(u+v)/ρ , 3 ρ 1 ρ−4 vtt − vxx + Φ(ρ, ρx , . . . ; x)v = 3 ev/ρ − e−(u+v)/ρ ρ
ρ−4 utt − uxx + Φ(ρ, ρx , . . . ; x)u =
(15)
to obtain reduction to the corresponding un-modulated system ∗
∗ +v ∗ )
v∗
−(u∗ +v ∗ )
u∗t∗ t∗ − u∗x∗ x∗ + δu∗ = eu − e−(u vt∗∗ t∗ − vx∗∗ x∗ + δv ∗ = e − e
,
.
(16)
In the case δ = 0, the latter constitutes the Demoulin system descriptive of an important subclass of surfaces in classical projective geometry. This system is integrable in the solitonic sense, and accordingly, in particular, admits invariance under a B¨acklund transformation (see e.g. [16] and cited literature). This invariance, conjugated with R∗ generates a novel invariance of the class of modulated Demoulin systems (15). In the single-component case u = v, (15) constitutes a modulated version of the classical Tzitzeica equation. It is remarked that a particular modulated version of the latter has been set down in [17]. 4. Ermakov-Painlev´ e modulation Here, modulations are considered wherein the auxiliary equation (2) in ρ constitutes a hybrid integrable Ermakov-Painlev´e II, Ermakov-Painlev´e III or Ermakov-Painlev´e IV equation [36, 37, 38, 39, 40, 41, 42]. These adopt the forms discussed in the following. 4.1. Ermakov-Painlev´e II This is the nonlinear equation ρxx − ρ3 +
(α + 12 )2 xρ =− 2 4ρ3
7
(17)
as derived in [36] via a symmetry reduction of an n+1-dimensional ‘resonant’ Manakov-type system incorporating a de Broglie-Bohm potential ∇2 |Ψ|/|Ψ|, namely ∂Ψ + ∇2 Ψ + ν(|Ψ|2 + |Ω|2 )Ψ = s(∇2 |Ψ|/|Ψ|)Ψ, i ∂t i
∂Ω + ∇2 Ω + ν(|Ψ|2 + |Ω|2 )Ω = s(∇2 |Ω|/|Ω|)Ω, ∂t
(18)
(∇2 := ∂ 2 /∂x21 + · · · + ∂ 2 /∂x2n ) .
It is recalled that certain single component resonant NLS equations involving a de Broglie-Bohm potential have been shown to admit solitonic fission or fusson phenomena if s > 1 [43, 44]. The hybrid Ermakov-Painlev´e II equation (17) has also been subsequently derived in [45] as a symmetry reduction of the cold plasma system of [46] and of the classical Korteweg capillarity system in [47]. Importantly, the hybrid Ermakov-Painlev´e II equation (17) may be obtained by setting Σ = ρ2 (19) in the integrable P34 equation Σxx
2 1 1 Σ2x 2 − xΣ + 2Σ − α + , = 2Σ 2 2Σ
(20)
which, in turn, is linked to the classical PII equation Yxx = 2Y 3 + xY + α
(21)
via the Hamiltonian system Yx = −Y 2 −
x +Σ , 2
1 Σx = 2Y Σ + α + . 2
(22)
In order to implement the reciprocal transformation R∗ it is required to evaluate Z Z 1 1 ∗ x = dx = dx (23) 2 ρ (x) Σ(x)
for positive solutions Σ of P34 . A B¨acklund transformation for PII may be adduced to generate iteratively an infinite sequence of solutions Σ for which the integral (23) can be evaluated (see [38]). Thus, let Yα be a solution of 8
PII corresponding to the parameter α and Σα an associated solution of P34 . Then, another pair of solutions {Yα+1 , Σα+1 } is given by Yα+1 = −Yα −
α + 12 , Σα
and, on setting x∗α
=
2 Σα+1 = −Σα + 2Yα+1 +x
Z
1 dx Σα (x)
