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Research ArticleSecond-Order Conditional Lie-Bäcklund Symmetry andDifferential Constraint of Radially Symmetric Diffusion System

Jianping Wang, Huijing Ba, Yaru Liu, Longqi He, and Lina Ji

Department of Information and Computational Science, He’nan Agricultural University, Zhengzhou 450002, China

Correspondence should be addressed to Lina Ji; [email protected]

Received 11 September 2020; Revised 7 December 2020; Accepted 5 January 2021; Published 18 January 2021

Academic Editor: Wen-Xiu Ma

Copyright © 2021 Jianping Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The classifications and reductions of radially symmetric diffusion system are studied due to the conditional Lie-Bäcklund symmetrymethod. We obtain the invariant condition, which is the so-called determining system and under which the radially symmetricdiffusion system admits second-order conditional Lie-Bäcklund symmetries. The governing systems and the admittedsecond-order conditional Lie-Bäcklund symmetries are identified by solving the nonlinear determining system. Exactsolutions of the resulting systems are constructed due to the compatibility of the original system and the admitteddifferential constraint corresponding to the invariant surface condition. For most of the cases, they are reduced to solvingfour-dimensional dynamical systems.

1. Introduction

In the latter part of the 19th century, Lie [1] introduced thenotion of continuous groups, now known as Lie groups andclassical symmetry, which at once unified a variety of specialtechniques designed to solve the ordinary differential equa-tion (ODE) including the integrating factor, reduction oforder, undetermined coefficients, Laplace transform, etc.These special methods were, in fact, all special cases of ageneral integration procedure based on the invariance ofthe differential equation (DE) under a continuous group ofsymmetry.

To date, various generalizations to the concept of sym-metry groups for nonlinear PDEs have been proposed.Noether [2] introduced the notion of the Lie-Bäcklundsymmetry. Ovsiannikov [3] developed the method of par-tially invariant solutions. Bluman and Cole [4] suggestedthe so-called nonclassical method. Fushchych et al. [5]presented the conditional symmetry, which is an extensionof the nonclassical method. Olver and Rosenau [6, 7] gen-eralized the nonclassical method to the weak symmetrymethod and the side condition method. Nucci [8] gavethe iteration of the nonclassical method. Fokas and Liu[9] and Zhdanov [10] independently carried out the con-

ditional Lie-Bäcklund symmetry (CLBS). There are a lotof symmetry-related methods such as the direct and mod-ified direct methods [11, 12], the ansatz-based method[13–15], the sign-invariant method [16–18], and theinvariant subspace method [19, 20].

The merging of the extension ideas of Lie-Bäcklundsymmetry and conditional symmetry was independentlycarried out by Fokas and Liu [9] and Zhdanov [10]. In[9], Fokas and Liu introduced the notion of generalizedconditional symmetry and constructed multishock andmultisoliton solutions of certain nonintegrable equations.The reduction theorem of conditional Lie-Bäcklund sym-metry (CLBS) which ensures that a PDE will be reducedto a system of ODEs was presented by Zhdanov in [10],where the corresponding symmetry reductions for thenonlinear heat conductivity equation were also displayed.Moreover, one-to-one correspondence relation betweenreducibility of a given evolution equation to a system ofODEs and its CLBS was established by Zhdanov in [21].In fact, generalized conditional symmetry [9], CLBS [10],and higher conditional symmetry [21] are always thesame.

It is proven that CLBS are very effective to construct exactsolutions and find symmetry reductions for different types of

HindawiAdvances in Mathematical PhysicsVolume 2021, Article ID 8891750, 17 pageshttps://doi.org/10.1155/2021/8891750

https://orcid.org/0000-0001-9250-1854https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2021/8891750

nonlinear diffusion equations [22–32]. CLBSs and symmetryreductions of nonintegrable equations can be referred to in[9, 33–35]. The reduction of initial-value problem for evolu-tion equations to Cauchy problem for system of ODEs whichcan be fully characterized in terms of CLBS is shown byZhdanov et al. in [21, 36–38]. The studies of CLBS for theevolution system are discussed by Andreytsev in [39] andby Sergyeyev in [40, 41], where the complete description ofthe evolution system which admits a given CLBS is provided.The reduction theorem for CLBS of the evolution system isconstructed in [42], which generalizes the one for scalarevolution equation [10]. The studies of classifications andreductions for the two-component diffusion system can bereferred to in [42–44]

