Applied Mathematics and Computation 237 (2014) 77–84

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Nonisospectral scattering problems and similarity reductions

http://dx.doi.org/10.1016/j.amc.2014.03.1070096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail address: pilar.gordoa@urjc.es (P.R. Gordoa).

P.R. Gordoa a,⇑, A. Pickering a, J.A.D. Wattis b

a Departamento de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, C/ Tulipán s/n, 28933 Móstoles, Madrid, Spainb School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

a r t i c l e i n f o

Keywords:Korteweg–de Vries hierarchyDispersive water wave hierarchyNonisospectral scattering problemsSimilarity reductionsPainlevé hierarchies

a b s t r a c t

We give mappings between hierarchies having nonisospectral scattering problems andhierarchies having isospectral scattering problems. Special cases of these changes ofvariables involve similarity variables, and similarity solutions of a hierarchy are seen tocorrespond to time-independent solutions of an equivalent hierarchy. We thus explainwhy the use of nonisospectral scattering problems and similarity reductions yield the samePainlevé hierarchies. As examples we consider the Korteweg–de Vries hierarchy and thedispersive water wave hierarchy.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

The realisation that there exists a connection between completely integrable partial differential equations and Painlevéequations dates back to 1977, when Ablowitz and Segur [1] noted, amongst other examples, that the scaling similarity reduc-tion of the modified Korteweg–de Vries (mKdV) equation gives rise to the second Painlevé equation (PII). This then allowedthem to use the Inverse Scattering Transform of the mKdV equation to provide a linear integral equation for a special case ofPII. Shortly afterwards Airault [2] defined a whole hierarchy of ordinary differential equations (ODEs), a PII hierarchy, by sim-ilarity reduction of the Korteweg–de Vries (KdV) and mKdV hierarchies. However, interest in Painlevé hierarchies did notreally take off until nearly twenty years later, when Kudryashov [3] derived both a first and (Airault’s) second Painlevé hier-archy. Since then, there has been a great deal of interest in Painlevé hierarchies and their properties, e.g., [4–27].

An alternative derivation of Painlevé equations, along with their underlying linear problems, was proposed in [28], withthe important observation that Painlevé equations arise as stationary reductions of partial differential equations (PDEs)having nonisospectral scattering problems. This idea, and extensions thereof, were used in [5] in order to derive Painlevéhierarchies and their underlying linear problems from hierarchies of PDEs having nonisospectral scattering problems. Thisapproach to deriving Painlevé hierarchies seems more readily applicable than the extension to completely integrablehierarchies of similarity reductions of their first members, a task that is not always so straightforward. Further applicationsof this approach to deriving Painlevé hierarchies, including discrete Painlevé hierarchies, can be found in [7,8,14,15,18–20].

In the present paper we show how certain 1þ 1-dimensional nonisospectral hierarchies and their scattering problemscan be mapped onto standard PDE hierarchies and their isospectral scattering problems. Special cases of these mappingsinvolve similarity variables, and similarity solutions of a standard hierarchy are seen to correspond to time-independentsolutions of an equivalent nonisospectral hierarchy. This then explains why the use of nonisospectral scattering problemsand similarity reductions yield the same Painlevé hierarchies. As examples we consider in Section 2 the KdV hierarchyand in Section 3 the dispersive water wave (DWW) hierarchy. Our final section is devoted to conclusions.

78 P.R. Gordoa et al. / Applied Mathematics and Computation 237 (2014) 77–84

2. The Korteweg–de Vries case

2.1. The Korteweg–de Vries hierarchy

We recall that the KdV hierarchy is a hierarchy of completely integrable PDEs given by

Ut ¼ Rn½U�Ux; R½U� ¼ @2x þ 4U þ 2Ux@

�1x ; ð2:1Þ

where for ease of notation we suppress the usual labelling of the flow times. The recursion operator R½U� is the quotientR½U� ¼ B1½U�B�1

0 ½U� of the two Hamiltonian operators

B1½U� ¼ @3x þ 4U@x þ 2Ux; B0½U� ¼ @x: ð2:2Þ

By integrable, we mean that the KdV hierarchy can be solved by the Inverse Scattering Transform (IST), having the corre-sponding scattering problem

wxx þ ðU � kÞw ¼ 0; ð2:3Þ

where k is the (constant) spectral parameter.Let us also recall for future use the following Lemma [29].