it may be shown that [38] . 1 3 ∗ ∗ xα+1 = α+ xα + ln (Σα+1 Σα ) α+ . 2 2
(24)
(25)
(26)
The importance of positive solutions of P34 arises naturally in a twoion electro-diffusion context wherein the scaled electric field is governed by PII and the associated ion concentrations by P34 . The physical requirement that the latter be positive has been examined in detail in [48] both for the rational solutions in terms of Yablonskii-Vorob’ev polynomials and for exact solutions involving the classical Airy function, in each case as generated by the iterative action of a B¨acklund transformation. 4.2. Ermakov-Painlev´e III This adopts the form 2 γρ4 δ ρx ρx 1 4 + 2 (αρ + β) + ρ= 3 ρxx − 2 − ρ ρx 2ρ x 2 2ρ
(27)
and is obtained by setting w = ρ2
(28)
in the classical PIII equation wxx =
wx2 wx 1 δ − + (αw2 + β) + γω 3 + . w x x w
(29)
The latter canonical nonlinear equation may be obtained, in particular, as a symmetry reduction of an integrable Ernst system as derived in [49] via the 2+1-dimensional solitonic system of [50, 51]. It was subsequently established in [52] that, remarkably, this generalised Ernst system incorporates a 9
Ermakov system reduction. It was this observation that originally motivated the study of hybrid integrable systems. Here, in view of the relation (28), in order to apply R∗ it is required to evaluate Z 1 dx (30) w(x)
where w is a positive solution of the PIII equation (29). In this connection, the latter admits rational solutions generated iteratively by a B¨acklund transformation as described in [53] whence the integral (30) may, in principle, be evaluated and regions of positivity of w(x) delimited analytically.
4.3. Ermakov-Painlev´e IV This nonlinear equation adopts the form [41] β 3 4 2 2 ρxx − ρ + 2xρ + x − α ρ = 3 4 2ρ
(31)
and may be obtained by setting σ = ρ2
(32)
in the canonical PIV equation σxx =
1 2 3 3 β σx + σ + 4xσ 2 + 2(x2 − α)σ + . 2σ 2 σ
(33)
The Ermakov-Painlev´e IV equation (31) was derived via symmetry reduction of an n-component, resonant, derivative NLS system in [39]. Here, the implementation of the reciprocal transformation R∗ requires evaluation of the integral Z 1 ∗ dx (34) x = σ(x)
for positive solutions σ(x) of the PIV equation. In this connection, it was recently recorded in [41] that a B¨acklund transformation for PIV , namely Σn+1 (x) =
[Σn,x (x) − Σ2n (x) − 2xΣn (x)]2 2Σn (x)[Σn,x (x) − Σ2n (x) − 2xΣn (x) + 4(n + 1)]
(35)
applied iteratively to the seed bound state solution ξ exp(−x2 ) , Σn (x) = √ π 1 − 21 ξ erfc(x) 10
0 < ξ < 1,
(36)