Olver [45] showed that the invariant surface conditioncorresponding to CLBS can be regarded as a differential con-straint (DC) compatible with the initial equation within theframework of empiric compatibility theory. The term “condi-tional” is explained by the fact that attaching an additionaldifferential equation called DC to the original PDE and theDC is right with the invariant surface equation correspond-ing to the admitted symmetry. In fact, conditional invariancecriterion is nothing but a compatibility condition of thecombined system including original PDE and additionalinvariant surface equation. The relations between conditionalsymmetry, reduction, and compatibility of the combined sys-tem were discussed in [45, 46]. The related papers addressingthis problem include [33, 47, 48]. CLBS can be reformulatedwithin the framework of the DC method. Olver [6, 7, 45],Kaptsov [49], Levi and Winternitz [50], and Pucci andSaccomandi [51] conclude that many reduction methodssuch as conditional symmetry, partial invariance, variableof separation, and direct method can be understood by usingthe technicalities of the method of DC. The method of CLBSprovides an appropriate symmetry background for themethod of DC. The base of symmetry reduction for CLBSis the fact that the corresponding invariant surface conditionis formally compatible with the governing system, which isextensively discussed in [48], where it is shown that the prob-lem of discussing the DC of the evolution system is equiva-lent to studying the CLBS of this system. The equivalencerelation between CLBS, DC, and direct reduction is also dis-cussed in our recent paper [43]

The procedure for determining whether or not a givenLie-Bäcklund vector field is conditionally invariant of theconsidered equation is straightforward; however, the deter-mination of the most general CLBSs admitted by a givenPDE is a very difficult, if not impossible, problem since theassociated determining system is an overdetermined nonlin-ear system of PDEs. Nevertheless, as is known, even findingparticular CLBSs can lead to new explicit solutions of theconsidered equation. In practice, the principle direction ofsuch research is to content oneself with finding CLBS inparticular cases, and these cases must be chosen using addi-tional considerations. It has been proven that CLBSs relatedwith invariant subspaces [22–24], sign-invariants [25–30],and separation of variables [31, 32] are very effective to studythe classifications and reductions of second-order nonlineardiffusion equations.

In this paper, we will study the second-order CLBS withthe characteristic

η1 = urr +H uð Þu2r +G r, uð Þur + F r, uð Þ,η2 = vrr + L vð Þv2r +M r, vð Þv1 +N r, vð Þ,

(ð1Þ

of a nonlinear radially symmetric diffusion system

ut =1

rn−1rn−1ukur

� �r+ P u, vð Þ,

vt =1

rn−1rn−1vlvr

� �r+Q u, vð Þ,

8>><>>: ð2Þ

which is equivalent to studying the second-order DC

η1 = urr +H uð Þu2r +G r, uð Þur + F r, uð Þ = 0,η2 = vrr + L vð Þv2r +M r, vð Þv1 +N r, vð Þ = 0,

(ð3Þ

of Equation (2). It is noted that k and l in (2) are botharbitrary real constants. The form of CLBS (1) generalizesthe one for a nonlinear diffusion system in [43], and thecorresponding second-order CLBS for the scalar diffusionequation can provide symmetry interpretation for first-order Hamilton-Jacobi sign-invariant for the consideredequation [25–30]. The discussion about second-order CLBS(1) of system (2) for the case of n = 1 is referred to [43].

The layout of this paper is listed here. Section 2 is devotedto necessary definitions and notations about CLBS and DC ofevolution system. Second-order CLBS (1) and DC (3) of sys-tem (2) are displayed in Section 3. Exact solutions of system(2) are constructed in Section 4. Conclusions and remarksare given in the last section.