Lemma 1. The change of variables ~U ¼ U þ C, where C is an arbitrary constant, in Rn½~U�~Ux, yields

Rn½~U�~Ux ¼Xn

j¼0

an;jCn�jRj½U�Ux; ð2:4Þ

where the coefficients an;j are determined recursively by

an;n ¼ 1; ð2:5Þan;j ¼ 4an�1;j þ an�1;j�1; j ¼ 1; . . . ;n� 1; ð2:6Þ

an;0 ¼4nþ 2

nan�1;0 ð2:7Þ

and where a0;0 ¼ 1.We now consider the relationship between a nonisospectral KdV hierarchy, i.e., having a scattering problem where the

spectral parameter is no longer constant, and a standard (isospectral) KdV hierarchy.

2.2. A nonisospectral Korteweg–de Vries hierarchy

We consider the nonisospectral hierarchy

us ¼ Rn½u�un þXn�1

j¼0

BjðsÞRj½u�un þ gn�1ðsÞð4uþ 2nunÞ þ gnðsÞ; ð2:8Þ

where R½u� is given by R½U� in (2.1) with U replaced by u and @x by @n. This hierarchy is a special case of a 2þ 1-dimensionalhierarchy considered in [5], and can also be found in [30]. This equation is of interest from the point of view of Painlevé hier-archies since its stationary reduction, where gn�1ðsÞ; gnðsÞ and all BjðsÞ are now constant, yields the equation

Rn½u�un þXn�1

j¼0

BjRj½u�un þ gn�1ð4uþ 2nunÞ þ gn ¼ 0: ð2:9Þ

This last equation is equivalent to Eq. (3.29) in [5] and gives rise to a generalized thirty-fourth Painlevé (P34) hierarchy and toa generalized first Painlevé (PI) hierarchy. These hierarchies arise in the cases gn�1 – 0 (when we can also assume gn ¼ 0) andgn�1 ¼ 0 respectively. Generalized and non-generalized P34 and PI hierarchies can be found in [3–5,9,11].

We now show that Eq. (2.8) can in fact be mapped to an isospectral KdV hierarchy generalized by the inclusion of lower-order flows.

Proposition 1. The nonisospectral hierarchy (2.8) is equivalent to the isospectral KdV hierarchy

Ut ¼ Rn½U�Ux þXn�1

i¼1

biðtÞRi½U�Ux ð2:10Þ

under a change of variables of the form

U ¼ gðtÞ2uðn; sÞ þmðtÞ; n ¼ gðtÞxþ hðtÞ; s ¼ kðtÞ: ð2:11Þ

P.R. Gordoa et al. / Applied Mathematics and Computation 237 (2014) 77–84 79

Proof. Using Lemma 1 we see that substituting (2.11) in (2.10) gives

Xn

p¼0

cpðtÞRp½u�un � gðtÞg0ðtÞð2uþ nunÞ �m0ðtÞ � gðtÞ2k0ðtÞus �Xn

j¼0

an;jmðtÞn�jgðtÞ2jþ3Rj½u�un

þXn�1

i¼1

biðtÞXi

j¼0

ai;jmðtÞi�jgðtÞ2jþ3Rj½u�un

!þ gðtÞg0ðtÞhðtÞ � gðtÞ2h0ðtÞh i

un � gðtÞg0ðtÞð2uþ nunÞ

�m0ðtÞ � gðtÞ2k0ðtÞus ¼ 0: ð2:12Þ

We recall that each ai;i ¼ 1, and so in particular cnðtÞ ¼ gðtÞ2nþ3.We solve the equations

cpðtÞ ¼ BpðkðtÞÞgðtÞ2nþ3; p ¼ n� 1; . . . ;1 ð2:13Þ

recursively for the coefficients bpðtÞ and the equation

c0ðtÞ ¼ B0ðkðtÞÞgðtÞ2nþ3 ð2:14Þ

for hðtÞ, where gðtÞ;mðtÞ and kðtÞ are determined by the equations

g0ðtÞ ¼ �2gn�1ðkðtÞÞgðtÞ2nþ2

; ð2:15Þm0ðtÞ ¼ �gnðkðtÞÞgðtÞ

2nþ3; ð2:16Þ

k0ðtÞ ¼ gðtÞ2nþ1: ð2:17Þ

Cancelling an overall factor of gðtÞ2nþ3 in (2.12) and writing the coefficients as functions of s then gives (2.8).The mapping between the scattering problems is given by the following:

Proposition 2. Extending the change of variables (2.11) with

wðx; tÞ ¼ uðn; sÞ; lðkðtÞÞ ¼ k�mðtÞgðtÞ2

; ð2:18Þ

the scattering problem (2.3) is transformed into the nonisospectral scattering problem

unn þ u� lðsÞ½ �u ¼ 0 ð2:19Þ

where lðsÞ satisfies the nonisospectral condition

dlds¼ 4gn�1ðsÞlðsÞ þ gnðsÞ: ð2:20Þ

Proof. Eq. (2.19) follows easily from the substitution of (2.11) and (2.18) in (2.3). Eq. (2.20) follows from differentiating thesecond equation of (2.18) with respect to t to obtain

dldsðkðtÞÞ

� �k0ðtÞ ¼ �2

k�mðtÞgðtÞ2

g0ðtÞgðtÞ �

m0ðtÞgðtÞ2

ð2:21Þ

and then using Eqs. (2.15)–(2.17) and the second equation of (2.18), and writing the result in terms of s.

2.3. Similarity reductions of the Korteweg–de Vries hierarchy

As mentioned in the Introduction, one method of deriving Painlevé hierarchies is as similarity reductions of completelyintegrable PDE hierarchies. However this process is not always so straightforward. For example, it is only recently that theaccelerating-wave reduction and generalized scaling reduction of the KdV equation have been extended to the KdV hierar-chy; see [29] and [31] respectively. Here we consider the relationship between the derivation of Painlevé hierarchies usingnonisospectral scattering problems and using similarity reductions; it turns out that the key to understanding this relation-ship lies in the transformation (2.11). This then allows us to explain why the same Painlevé hierarchies are found using thesetwo techniques.

We consider two special cases of (2.8) such that the stationary reduction (2.9) gives rise to a generalized PI hierarchy or ageneralized P34 hierarchy. The two cases that we consider have all BkðsÞ constant, and in addition gn�1ðsÞ ¼ 0 and gnðsÞconstant, and gnðsÞ ¼ 0 and gn�1ðsÞ constant, respectively, and are both nonisospectral equations.

80 P.R. Gordoa et al. / Applied Mathematics and Computation 237 (2014) 77–84

Corollary 1. For the special case of Eq. (2.10) and transformation (2.11) having gðtÞ ¼ 1;mðtÞ ¼ �gnt ðgn constantÞ kðtÞ ¼ t andhðtÞ and all bkðtÞ such that all BkðtÞ, k ¼ 0;1; . . . n� 1 are constant, the corresponding equivalent nonisospectral equation is

ut ¼ Rn½u�un þXn�1

j¼0

BjRj½u�un þ gn: ð2:22Þ

Proof. A simple substitution in Proposition 1. We note in particular that since gðtÞ is constant, gn�1ðtÞ ¼ 0 (see (2.15)).

Remark 1. The stationary reduction of (2.22) yields (2.9) with gn�1 ¼ 0,

Rn½u�un þXn�1

j¼0

BjRj½u�un þ gn ¼ 0 ð2:23Þ

and the substitution used to obtain this last from the corresponding case of (2.10), that is, the special case of the transfor-mation (2.11) given in Corollary 1 but now with u a function of n only, then corresponds precisely to the similarity reductionof this case of (2.10) given in [29] which yields the generalized PI hierarchy, this last being obtained by integrating (2.23) (togive for example (2.13) in [29]).That is, the change of variables of Corollary 1 explains why the stationary reduction of thenonisospectral hierarchy (2.22) can also be obtained as a similarity reduction of the corresponding case of the standard hier-archy (2.10).Both techniques then yield the same Painlevé hierarchy.It is interesting to note that these similarity solutions ofthis case of the hierarchy (2.10) correspond to time-independent solutions of the equivalent hierarchy (2.22).These resultsrepresent a generalization to the case of hierarchies—a generalization which relies on Lemma 1 published recently in [29]—ofthe special case n ¼ 1 considered in [28], where it was shown that the accelerating-wave reduction of the KdV equation,which gives PI , could be extended to give a nonisospectral equation having PI as stationary reduction.Similarly, Proposition1 represents an extension to the case of hierarchies of the results in [32–34], where generalizations and special cases of (2.8)for n ¼ 1 were considered along with the question of mappings to the KdV equation; again, this generalization relies on Lem-ma 1 in order to precisely formulate Eqs.(2.13) and(2.14). We recall that from the point of view of Painlevé hierarchies, theprincipal interest of the hierarchy (2.8) is its reduction to (2.9), and that for this last there is no mapping from the casegn�1; gn arbitrary to the case gn�1 ¼ gn ¼ 0.