corresponding to parameters α = 1 and β = 0 generates an infinite sequence of bound state solutions corresponding to the parameters α = 2n + 1 and β = 0. The bound state solution (36) is positive for all x while the subsequent bound state solutions as generated by the iterated action of the B¨acklund transformation (35) are positive on regions of the x-axis separated by zeros. This phenomenon is illustrated graphically in [41]. Moreover, the integral (34) can be evaluated iteratively for the bound state solutions Σn (x) of PIV by application of the relation Z Z dx 2 dx = + . (37) 2 Σn+1 (x) Σn (x) Σn,x (x) − Σn (x) − 2xΣn (x) 5. An integrable modulated Manakov systems. A symmetry reduction Under the reciprocal transformation R∗ given by (9) and modulation now determined by the generalised Ermakov equation ρxx + Φ(ρ, ρx , . . . ; x)ρ =
δ , ρ3
(38)
it is readily seen that the class of heterogeneous Manakov systems i
∂u + ρ4 uxx + ρ−2 (|u|2 + |v|2 )u + ρ4 Φ(ρ, ρx , . . . ; x)u = 0, ∂t
∂v i + ρ4 vxx + ρ−2 (|u|2 + |v|2 )v + ρ4 Φ(ρ, ρx , . . . ; x)v = 0 ∂t
(39)
is reduced to the canonical integrable Manakov system i
∂u∗ + u∗x∗ x∗ + (|u∗ |2 + |v ∗ |2 )u∗ + δu∗ = 0, ∂t∗
∂v ∗ i ∗ + vx∗∗ x∗ + (|u∗ |2 + |v ∗ |2 )v ∗ + δv ∗ = 0. ∂t
(40)
It is interesting to note that in [19], in a rogue wave propagation context, a Manakov-type system with certain modulated dispersion and nonlinearity terms together with an external potential was set down which admits reduction to the canonical integrable two-component Manakov system. The latter 11
has an extensive literature both in nonlinear optics and the theory of BoseEinstein condensates. Moreover, it was recorded in [54] (see, also, [55]) that the Manakov system (40) remains integrable if one introduces an additional linear coupling, namely, i
∂u∗ + u∗x∗ x∗ + (|u∗ |2 + |v ∗ |2 )u∗ + cv ∗ + δu∗ = 0, ∂t∗
(41) ∂v ∗ ∗ ∗ 2 ∗ 2 ∗ ∗ ∗ i ∗ + vx∗ x∗ + (|u | + |v | )v + cu + δv = 0, ∂t where c is a constant. This corresponds to a heterogeneous Manakov-type system with additional coupling terms cv and cu in (39)1 and (39)2 respectively. In the case when the auxiliary equation (38) is taken as the classical Ermakov equation (see e.g. [12] and [56, 57]) ρxx + ω(x)ρ =
δ , ρ3
(42)
the integration of the reciprocal relation in (5) to determine x∗ may be achieved by use of the well-known nonlinear superposition principle for (42). This is readily derived by Lie group methods as in [58, 59] and states that p ρ = c1 α2 (x) + 2c2 α(x)β(x) + c3 β 2 (x) (43)
where α(x), β(x) are two linearly independent solutions of (42)|δ=0 with corresponding constant Wronskian W = αβx − βαx and with the constants ci such that c1 c3 − c22 = δ/W 2 . (44) Thus,
∗
x =
Z
Z
dx c1 α2 (x) + 2c2 α(x)β(x) + c3 β 2 (x) Z β(x)/α(x) 1 dζ = = S(β/α). W c1 + 2c2 ζ + c3 ζ 2
dx = ρ2 (x)
(45)
5.1. A symmetry reduction. Application of the Ermakov invariant Here, a symmetry reduction of the canonical Manakov system (40) is sought with u∗ = q1∗ (x∗ ) exp[−iσt∗ + ir1∗ (x∗ )],
v ∗ = q2∗ (x∗ ) exp[−iσt∗ + ir2∗ (x∗ )] 12
(46)
whence, on substitution, it is seen that ∗ ∗2 ∗2 ∗ ∗ ∗ ∗2 q1x ∗ x∗ + (q1 + q2 )q1 + (δ + σ)q1 − q1 r1x∗ = 0,
∗ ∗2 ∗ ∗2 ∗ ∗2 ∗ q2x ∗ x∗ + (q1 + q2 )q2 + (δ + σ)q2 − q2 r2x∗ = 0,
∗ ∗ ∗ ∗ r1x ∗ x∗ /r1x∗ = −2q1x∗ /q1 ,
(47)
∗ ∗ ∗ ∗ r2x ∗ x∗ /r2x∗ = −2q2x∗ /q2 .