2. Preliminaries

The one-parameter Lie-Bäcklund group of infinitesimaltransformations

~u ið Þ = u ið Þ + εη ið Þ r, t, u ið Þ, u ið Þ1 ,⋯,uið Þli

� �+O ε2

� �,

~u ið Þt = u ið Þ + εDtη ið Þ r, t, u ið Þ, uið Þ1 ,⋯,u

ið Þli

� �+O ε2

� �,

~u ið Þr = u ið Þ + εDrη ið Þ r, t, u ið Þ, uið Þ1 ,⋯,u

ið Þli

� �+O ε2

� �,

⋯ð4Þ

is generated by the Lie-Bäcklund vector field (LBVF)

V = 〠m

i=1η ið Þ r, t, u ið Þ, u ið Þ1 ,⋯,u

ið Þli

� � ∂∂u ið Þ

"

+Drη ið Þ r, t, u ið Þ, uið Þ1 ,⋯,u

ið Þli

� � ∂∂u ið Þ1

+Dtη ið Þ r, t, u ið Þ, uið Þ1 ,⋯,u

ið Þli

� � ∂∂u ið Þt

+D2rη ið Þ r, t, u ið Þ, uið Þ1 ,⋯,u

ið Þli

� � ∂∂u ið Þ2

+⋯#:

ð5Þ

2 Advances in Mathematical Physics

Definition 1 (see [3]). The evolutionary vector field (5) is saidto be a Lie-Bäcklund symmetry of the evolution system

u ið Þt = F ið Þ t, r, u 1ð Þ, u1ð Þ1 ,⋯,u

1ð Þk1,⋯,u mð Þ, u mð Þ1 ,⋯,u

mð Þkm

� �,

i = 1, 2,⋯,m,ð6Þ

if

V u ið Þt − F ið Þ� ��

M= 0, i = 1, 2,⋯; ;m, ð7Þ

where M denotes the set of all differential consequences ofthe system (6).

Definition 2 (see [9, 10]). The evolutionary vector field (5) issaid to be a CLBS of (6) if

V u ið Þt − F ið Þ� ��

M∩Lr= 0, i = 1, 2,⋯,m, ð8Þ

where Lr denotes the set of all differential consequences ofinvariant surface condition

η ið Þ r, t, u ið Þ, u ið Þ1 ,⋯,uið Þli

� �= 0, i = 1, 2,⋯,m, ð9Þ

with respect to r.In fact, the invariant criterion (8) is reduced to

V u ið Þt − F ið Þ� �

M∩Lr= Dtη ið Þ − 〠

m

l=1〠kl

j=1Djxη

lð ÞF ið Þu lð Þj

" #M∩Lx

=Dtη ið Þ��M∩Lr

= 0:ð10Þ

The fact that LBVF (5) is a CLBS of system (1) leads to thecompatibility of the invariant surface condition (9) and thegoverning system (1). For evolution system (1), the invariantcriterion (10) is exactly the sufficient condition which is usedto construct the DC of (1).

Definition 3 (see [52]). The differential constraints (9) andthe evolution system (1) satisfy the compatibility condi-tion if

Dtηið Þ��Mr∩Lr

= 0, i = 1, 2,⋯,m, ð11Þ

where Mr denotes the set of all differential consequencesof the system (6) with respect to r.

The calculation of CLBS admitted can be divided intofour steps. Firstly, we need to compute total derivatives ofηðiÞ = 0 with respect to t for i = 1, 2,⋯,m, which yields

ηið Þt + η

ið Þu ið Þu

ið Þt + η

ið Þu ið Þ1u ið Þ1t +⋯+η

ið Þu ið Þliu ið Þlit = 0: ð12Þ

It is noted that the subscripts denote partial derivativeswith respect to the indicated variables. The next step is to

eliminate all derivatives uðiÞjt by using differential conse-

quences DjrðuðiÞt − FðiÞÞ = 0, which implies

ηið Þt + η

ið Þu ið ÞF

ið Þ + η ið Þu ið Þ1DxF

ið Þ+⋯+η ið Þu ið ÞliDlix F

ið Þ = 0: ð13Þ

In fact, the left side of (13) can be simplified as a polyno-

mial about uðiÞ1 , uðiÞ2 ,⋯, u

ðiÞli−1 by substituting all higher-order

derivatives of uðiÞ with respect to r by lower-order ones dueto differential consequences Djrη

ðiÞ = 0 for j = 0, 1, 2,⋯. Con-sequently, equating the coefficients of resulting polynomialsto zero will yield the so-called determining system for theundetermined parts in the governing system (6) and thecorresponding characteristic system (9). Finally, solving thissystem leads to the form of CLBS (9) admitted by the system(6). In general, solving the resulting nonlinear determiningsystem is as difficult as solving the original evolution system.However, even finding one particular CLBS admitted by theconsidered evolution system (6) will lead to the correspond-ing reductions of the original system. Thus, we will find par-ticular solutions of the overdetermined system to determinethe CLBS admitted by the system (6), which are proved tobe very powerful to study classifications and reductions ofscalar nonlinear diffusion equations [22–32].