Corollary 2. For the special case of Eq. (2.10) and transformation (2.11) having gðtÞ ¼ 1=½2ð2nþ 1Þgn�1t�1=ð2nþ1Þ (gn�1 constant),mðtÞ ¼ d (constant), kðtÞ ¼ logðtÞ=½2ð2nþ 1Þgn�1� and hðtÞ and all bkðtÞ such that all BkðsÞ, k ¼ 0;1; . . . n� 1 are constant, thecorresponding equivalent nonisospectral equation is

us ¼ Rn½u�un þXn�1

j¼0

BjRj½u�un þ gn�1ð4uþ 2nunÞ: ð2:24Þ

Proof. A simple substitution in Proposition 1. We note in particular that since mðtÞ is constant, gnðsÞ ¼ 0 (see (2.16)).

Remark 2. The stationary reduction of (2.24) yields (2.9) with gn ¼ 0,

Rn½u�un þXn�1

j¼0

BjRj½u�un þ gn�1ð4uþ 2nunÞ ¼ 0 ð2:25Þ

and the substitution used to obtain this last from the corresponding case of (2.10), that is, the special case of the transfor-mation (2.11) given in Corollary 2 but now with u a function of n only, then corresponds precisely to the generalized scalingreduction of this case of (2.10) given in [31] which yields the generalized P34 hierarchy, this last being obtained as a firstintegral of (2.25) (and given for example as (2.10) in [31]). That is, the change of variables of Corollary 2, which involves scal-ing similarity variables, explains why the stationary reduction of the nonisospectral hierarchy (2.24) can also be obtained asa generalized scaling reduction of the corresponding case of the standard hierarchy (2.10). Both techniques then yield thesame Painlevé hierarchy. It is interesting to note that scaling similarity solutions of this case of the hierarchy (2.10) corre-spond to time-independent solutions of the equivalent hierarchy (2.24).

3. The dispersive water wave case

3.1. The dispersive water wave hierarchy

The DWW hierarchy is a two-component completely integrable hierarchy in U ¼ ðU;VÞT given by [35]

P.R. Gordoa et al. / Applied Mathematics and Computation 237 (2014) 77–84 81

Ut ¼ Rn½U�Ux; R½U� ¼ 12

@xU@�1x � @x 2

2V þ Vx@�1x U þ @x

!; ð3:1Þ

where once again for ease of notation we suppress the usual labelling of the flow times. The recursion operator R is thequotient R½U� ¼ B2½U�B�1

1 ½U� of the two Hamiltonian operators

B2½U� ¼12

2@x @xU � @2x

U@x þ @2x V@x þ @xV

!ð3:2Þ

and

B1½U� ¼0 @x

@x 0

� �: ð3:3Þ

As is well-known, the DWW hierarchy has an energy-dependent scattering problem,

wxx ¼ k� 12

U� �2

þ 12

Ux � V

" #w; ð3:4Þ

where k is the spectral parameter.We recall for future use the following Lemma [29].

Lemma 2. The change of variables ~U ¼ ðU þ C;VÞT , where C is an arbitrary constant, in Rn½~U�~Ux, yields

Rn½~U�~Ux ¼Xn

j¼0

an;jCn�jRj½U�Ux; ð3:5Þ

where the coefficients an;j are determined recursively by

an;n ¼ 1; ð3:6Þ

an;j ¼12an�1;j þ an�1;j�1; j ¼ 1; . . . ;n� 1; ð3:7Þ

an;0 ¼12

nþ 1n

� �an�1;0 ð3:8Þ

and where a0;0 ¼ 1.We now turn to the relationship between a nonisospectral DWW hierarchy and a standard DWW hierarchy.