The latter pair of relations shows that ∗ ∗2 r1x ∗ = λ/q1 ,
∗ ∗2 r2x ∗ = µ/q2
(48)
whence, the nonlinear coupled system ∗ ∗2 ∗2 ∗ ∗ q1x ∗ x∗ + (q1 + q2 )q1 + (δ + σ)q1 = ∗ q2x ∗ x∗
+
(q1∗2
+
q2∗2 )q2∗
+ (δ +
σ)q2∗
λ2 , q1∗3
µ2 = ∗3 q2
(49)
results. It is remarked that a similarity ansatz of the type (46) has previously been applied to a Manakov system in [60] and [61]. In the latter paper, it was noted that the nonlinear coupled system derived is descriptive of the motion of particles interacting with a quartic potential and an inverse square potential. Moreover, for a particular choice of parameters, a Lax representation was obtained. Here, it is shown that the system (49) is amenable to exact solution via application of its Hamiltonian together with its admitted Ermakov invariant. The latter adopts the form " ∗ 2 ∗ 2 # 1 q q 2 2 ∗ ∗ ∗ 2 (q1∗ q2x + µ2 1∗ , (50) E= ∗ − q2 q1x∗ ) + λ ∗ 2 q1 q2 while the Hamiltonian is given by (q1∗2 + q2∗2 )2 1 ∗2 λ2 µ2 ∗2 ∗2 ∗2 H= q ∗ + q2x∗ + + (δ + σ)(q1 + q2 ) + ∗2 + ∗2 . (51) 2 1x 2 q1 q2 On application of the identity 2 2 ∗ ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗2 (q1∗2 + q2∗2 )(q1x ∗ + q2x∗ ) − (q1 q2x∗ − q2 q1x∗ ) ≡ (q1 q1x∗ + q2 q2x∗ )
13
(52)
and use of the pair of invariants E and H, it is seen that Σ∗2 λ2 µ2 ∗ ∗ − (δ + σ)Σ − ∗2 − ∗2 Σ 2H − 2 q1 q2 " ∗ 2 ∗ 2 # q q 1 = Σ∗2 − 2E − λ2 2∗ − µ2 1∗ ∗, q1 q2 4 x
(53)
where Σ∗ = q1∗2 + q2∗2 .
(54)
On reduction, (53) yields 1 Σ∗2 ∗ ∗ − (δ + σ)Σ − 2E + λ2 + µ2 = Σ∗2 Σ 2H − ∗ 2 4 x
(55)
which serves to determine Σ∗ in terms of elliptic functions. Moreover, the Ermakov invariant relation (50) shows that ∗ 2 ∗ 2 ∗ 2 q1 q2 q1 −1 2 2 ∗2 ∗2 d tan = 2E − λ −µ , (56) (q1 + q2 ) ∗ ∗ dx q2 q1 q2∗ whence, on introduction of the new independent variable x according to dx = Σ∗−1 dx,
(57)
it follows that q1∗ /q2∗ is determined via the tractable integral Z dΛ p =x 2 2E − λ cot2 Λ − µ2 tan2 Λ where
−1
Λ = tan
q1∗ q2∗
.
(58)
(59)
With Σ∗ and Λ determined in turn by (55) and (58), q1∗ and q2∗ in the symmetry reduction are now given by q1∗ = ±Σ∗1/2 sin Λ,
q2∗ = ±Σ∗1/2 cos Λ
(60)
and r1∗ , r2∗ by integration of the relations ∗ 2 ∗ r1x ∗ = λ/(Σ sin Λ),
∗ ∗ 2 r2x ∗ = µ/(Σ cos Λ).