In full analog with the classical symmetry reductionmethod, conditional symmetries can also be used to performdimensional reductions of the governing system. The invari-ant solutions are defined by the characteristic system (9). Thegeneral solution to system (9) can be (locally) written as

u ið Þ = f ið Þ t, r, ϕ ið Þ1 tð Þ, ϕ ið Þ2 tð Þ,⋯,ϕ ið Þli tð Þ� �

, ð14Þ

where ϕðiÞj ðtÞ are arbitrary smooth functions. ηðiÞ in (9)depends only on uðiÞ and its higher-order derivatives, so theansatz uðiÞ in (14) is independent with each other. Conse-quently, the studies of direct reductions, DCs and CLBSsfor evolution system are about the same as that for scalar evo-lution equation. Thus, the equivalence relation [21, 45, 48]between direct reductions, DCs and CLBSs for scalar evolu-tion equation holds true for the case of evolution system. Itis concluded that reducibility of the evolution system (1) tothe ODE system due to the ansatz (14), CLBS (5) and DC(9) of system (1) is, in some sense, equivalent [42, 43].

3Advances in Mathematical Physics

3. CLBS (1) of System (2)

Since the diffusion system (2) admits of CLBS with the char-acteristic (1), the invariant criterion (10) can be simplified as

Dtη1jM∩Lr = −u3H ′′ + 4uH − 3kð Þu2H ′ + 4k 1 − kð ÞuH

h+ 5ku2H2 − 2u3H3 + k k − 1ð Þ k − 2ð Þ

iuk−3u4r

+ −u2Guu + 2uH − 3kð ÞuGu�

+ 4u2H ′ + 9kuH − 4u2H2 + 5k − 5k2� �

G

+ k n − 1ð Þr

k − 1 − uHð Þ�uk−2u3r

+ Puu +H ′P +HPu

− uk−2 u2Fuu + 3kuFu − 4u2H ′F�

− 2 4kuH − 2u2H2 + 3k − 3k2� �

F

+ 2 2k − uHð Þu Gr −G2� �

− 2u2GGu + 2u2Gru

+ 2k n − 1ð Þur

G + 1r

��u2r

+ 2 Puv +HPvð Þurvr + Pvv − PvLð Þv2r+ G −Mð ÞPvvr+ PGu + k 7FG − 4Fr −

3 n − 1ð Þr

F� �

uk−1

+ 2FGu + 2GGr −Grr − 2Fru − 4HFG�

+ n − 1r

2r2

+ 1rG −Gr

��uk�ur + FuP − PuF

− PvN + 3kuk−1F2

+ 2 FGr −HF2� �

− Frr +n − 1r

2rF − Fr

�� uk

= 0,

Dtη2jM∩Lr = −v3L′′ + 4vL − 3lð Þv2L′ + 4l 1 − lð ÞvL + 5lv2L2

h− 2v3L3 − v3L′′ + 4vL − 3lð Þv2L′ + 4l 1 − lð ÞvL+ 5lv2L2 − 2v3L3 + l l − 1ð Þ l − 2ð Þ

ivl−3v4r

+ −v2Mvv + 2vL − 3lð ÞvMv�

+ 4v2L′ + 9lvL − 4v2L2 + 5l − 5l2� �

M

+ l n − 1ð Þr

l − 1 − vLð Þ�vl−2v3r

+ Qvv +QL′ + LQv − vl−2 v2Nvv + 3lvNv�

− 4v2L′N − 2 4lvL − 2v2L2 + 3l − 3l2� �

N

+ 2 2l − vLð Þv Mr −M2� �

− 2v2MMv + 2v2Mrv

+ 2l n − 1ð Þvr

M + 1r

��v2r + 2 Quv + LQuð Þurvr

+ Quu −QuHð Þu2r + M − Gð ÞQuur

+ QMv + l 7MN − 4Nr −3 n − 1ð Þ

rN

� �vl−1

+ 2NMv + 2MMr −Mrr − 2Nrv − 4LMN�

+ n − 1r

2r2

+ 1rM −Mr

��vl�vr +NvQ −QuF

−QvN + 3lvl−1N2 + 2 NMr − LN2� �

−Nrr

�

+ n − 1r

2rN −Nr

��vl = 0: ð15Þ

The vanishing of all the coefficients of the two polyno-mials about ur and vr yields the nonlinear system of deter-mining equations