3.2. A nonisospectral dispersive water wave hierarchy

We consider the nonisospectral DWW hierarchy in u ¼ ðu;vÞT ,

us ¼ Rn½u�un þXn�1

j¼0

BjðsÞRj½u�un þ12

gnðsÞðnuÞn

2v þ nvn

� �þ gnþ1ðsÞ

10

� �; ð3:9Þ

where R½u� is given by R½U� in (3.1) with U replaced by u and @x by @n. This hierarchy is a special case of a hierarchy in 2þ 1dimensions considered in [8] and is of interest from the point of view of the derivation of Painlevé hierarchies since its sta-tionary reduction

Rn½u�un þXn�1

j¼0

BjRj½u�un þ12

gn

ðnuÞn2v þ nvn

� �þ gnþ1

10

� �¼

00

� �ð3:10Þ

—where gnðsÞ; gnþ1ðsÞ and all BjðsÞ are now constant—gives rise to a PII hierarchy and to a PIV hierarchy. This last equation isequivalent to the special case gn�1 ¼ 0 of (53) in [8]. The PII hierarchy and PIV hierarchy correspond respectively to the casesgn ¼ 0 and gn – 0 (when we may also assume gnþ1 ¼ 0). For PII and PIV hierarchies see [8,10–12,16,17].

We now show that Eq. (3.9) can be mapped to an isospectral DWW hierarchy generalized by the inclusion of lower orderflows.

Proposition 3. The nonisospectral hierarchy (3.9) is equivalent to the isospectral DWW hiertarchy

Ut ¼ Rn½U�Ux þXn�1

i¼1

ciðtÞRi½U�Ux ð3:11Þ

under a change of variables of the form

82 P.R. Gordoa et al. / Applied Mathematics and Computation 237 (2014) 77–84

U ¼ gðtÞuðn; sÞ þmðtÞ; V ¼ gðtÞ2vðn; sÞ; n ¼ gðtÞxþ hðtÞ; s ¼ kðtÞ: ð3:12Þ

Proof. Using Lemma 2 we see that substituting (3.12) in (3.11) gives

Xn

p¼0

CpðtÞRp½u�un �g0ðtÞðnuÞn

gðtÞg0ðtÞð2v þ nvnÞ

� ��m0ðtÞ

10

� ��

gðtÞk0ðtÞus

gðtÞ2k0ðtÞvs

!

�Xn

j¼0

an;jmðtÞn�jGjþ2ðtÞRj½u�un þXn�1

i¼1

ciðtÞXi

j¼0

ai;jmðtÞi�jGjþ2ðtÞRj½u�un

!� ½gðtÞh0ðtÞ � g0ðtÞhðtÞ�un

gðtÞ½gðtÞh0ðtÞ � g0ðtÞhðtÞ�vn

!

�g0ðtÞðnuÞn

gðtÞg0ðtÞð2v þ nvnÞ

� ��m0ðtÞ

10

� ��

gðtÞk0ðtÞus

gðtÞ2k0ðtÞvs

!¼

00

� �;

ð3:13Þ

where

GjðtÞ ¼gðtÞj 0

0 gðtÞjþ1

!: ð3:14Þ

We recall that each ai;i ¼ 1, and so in particular CnðtÞ ¼ Gnþ2ðtÞ.We solve the equations

CpðtÞ ¼ BpðkðtÞÞGnþ2ðtÞ; p ¼ n� 1; . . . ;1 ð3:15Þ

recursively for the coefficients cpðtÞ and the equation

C0ðtÞ ¼ B0ðkðtÞÞGnþ2ðtÞ ð3:16Þ

for hðtÞ, where gðtÞ;mðtÞ and kðtÞ are determined by the equations

g0ðtÞ ¼ �12

gnðkðtÞÞgðtÞnþ2

; ð3:17Þ

m0ðtÞ ¼ �gnþ1ðkðtÞÞgðtÞnþ2

; ð3:18Þk0ðtÞ ¼ gðtÞnþ1

: ð3:19Þ

Multiplying by the inverse of Gnþ2ðtÞ in (3.13) and writing the coefficients as functions of s then gives (3.9).As in the KdV case, we can give the mapping between the corresponding isospectral and nonisospectral scattering

problems:

Proposition 4. Extending the change of variables (3.12) with

wðx; tÞ ¼ uðn; sÞ; lðkðtÞÞ ¼ 2k�mðtÞ2gðtÞ ; ð3:20Þ

the scattering problem (3.4) is transformed into the nonisospectral scattering problem

unn ¼ lðsÞ � 12

u� �2

þ 12

un � v" #

u ð3:21Þ

where lðsÞ satisfies the nonisospectral condition

dlds¼ 1

2gnðsÞlðsÞ þ

12

gnþ1ðsÞ: ð3:22Þ

Proof. Eq. (3.21) follows easily from the substitution of (3.12) and (3.20) in (3.4). Eq. (3.22) follows from differentiating thesecond equation of (3.20) with respect to t to obtain

dldsðkðtÞÞ

� �k0ðtÞ ¼ �2k�mðtÞ

2gðtÞg0ðtÞgðtÞ �

12

m0ðtÞgðtÞ ð3:23Þ

and then using Eqs. (3.17)–(3.19) and the second equation of (3.20), and writing the result in terms of s.