14
(61)
The reciprocal transformation R∗ now delivers the associated class of solutions of the heterogeneous Manakov system (39) with u = ρu∗ (x∗ ), v = ρv ∗ (x∗ ), Z ∗ x = ρ−2 dx,
(62)
where, if the auxiliary equation (42) is the classical Ermakov equation (42) then x∗ is given by (45). In particular, if ω = const = 2 in (42) then, with α = cos x, β = sin x, p ρ = c1 cos2 x + 2c2 sin x cos x + c3 sin2 x, (63) c1 c3 − c22 = δ/2 . Thus, in this case, an integrable periodically-modulated Manakov system (39) results. 5.2. A spatially Ermakov-modulated Manakov soliton We now illustrate the periodic spatial modulation (62), (63) governed by the classical Ermakov equation by applying it to the standard bright onesoliton solution of the Manakov system (40) [18]. Thus, it may be verified directly that i(k3 x∗ +(k2 −k2 +δ)t) ∗ 2 3 k1 eik4 e u , 2k22 = k12 + kˆ12 , (64) = ˆ ikˆ4 ∗ ∗ v k1 e cosh[k2 (x − 2k3 t)] wherein the remaining constants are arbitrary but real, constitutes a solution of the Manakov system (40). Accordingly, the canonical quantity Σ∗ = |u∗ |2 + |v ∗ |2 =
2k22 cosh2 [k2 (x∗ − 2k3 t)]
(65)
exhibits the typical sech2 -profile of a soliton. The propagation of this soliton is depicted in Figure 1 for k2 = 1 and k3 = 1/2. The modulated analogue of the soliton solution (64) which satisfies the periodically-modulated Manakov system (39), (42), (63) is now generated via the relations (62). In particular, one obtains the modulated bright soliton Σ = |u|2 + |v|2 =
2k22 [ρ(x)]2 . cosh2 [k2 (x∗ (x) − 2k3 t)] 15
(66)
Figure 1: The bright one-soliton solution (65) of the Manakov system (40) for k2 = 1 and k3 = 1/2.
*
Figure 2: The relation x∗ = x∗ (x) for c1 = 1/3, c2 = 0, c3 = 3. Here, = 1 (red) and = 6 (blue) respectively.
Integration of the relation (62)3 yields, up to an additive constant, ! c tan x + c x 1 1 π 3 2 ∗ arctan p + , (67) x = p + p 2 c1 c3 − c22 c1 c3 − c22 c1 c3 − c22 π
where b · c denotes the standard floor function. In the above, we have assumed that c3 > 0. In the case, c1 = 1/3, c2 = 0 and c3 = 3, the dependence of x∗ on x is depicted in Figure 2 for = 1 and = 6. It is evident that the modulated soliton Σ∗ travels “on average” at the same speed as the Manakov soliton Σ. However, the shape of the soliton Σ∗ is modulated in two different ways. On the one hand, its amplitude is modulated due to the factor ρ2 . 16
On the other hand, it is deformed “horizontally” due to the dependence of sech2 on x∗ (x) − 2k3 t which slightly deviates form x − 2k3 t. The larger , the less pronounced this distortion becomes. Figure 3 depicts the modulated Manakov soliton Σ∗ for the choice of parameters adopted in the preceding and = 1, 2, 3, 4, 5, 6. 6. Conclusion Here, a novel kind of reciprocal transformation has been used to reduce certain spatially-modulated two-component systems of partial differential equations to their unmodulated counterparts. The reciprocal reduction method has its genesis in an autonomisation procedure as originally presented in [62] for the physically important coupled Ermakov-Ray-Reid system (see, e.g., [12]). The method may be applied, mutatis mutandis, to the temporallymodulated analogues of these systems. In addition, such reduction may be readily extended to multi-component systems of physical interest. Thus, in particular under the reciprocal transformation ψi∗ = ψi /ρ dx∗ = ρ−2 dx,
(68)
t∗ = t
with −ρxx + Φ(ρ, ρx , . . . ; x)ρ =
δ ρ3
(69)
it is seen that the modulated Toda lattice system ρ−4 ψn,tt − ψn,xx + Φ(ρ, ρx , . . . ; x)ψn =
1 [−eψn+1 /ρ + 2eψn /ρ − eψn−1 /ρ ] (70) ρ3
is reduced to the corresponding unmodulated system ∗
∗
∗
∗ ψn+1 ∗ ∗ + 2eψn − eψn−1 . ψn,t ∗ t∗ − ψn,x∗ x∗ + δψn = −e
(71)
Thus, if δ = 0 the standard integrable Toda lattice system results [63]. Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 17
Figure 3: The modulated Manakov soliton (66), (67) for = 1, 2, 3, 4, 5, 6.