−u3H ′′ + 4uH − 3kð Þu2H ′ + 4k 1 − kð ÞuH + 5ku2H2− 2u3H3 + k k − 1ð Þ k − 2ð Þ = 0,

−u2Guu + 2uH − 3kð ÞuGu + 4u2H ′ + 9kuH − 4u2H2�

+ 5k − 5k2�G + k n − 1ð Þ

rk − 1 − uHð Þ = 0,

Puu +H ′P +HPu − uk−2 u2Fuu + 3kuFu − 4u2H ′F�

− 2 4kuH − 2u2H2 + 3k − 3k2� �

F + 2 2k − uHð Þu Gr −G2� �

− 2u2GGu + 2u2Gru +2k n − 1ð Þu

rG + 1

r

��= 0,

Puv +HPv = 0, Pvv − PvL = 0, G −Mð ÞPv = 0, PGu

+ k 7FG − 4Fr −3 n − 1ð Þ

rF

� �uk−1

+ 2FGu + 2GGr −Grr − 2Fru − 4HFG�

+ n − 1r

2r2

+ 1rG −Gr

��uk = 0,

FuP − PuF − PvN + 3kuk−1F2 + 2 FGr −HF2� �

− Frr

�

+ n − 1r

2rF − Fr

��uk = 0 − v3L′′ + 4vL − 3lð Þv2L′

+ 4l 1 − lð ÞvL + 5lv2L2 − 2v3L3 + l l − 1ð Þ l − 2ð Þ = 0,


−v2Mvv + 2vL − 3lð ÞvMv

+ 4v2L′ + 9lvL − 4v2L2 + 5l − 5l2� �

M

+ l n − 1ð Þr

l − 1 − vLð Þ = 0,

Qvv +QL′ + LQv − vl−2 v2Nvv + 3lvNv − 4v2L′N�

− 2 4lvL − 2v2L2 + 3l − 3l2� �

N + 2 2l − vLð Þv Mr −M2� �

− 2v2MMv + 2v2Mrv +2l n − 1ð Þv

rM + 1

r

��= 0,

Quv + LQu = 0,Quu −QuH = 0, M −Gð ÞQu = 0,QMv

+ l 7MN − 4Nr −3 n − 1ð Þ

rN

� �vl−1

+ 2NMv + 2MMr −Mrr − 2Nrv − 4LMN�

+ n − 1r

2r2

+ 1rM −Mr

��vl = 0,NvQ −QuF −QvN

+ 3lvl−1N2 + 2 NMr − LN2� �

−Nrr

�

+ n − 1r

2rN −Nr

��vl = 0: ð16Þ

It is impossible to present the general solutions of thenonlinear determining system since system (16) is a couplednonlinear system of PDEs. The workable way is to constructparticular solutions of the nonlinear determining system(16). The determined CLBS (1) will lead to symmetry reduc-tions of the governing system (2).

Setting HðuÞ = h/u into the first one of system (16), wecan obtain the algebraic equation

h − kð Þ h − k + 1ð Þ h − k2 + 1

�

= 0: ð17Þ

Thus, we can list three special solutions

H uð Þ = ku, k − 1

u, k/2 − 1

u: ð18Þ

The similar procedure gives

L vð Þ = lv, l − 1

v, l/2 − 1

v: ð19Þ

We just consider six different cases

H uð Þ = ku, L vð Þ = l

v,

H uð Þ = ku, L vð Þ = l − 1

v,

H uð Þ = ku, L vð Þ = l/2 − 1

v,

H uð Þ = k − 1u

, L vð Þ = l − 1v

,

H uð Þ = k − 1u

, L vð Þ = l/2 − 1v

,

H uð Þ = k/2 − 1u

, L vð Þ = l/2 − 1v

,

ð20Þ

for further study due to the symmetrical form of system (2)and the admitted CLBS (1).