3.3. Similarity reductions of the dispersive water wave hierarchy

We now consider once again the relationship between the derivation of Painlevé hierarchies using nonisospectralscattering problems and using similarity reductions; we take as examples the extensions to the DWW hierarchy of the

P.R. Gordoa et al. / Applied Mathematics and Computation 237 (2014) 77–84 83

accelerating-wave reduction and generalized scaling reduction of the DWW equation, given in [29] and [31] respectively.Similarly to the KdV case, it is the change of variables (3.12) that provides an understanding of this relationship, and allowsus to explain why the same Painlevé hierarchies are found using these two techniques.

We consider two special cases of (3.9) such that the stationary reduction (3.10) gives rise to a PII hierarchy or a PIV hier-archy. The two cases that we consider have all BkðsÞ constant, and in addition gnðsÞ ¼ 0 and gnþ1ðsÞ constant, and gnþ1ðsÞ ¼ 0and gnðsÞ constant, respectively, and are both nonisospectral equations.

Corollary 3. For the special case of Eq. (3.11) and transformation (3.12) having gðtÞ ¼ 1;mðtÞ ¼ �gnþ1t ðgnþ1 constantÞ, kðtÞ ¼ tand hðtÞ and all ckðtÞ such that all BkðtÞ, k ¼ 0;1; . . . n� 1 are constant, the corresponding equivalent nonisospectral equation is

ut ¼ Rn½u�un þXn�1

j¼0

BjRj½u�un þ gnþ110

� �: ð3:24Þ

Proof. A simple substitution in Proposition 3. We note in particular that since gðtÞ is constant, gnðtÞ ¼ 0 (see (3.17)).

Remark 3. We make similar remarks to those made in the KdV case. The stationary reduction of (3.24) yields (3.10) withgn ¼ 0,

Rn½u�un þXn�1

j¼0

BjRj½u�un þ gnþ110

� �¼

00

� �ð3:25Þ

and the substitution used to obtain this last from the corresponding case of (3.11), that is, the special case of the transfor-mation (3.12) given in Corollary 3 but now with u and v functions of n only, then corresponds precisely to the similarityreduction of this case of (3.11) given in [29] which yields a PII hierarchy (this last being obtained by integrating each equationof (3.25) to obtain for example (3.7) in [29]). Thus the change of variables of Corollary 3 explains why the stationary reduc-tion of the nonisospectral hierarchy (3.24) can also be obtained as a similarity reduction of the corresponding case of thestandard hierarchy (3.11). Both techniques then yield the same Painlevé hierarchy. Again we note that these similarity solu-tions of this case of the hierarchy (3.11) correspond to time-independent solutions of the equivalent hierarchy (3.24).

Corollary 4. For the special case of Eq. (3.11) and transformation (3.12) having gðtÞ ¼ 1= 12 ðnþ 1Þgnt� �1=ðnþ1Þ ðgn constantÞ,

mðtÞ ¼ d ðconstantÞ, kðtÞ ¼ logðtÞ= 12 ðnþ 1Þgn

� �and hðtÞ and all ckðtÞ such that all BkðsÞ; k ¼ 0;1; . . . n� 1 are constant, the cor-

responding equivalent nonisospectral equation is

us ¼ Rn½u�un þXn�1

j¼0

BjRj½u�un þ12

gn

ðnuÞn2v þ nvn

� �: ð3:26Þ

Proof. A simple substitution in Proposition 3. We note in particular that since mðtÞ is constant, gnþ1ðsÞ ¼ 0 (see (3.18)).