18
References [1] J. Belmonte-Beita, V.M. P´erez-Garcia and V. Vekslechnik, Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities, Physical Review Letters 98, 064102-1–064102-4 (2007). [2] J. Belmonte-Beita, V.M. P´erez-Garcia, V. Vekslechik and V.V. Konotop, Localised nonlinear waves in systems with time and space-modulated nonlinearities, Physical Review Letters 100, 064102-1–064102–4 (2008). [3] X.Y. Tang and P.K. Shukla, Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schr¨odinger equation with external potential, Phys. Rev. A 76, 01312 (2007). [4] J.F. Zhang, Yi S. Li, J. Meng, L. Wo and B.A. Malomed, Matter-wave solitons and finite-amplitude Bloch waves in optical lattices with a spatially modulated nonlinearity, Phys. Rev. A 82, 033614 (2010). [5] W.P. Zhong, M.R. Belic, B.A. Malomed and T. Huang, Solitary waves in the nonlinear Schr¨odinger equation with Hermite-Gaussian modulation of the local linearity, Phys. Rev. E 84, 046611 (2011). [6] W.P. Zhong, M.R. Belic and T. Huang, Solitary waves in the nonlinear Schr¨odinger equation with spatially modulated Bessel nonlinearity, J. Opt. Soc. An. B30, 1276–1283 (2013). [7] C. Rogers, G. Saccomandi and L. Vergori, Ermakov-modulated nonlinear Schr¨odinger models. Integrable reduction, J. Nonlinear Mathematical Physics 23, 108–126 (2016). [8] F.C. Karal and J.B. Keller, Elastic wave propagation in homogeneous and inhomogeneous media, J. Acoust. Soc. America 31, 694–705 (1959). [9] D.W. Barclay, T.B. Moodie and C. Rogers, Cylindrical impact waves in inhomogeneous Maxwellian visco-elastic media, Acta Mechanica 29, 93–117 (1978). [10] D.L. Clements, C. Atkinson and C. Rogers, Antiplane crack problems for an inhomogeneous elastic material, Acta Mechanica 29, 199–211 (1978).
19
[11] D.L. Clements and C. Rogers, On the Bergman operator method and anti-plane contact problems involving an inhomogeneous half-space, SIAM J. Appl. Math. 34, 764–773 (1978). [12] C. Rogers and W.K. Schief, Ermakov-type systems in nonlinear physics and continuum mechanics, in Nonlinear Systems and Their Remarkable Mathematical Structures (Ed. N. Euler) CRC Press (2018) 541–576. [13] B.A. Malomed, Motion of a kink in a spatially modulated sine-Gordon system Phys. Lett. A 144, 351–356 (1990). [14] B.A. Malomed and M.I. Tribelsky, Emission of waves by moving kinks in a spatially modulated sine-Gordon system, Physical Review B41, 11271– 11281 (1990). [15] A. S´anchez, A.R. Bishop and F. Dominguez-Adame, Kink stability, propagation and length scale competition in the periodically modulated sine-Gordon equation, Phys. Rev. E 49, 4603–4615 (1994). [16] E.V. Ferapontov and W.K. Schief, Surfaces of Demoulin: differential geometry, B¨acklund transformation and integrability, J. Geom. Phys. 30, 343–363 (1999). [17] M. Dunajski and P. Plansangkate, Strominger-Yau-Zaslow geometry, affine spheres and Painlev´e III, Comm. Math. Phys. 290, 997–1024 (2009). [18] S.V. Manakov, On the theory of two-dimensional stationary focussing of electromagnetic waves, Sov. Phys. JETP 38, 248–253 (1974). [19] W.P. Zhong, M.R. Belic and B.A. Malomed, Rogue waves in a twocomponent Manakov system with variable coefficients and an external potential, Phys. Rev. E 92, 053201 (2015). [20] T. Kanna, R.B. Mareesaran and F. Mertens, Non-autonomous bright solitons and their stability in Rabi coupled binary Bose-Einstein condensates, J. Phys. Commun 1, 045005 (2017). [21] C. Rogers, M.P. Stallybrass and D.L. Clements, On two phase filtration under gravity and with boundary infiltration: application of a B¨acklund transformation, J. Nonlinear Analysis, Theory, Methods and Applications, 7, 785–799 (1983). 20