Since system (2) is a coupled one, the sixth one and thefourteenth one of the determining system (16) implies

M r, uð Þ = G r, vð Þ = A rð Þ: ð21Þ

As a consequence, AðrÞ = ðn − 1Þ/r is finally determineddue to the second one and the tenth one of system (16) forcase ðiÞ. Substituting HðuÞ = k/u and LðvÞ = l/v into the fifthone and the thirteenth one of the determining system (16),we get Pðu, vÞ and Qðu, vÞ, respectively, which satisfy thelinear differential equation

Pvv −lvPv = 0,

Quu −kuQu = 0:

ð22Þ

Solving the differential equation about Pðu, vÞ andQðu, vÞ, we know that three subcases

P u, vð Þ = P1 uð Þ + P2 uð Þv1+l,Q u, vð Þ =Q1 vð Þ+Q2 vð Þu1+k k ≠ −1, l ≠ −1ð Þ,

P u, vð Þ = P1 uð Þ + P2 uð Þv1+l,Q u, vð Þ =Q1 vð Þ+Q2 vð Þ ln uð Þ k = −1, l ≠ −1ð Þ,

P u, vð Þ = P1 uð Þ + P2 uð Þ ln vð Þ,Q u, vð Þ =Q1 vð Þ+Q2 vð Þ ln uð Þ k = −1, l = −1ð Þ,

ð23Þ

will be considered. For the case of k ≠ −1, l ≠ −1, P2ðuÞ =p1u

−k and Q2ðvÞ = p2v−l can be obtained by solving thefourth one and the twelfth one of system (16). Then, we


Table1:CLB

S(1)of

RDsystem

(2).

No.

RDsystem

(2)

CLB

S(1)

1u t

=1

rn−1

rn−1uku r

�� r+

a 1u+b 1u−

k+p 1u−

k v1+

l ,

v t=

1rn

−1rn

−1vlv r

�� r+

a 2v+b 2v−

l+p 2v−

l u1+

k :

8 < :η 1

=u r

r+

k uu2 r

+n−1

ru r,

η 2=v r

r+

l vv2 r

+n−1

rv r:

(

2u t

=1

rn−1

rn−1uku r

�� r+

a 1u−

k−

1+l

ðÞp 1

s 21+k

ðÞs 1

u+s 1+p 1u−

k v1+

l ,

v t=

1rn

−1rn

−1vlv r

�� r+

a 2v−

l−

1+k

ðÞp 2

s 11+l

ðÞs 2

v+s 2+p 2v−

l u1+

k :

8 < :η 1

=u r

r+

k uu2 r

+n−1

ru r

+s 1u−

k ,

η 2=v r

r+

l vv2 r

+n−1

rv r+s 2v−

l :

(

3u t

=1

rn−1

rn−1uku r

�� r+

a 1u+

1+k

ðÞa

1s1+

1+l

ðÞp 1

s 31+l

ðÞs 2

u−k+

1+l

ðÞs 2

1+ku1

+k+s 1+p 1u−

k v1+

l ,

v t=

1rn

−1rn

−1vlv r

�� r+

a 2v+

1+k

ðÞp 2

s 1+

1+l

ðÞa

2s3

1+l

ðÞs 2

v−l+s 2v1

+l+s 3+p 2v−

l u1+

k :

8 < :η 1

=u r

r+

k uu2 r

+n−1

ru r

+1+l

ðÞs 2

1+ku+s 1u−

k ,

η 2=v r

r+

l vv2 r

+n−1

rv r+s 2v+s 3v−

l :

(

4u t

=1

rn−1

rn−1u−

1/2 u

r

�� r+

a 1u+b 1

ffiffiffiffiffi uvp,

v t=

1rn

−1rn

−1v−

1/2 v

r

�� r+

a 2v+b 2

ffiffiffiffiffi uvp:

(η 1

=u r

r−

1 2uu2 r

+n−1

ru r

+4n−4

ðÞ

r2u,

η 2=v r

r−

1 2vv2 r

+n−1

rv r+

4n−4

ðÞ

r2v:

8 < :

5u t

=1

rn−1

rn−1u−

1/2 u

r

�� r+

a 1u+s 1+2s

3ffiffiffi up+

s2 3−a 1s 1

s 2

ffiffiffiffiffi uvp,

v t=

1rn

−1rn

−1v−

1/2 v

r

�� r+

a 2v+s 2+2s

3ffiffiffi vp +

s2 3−a 2s 2

s 1

ffiffiffiffiffi uvp:

8 < :η 1

=u r

r−

1 2uu2 r

+n−1

ru r

+s 3u+s 1

ffiffiffi up+

4n−4

ðÞ

r2u,

η 2=v r

r−

1 2vv2 r

+n−1

rv r+s 3v+s 2

ffiffiffi vp +4n−4

ðÞ

r2v:

8 < :

6u t

=1

rn−1

rn−1u−

1 ur

�� r+

a 1u+b 1uln

u ðÞ+

p 1uv

1+l ,

v t=

1rn

−1rn

−1vlv r

�� r+

a 2v+b 2v−

l+p 2v−

lln

u ðÞ:

8 < :η 1

=u r

r−

1 uu2 r

+n−1

ru r,

η 2=v r

r+

l vv2 r

+n−1

rv r:

(

7u t

=1

rn−1

rn−1u−

1 ur

�� r+

a 1uln

u ðÞ+

1+l

ðÞp 1

s 1s 2

u+s 2ln

u ðÞ+

p 1uv

1+l ,

v t=

1 rn−1

rn−1vlv r

�� r+

a 2v+

1+l

ðÞa

2s1

s 2v−

l+

s 2 1+lv

1+l+s 1+p 2v−

l u1+

k :

8 < :η 1

=u r

r−

1 uu2 r

+n−1

ru r

+s 2uln

u ðÞ,

η 2=v r

r+

l vv2 r

+n−1

rv r+

s 2 1+lv

+s 1v−

l :

(

8u t

=1

rn−1

rn−1u−

1 ur

�� r+

a 1u−

1+l

ðÞp 1

s 1−a 1s 2

s 2uln

u ðÞ+

s 2ln

u ðÞ+

s 3+p 1uv

1+l ,

v t=

1 rn−1

rn−1vlv r

�� r+

a 2v+p 2v−

l+

s 2 1+lv

1+l+s 1−

1+l

ðÞa

2s1−p 2s 2

s 3v−

l :

8 < :η 1

=u r

r−

1 uu2 r

+n−1

ru r

+s 2uln

u ðÞ+

s 3u,

η 2=v r

r+

l vv2 r

+n−1

rv r+

s 2 1+lv

+s 1v−

l :

(

u t=

1rn

−1rn

−1u−

1 ur

�� r+

a 1u+b 1uln

u ðÞ+

p 1uln

v ðÞ,

v t=

1rn

−1rn

−1v−

1 vr

�� r+

a 2v+b 2vln

v ðÞ+p 2vln

u ðÞ:

(η 1

=u r

r−

1 uu2 r

+n−1

ru r,

η 2=v r

r−

1 vv2 r+

n−1

rv r:

(

10u t

=1

rn−1

rn−1u−

1 ur

�� r+

a 1u+b 1uln

u ðÞ−

b 1s 3 s 1uln

v ðÞ+s 3,

v t=

1rn

−1rn

−1v−

1 vr

�� r+

a 2v+b 2vln

v ðÞ−

b 2s 1 s 3vln

u ðÞ+

s 1:

8 < :η 1

=u r

r−

1 uu2 r

+n−1

ru r

+s 3u,

η 2=v r

r−

1 vv2 r+

n−1

rv r+s 1v:

(

11u t

=1

rn−1

rn−1u−

1 ur

�� r+

b 1s 3

+p 1s 1

s 2u+b 1uln

u ðÞ+

s 2ln

u ðÞ+

s 3+p 1uln

v ðÞ,

v t=

1rn

−1rn

−1v−

1 vr

�� r+

b 2s 1

+p 2s 3

s 2v+b 2vln

v ðÞ+s 2ln

v ðÞ+s 1+p 2vln

u ðÞ:

8 < :η 1

=u r

r−

1 uu2 r

+n−1

ru r

+s 2uln

u ðÞ+

s 3u,

η 2=v r

r−

1 vv2 r+

n−1

rv r+s 2vln

v ðÞ+s 1v:

(

12u t

=1 rru

−1/2u r

�� r+

a 1u+p 1

ffiffi up v,v t=

1 rrv

−1v r

�� r+

a 2v+p 2v2

ffiffiffi up :8 < :

η 1=u r

r−

1 2uu2 r

+1 ru

r−

8 r2u,

η 2=v r

r−

2 vv2 r+

1 rvr+

4 r2v:

(


Table1:Con

tinu

ed.