Remark 4. Again we make remarks similar to those made in the KdV case. The stationary reduction of (3.26) yields (3.10)with gnþ1 ¼ 0,

Rn½u�un þXn�1

j¼0

BjRj½u�un þ12

gn

ðnuÞn2v þ nvn

� �¼

00

� �ð3:27Þ

and the substitution used to obtain this last from the corresponding case of (3.11), that is, the special case of the transfor-mation (3.12) given in Corollary 4 but now with u and v functions of n only, then corresponds precisely to the generalizedscaling reduction of this case of (3.11) given in [31] which yields a PIV hierarchy, this last being obtained as a first integral of(3.27) (and given for example as (3.12) and (3.13) in [31]). That is, the change of variables of Corollary 4, which involves scal-ing similarity variables, explains why the stationary reduction of the nonisospectral hierarchy (3.26) can also be obtained asa generalized scaling reduction of the corresponding case of the standard hierarchy (3.11). Both techniques then yield thesame Painlevé hierarchy. It is interesting to note that scaling similarity solutions of this case of the hierarchy (3.11) corre-spond to time-independent solutions of the equivalent hierarchy (3.26).

4. Conclusions

We have shown that certain 1þ 1-dimensional nonisospectral hierarchies and their scattering problems can be mappedonto standard PDE hierarchies and their isospectral scattering problems. Special cases of these transformations involve sim-ilarity variables, and time-independent solutions of corresponding nonisospectral hierarchies are then seen to correspond to

84 P.R. Gordoa et al. / Applied Mathematics and Computation 237 (2014) 77–84

similarity reductions of standard hierarchies. This explains why the use of nonisospectral scattering problems and similarityreductions yield the same Painlevé hierarchies. For example, in the KdV case, we have seen why the ODE hierarchies (2.23)and (2.25), which may be derived as stationary reductions of nonisospectral hierarchies and which yield a generalized PI

hierarchy and a generalized P34 hierarchy respectively, are precisely those obtained in [29] and [31] by respectively extend-ing the accelerating-wave and generalized scaling reductions of the KdV equation to corresponding standard (isospectral)hierarchies. Similarly, in the DWW case, we have seen why the ODE hierarchies (3.25) and (3.27), which may be derivedas stationary reductions of nonisospectral hierarchies and which yield a PII hierarchy and a PIV hierarchy respectively, areprecisely those obtained in [29] and [31] by respectively extending the accelerating-wave and generalized scaling reductionsof the DWW equation to corresponding standard (isospectral) hierarchies. Further examples of mappings between noniso-spectral and isospectral scattering problems, and of special cases of such transformations related to similarity reduction, willbe considered in future papers.

Acknowledgements

The work of PRG and AP is supported in part by the Ministry of Economy and Competitiveness of Spain under contractMTM2012-37070. They are grateful to JADW for hosting their visit to the University of Nottingham in July 2013.

References

[1] M.J. Ablowitz, H. Segur, Exact linearization of a Painlevé transcendent, Phys. Rev. Lett. 38 (1977) 1103–1106.[2] H. Airault, Rational solutions of Painlevé equations, Stud. Appl. Math. 61 (1979) 31–53.[3] N.A. Kudryashov, The first and second Painlevé equations of higher order and some relations between them, Phys. Lett. A 224 (1997) 353–360.[4] A.N.W. Hone, Non-autonomous Hénon–Heiles systems, Phys. D 118 (1998) 1–16.[5] P.R. Gordoa, A. Pickering, Nonisospectral scattering problems: a key to integrable hierarchies, J. Math. Phys. 40 (1999) 5749–5786.[6] C. Cresswell, N. Joshi, The discrete first, second and thirty-fourth Painlevé hierarchies, J. Phys. A 32 (1999) 655–669.[7] P.R. Gordoa, A. Pickering, On a new non-isospectral variant of the Boussinesq hierarchy, J. Phys. A 33 (2000) 557–567.[8] P.R. Gordoa, N. Joshi, A. Pickering, On a generalized 2þ 1 dispersive water wave hierarchy, Publ. Res. Inst. Math. Sci. (Kyoto) 37 (2001) 327–347.[9] A. Pickering, Coalescence limits for higher order Painlevé equations, Phys. Lett. A 301 (2002) 275–280.

[10] A. Pickering, Painlevé hierarchies and the Painlevé test, Teoret. i Matem. Fizika 137 (2003) 445–456 (Theoret. and Math. Phys. 137 (2003) 1733–1742).[11] N.A. Kudryashov, Amalgamations of Painlevé equations, J. Math. Phys. 44 (12) (2003) 6160–6178.[12] N.A. Kudryashov, On the fourth Painlevé hierarchy, Theor. Math. Phys. 134 (2003) 86–93.[13] T. Kawai, T. Koike, Y. Nishikawa, Y. Takei, On the Stokes geometry of higher order Painlevé equations, Analyse complexe, systèmes dynamiques,

sommabilité des séries divergentes et théories galoisiennes II, Astérisque No. 297 (2004) 117–166.[14] P.R. Gordoa, A. Pickering, Z.N. Zhu, A non-isospectral extension of the Volterra hierarchy to 2þ 1 dimensions, J. Math. Phys. 46 (2005) 103509.[15] P.R. Gordoa, A. Pickering, Z.N. Zhu, Non-isospectral lattice hierarchies in 2þ 1 dimensions and generalized discrete Painlevé hierarchies, J. Nonlinear