[22] C. Rogers, Application of a reciprocal transformation to a two phase Stefan problem, J. Phys. A: Math. Gen. 18, L105–L109 (1985). [23] C. Rogers, On a class of moving boundary problems in nonlinear heat conduction: application of a B¨acklund transformation, Int. J. Nonlinear Mechanics 21, 249–256 (1986). [24] C. Rogers and B. Yu Guo, A note on the onset of melting in a class of simple metals. Condition on the applied boundary flux, Acta Mathematica Scientia 8, 425–430 (1988). [25] A.S. Fokas, C. Rogers and W.K. Schief, Evolution of methacrylate distribution during wood saturation. A nonlinear moving boundary problem, Appl. Math. Lett. 18, 321–328 (2005). [26] A.S. Fokas, A unifed transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London A 453, 1411–1443 (1997). [27] C. Rogers and P. Broadbridge, On a nonlinear moving boundary problem with heterogeneity: application of a B¨acklund transformation, Zeit. angew. Math. Phys. 39, 122–128 (1988). [28] L.A. Richards, Capillary conduction of liquids through porous mediums, J. Applied Physics 1, 318–333 (1931). [29] C. Rogers, On Stefan-type moving boundary problems with heterogeneity: canonical reduction via conjugation of reciprocal transformations, Acta Mechanica 230, 839–850 (2019). [30] C. Rogers, Moving boundary problems for heterogeneous media. Integrability via conjugation of reciprocal and integral transformations, J. Nonlinear Math. Phys., 26, 313–325 (2019). [31] J. Flavin and C. Rogers, Upper estimates for a moving boundary problem for a resonant nonlinear Schr¨odinger equation, Stud. Appl. Math. 121, 189–198 (2008). [32] S. Homma and S. Takeno, Topological solitons and modulated structure of bases in DNA double helices: A dynamic plane base-rotator model, Prog. Theor. Phys. 70, 308–311 (1983); A coupled base-rotator model for structure and dynamics of DNA: Local fluctuations in helical twist angles and topological solitons, ibid. 72, 679–693 (1984). 21
[33] C.T. Zhang, Soliton excitations in deoxyribonucleic acid (DNA) double helices, Physical Rev. A 35, 886–891 (1987). [34] S. Yomosa, Solitary excitations in deoxyribonuclei acid (DNA) double helices, Phys. Rev. A 30, 474–480 (1984). [35] O.M. Braun and Y.S. Kivshar, The Frenkel-Kontorova Model: Concepts, Methods and Applications, Springer (2004). [36] C. Rogers, A novel Ermakov-Painlev´e II system: N+1-dimensional coupled NLS and elastodynamic reductions, Stud. Appl. Math. 133, 214–231 (2014). [37] P. Amster and C. Rogers, On a Ermakov-Painlev´e II reduction in threeion electrodiffusion. A Dirichlet boundary value problem, Discrete and Continuous Dynamical Systems 35, 3277–3292 (2015). [38] C. Rogers and W.K. Schief, On Ermakov-Painlev´e II systems. Integrable reduction, Meccanica 51, 2967–2974 (2016). [39] C. Rogers, Hybrid Ermakov-Painlev´e IV systems, J. Nonlinear Math. Phys. 21, 628–642 (2014). [40] P. Amster and C. Rogers, On Dirichlet two-point boundary value problems for the Ermakov-Painlev´e IV equation, J. Applied Math. & Computing 48, 71–81 (2015). [41] C. Rogers, A.P. Bassom and P.A. Clarkson, Ermakov-Painlev´e IV systems. Canonical reduction, J. Math. Anal. Appl. 462, 1225–1241 (2018). [42] C. Rogers, On hybrid Ermakov-Painlev´e systems. Integrable reduction, J. Nonlinear Math. Phys. 24, 239–249 (2017). [43] O.K. Pashaev and J.H. Lee, Resonance solitons as black holes in a Madelung fluid, Mod. Phys. Lett. A 17, 1601–1619 (2002). [44] O.K. Pashaev, J.H. Lee and C. Rogers, Soliton resonances in a generalised nonlinear Schr¨odinger equation, J. Phys. A: Math. & Theor. 41, 452001–452009 (2008).