Math. Phys. 12 (Suppl. 2) (2005) 180–196.[16] P.R. Gordoa, N. Joshi, A. Pickering, Bäcklund transformations for fourth Painlevé hierarchies, J. Differ. Equ. 217 (2005) 124–153.[17] P.R. Gordoa, N. Joshi, A. Pickering, Second and fourth Painlevé hierarchies and Jimbo–Miwa linear problems, J. Math. Phys. 47 (2006) 073504.[18] P.R. Gordoa, A. Pickering, Z.N. Zhu, A 2þ 1 non-isospectral integrable lattice hierarchy related to a generalized discrete second Painleve hierarchy,

Chaos Solitons Fract. 29 (2006) 862–870.[19] P.R. Gordoa, A. Pickering, Z.N. Zhu, New 2þ 1 dimensional nonisospectral Toda lattice hierarchy, J. Math. Phys. 48 (2007) 023515.[20] Z.N. Zhu, A (2+1)-dimensional nonisospectral relativistic Toda hierarchy related to the generalized discrete Painlevé hierarchy, J. Phys. A 40 (2007)

7707–7719.[21] T. Koike, On the Hamiltonian structures of the second and the fourth Painlevé hierarchies, and the degenerate Garnier systems. Algebraic, analytic and

geometric aspects of complex differential equations and their deformations. Painlevé hierarchies, 99–127, RIMS Kôkyûroku Bessatsu, B2, Res. Inst.Math. Sci. (RIMS), Kyoto, 2007.

[22] A.H. Sakka, Schlesinger transformations for the second members of PII and PIV hierarchies, J. Phys. A 40 (2007) 7687–7697.[23] A.H. Sakka, On the special solutions of second and fourth Painlevé hierachies, Phys. Lett. A 373 (2009) 611–615.[24] P.R. Gordoa, A. Pickering, A method of reduction of order for discrete systems, J. Math. Phys. 50 (2009) 053513.[25] U. Mugan, A. Pickering, The Cauchy problem for the second member of a PIV hierarchy, J. Phys. A 42 (2009) 085203.[26] A. Pickering, On the nesting of Painlevé hierarchies: a Hamiltonian approach, Chaos Solitons Fract. 45 (2012) 935–941.[27] J.M. Conde, P.R. Gordoa, A. Pickering, Bäcklund transformations for new fourth Painlevé hierarchies, Commun. Nonlinear Sci. Numer. Simul. 19 (2014)

448–458.[28] D. Levi, O. Ragnisco, M.A. Rodríguez, On non-isospectral flows, Painlevé equations and symmetries of differential and difference equations, Teoret. i

Matem. Fizika 93 (1992) 473–480 (Theor. Math. Phys. 93 (1993) 1409–1414).[29] P.R. Gordoa, A. Pickering, A new derivation of Painlevé hierarchies, Appl. Math. Comput. 218 (2011) 3942–3949.[30] D. Levi, O. Ragnisco, Non-isospectral deformations and Darboux transformations for the third order spectral problem, Inverse Prob. 4 (1988) 815–828.[31] P.R. Gordoa, U. Mugan, A. Pickering, Generalized scaling reductions and Painlevé hierarchies, Appl. Math. Comput. 219 (2013) 8104–8111.[32] R.O. Popovych, O.O. Vaneeva, More common errors in finding exact solutions of nonlinear differential equations: part I, Commun. Nonlinear Sci.

Numer. Simul. 15 (2010) 3887–3899.[33] V. Listopadova, O. Magda, V. Pobyzh, How to find solutions, Lie symmetries and conservation laws of forced Korteweg–de Vries equations in optimal

way, Nonlinear Anal. Real World Appl. 14 (2013) 202–205.[34] A. Pickering, P.R. Gordoa, J.A.D. Wattis, Behaviour of the extended Volterra lattice, Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 589–600.[35] B.A. Kupershmidt, Mathematics of dispersive water waves, Commun. Math. Phys. 99 (1985) 51–73.