22
[45] C. Rogers and P.A. Clarkson, Ermakov-Painlev´e II reduction in cold plasma physics. Application of a B¨acklund transformation, J. Nonlinear Mathematical Physics 25, 247–261 (2018). [46] J.H. Lee, O.K. Pashaev, C. Rogers and W.K. Schief, The resonant nonlinear Schr¨odinger equation in cold plasma physics. Application of B¨acklund-Darboux transformations and superposition principles, J. Plasma Phys. 73, 257–272 (2007). [47] C. Rogers and P.A. Clarkson, Ermakov-Painlev´e II symmetry reduction of a Korteweg capillarity system, Symmetry, Integrability and Geometry: Methods and Applications 13, 018 (2017). [48] L.K. Bass, J.J.C. Nimmo, C. Rogers and W.K. Schief, Electrical structures of interfaces. A Painlev´e II model, Proc. Roy. Soc. London Ser. A 466, 2117–2136 (2010). [49] W.K. Schief, On a 2+1-dimensional integrable Ernst-type equation, Proc. Roy. Soc. London A 446, 381–398 (1994). [50] B.G. Konopelchenko and C. Rogers, On 2+1-dimensional nonlinear systems of Loewner-type, Phys. Lett. A158, 391–397 (1991). [51] B.G. Konopelchenko and C. Rogers, On generalised Loewner systems: novel integrable equations in 2+1-dimensions, J. Math. Phys. A34, 214– 242 (1993). [52] W.K. Schief, C. Rogers and A. Bassom, Ermakov systems with arbitrary order and dimension. Structure and linearisation, J. Phys. A: Math. Gen. 29, 903–911 (1996). [53] A.E. Milne and P.A. Clarkson, Rational solutions and B¨acklund transformations for the third Painlev´e equation, in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations (Ed. P.A. Clarkson), NATO ASI Series: Mathematical and Physical Sciences, Kluwer Academic Publishers 413, 341–352 (1993). [54] M.V. Tratnik and J.E. Sipe, Bound solitary waves in a birefringent optical fiber, Phys. Rev. A 38, 2011–2017 (1988)
23
[55] B. Deconinck, P.G. Kevrekidis, H.E. Nistazakis and D.J. Frantzeskakis, Linearly coupled Bose-Einstein condensates: From Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A 70, 063605 (2004). [56] K. Andriopoulos and P. Leach, The Ermakov equation: A commentary, Appl. Anal. Discrete Math. 2, 146–157 (2008). [57] Ya. Prykarpatskyy, Steen-Ermakov-Pinney equation and integrable nonlinear deformation of the one-dimensional Dirac equation, J. Mathematical Sciences 231, 820–826 (2018). [58] C. Rogers and U. Ramgulam, A nonlinear superposition principle and Lie group invariance: application in rotating shallow water theory, Int. J. Nonlinear Mech. 24, 229–236 (1989). [59] C. Rogers, W.K. Schief and P. Winternitz, Lie theoretical generalisation and discretisation of the Pinney equation, J. Math. Anal. Appl. 216, 246–264 (1997). [60] P.L. Christiansen, J.C. Eilbeck, V.Z. Enolski and N.A. Kostov, Quasiperiodic solutions of the coupled nonlinear Schr¨odinger equations, Proc. R. Soc. London A 451, 685–700 (1995). [61] P.L. Christiansen, J.C. Eilbeck, V.Z. Enolski and N.A. Kostov, Quasiperiodic and periodic solutions for coupled nonlinear Schr¨odinger equations of Manakov type, Proc. R. Soc. London A 456, 2263–2281(2000). [62] C. Athorne, C. Rogers, U. Ramgulam and A.Osbaldestin, On linearisation of the Ermakov system, Phys. Lett. A 143, 207–212 (1990). [63] A.V. Mikhailov, Integrability of the two-dimensional generalization of Toda chain, JETP Lett. 30, 414–418 (1979).
24