No.

RDsystem

(2)

CLB

S(1)

13u t

=1

rn−1

rn−1uku r

�� r+

a 1u+b 1u1

−k+p 1u1

−kvl,

v t=

1rn

−1rn

−1vlv r

�� r+

a 2v+b 2v1

−l+p 2v1

−luk

8 < :η 1

=u r

r+

k−1

uu2 r

−1 ru

r,

η 2=v r

r+

l−1

vv2 r

−1 rv

r:

(

14

u t=

1rn

−1rn

−1u2

3−n

ðÞ

n−4u r

�� r+

p 1s 2

3−n

ðÞ+

s2 1½

�n−3

ðÞs 1

u+p 1u3

n−10

n−4v2

3−n

ðÞ

n−4,

v t=

1 rn−1

rn−1v2

3−n

ðÞ

n−4v r

�� r+

p 2s 1

3−n

ðÞ+

s2 2½

�n−3

ðÞs 2

v+p 2v3

n−10

n−4u2

3−n

ðÞ

n−4:

8 > > < > > :η 1

=u r

r−

3n−10

n−4

ðÞuu2 r

+3n−3

ðÞ

ru r

+s 1u

3n−1

0ð

Þ/n−

4ð

Þ −n−4

ðÞ2

r2u,

η 2=v r

r−

3n−10

n−4

ðÞvv2 r

+3n−3

ðÞ

rv r+s 2v

3n−1

0ð

Þ/n−

4ð

Þ −n−4

ðÞ2

r2v:

8 > < > :

15u t

=1

rn−1

rn−1ulu r

�� r+

a 1u+p 1u1

−lvl,

v t=

1rn

−1rn

−1vlv r

�� r+

a 2v+p 2v1

−lul:

8 < :η 1

=u r

r+

l−1

uu2 r

+n−3

ðÞl−1

l+1

ðÞr

u r−

2n−2

ðÞ

l+1

ðÞr2

u,

η 2=v r

r+

l−1

vv2 r

+n−3

ðÞl−1

l+1

ðÞr

v r−

2n−2

ðÞ

l+1

ðÞr2

v:

8 < :

16u t

=1 rru

−1u r

�� r+

a 1u+

p 1u2 v,

v t=

1 rrv

−1v r

�� r+

a 2v+

p 2v2 u:

8 < :η 1

=u r

r−

2 uu2 r

+s ru r

+2s+1

ðÞ

r2u,

η 2=v r

r−

2 vv2 r+

s rv r+

2s+1

ðÞ

r2v:

8 < :

17

u t=

1rn

−1rn

−1u2

3−n

ðÞ

n−4u r

�� r+

a 1u+p 1u3

n−10

n−4v2

3−n

ðÞ

n−4,

v t=

1rn

−1rn

−1v2

3−n

ðÞ

n−4v r

�� r+

a 2v+p 2v3

n−10

n−4u2

3−n

ðÞ

n−4:

8 > < > :η 1

=u r

r−

3n−10

n−4

ðÞuu2 r

+3n−3

ðÞ

ru r

−n−4

ðÞ2

r2u,

η 2=v r

r−

3n−10

n−4

ðÞvv2 r

+3n−3

ðÞ

rv r−

n−4

ðÞ2

r2v:

8 > < > :18

u t=

1rn

−1rn

−1uku r

�� r+

a 1u+b 1u1

−k+p 1u1

−kv−

2/n+

2ð

Þ ,

v t=

1rn

−1rn

−1v−

4/n+

2ð

Þ vr

�� r+

a 2v+b 2v

n+4

ðÞ/

n+2

ðÞ +

p 2v

n+4

ðÞ/

n+2

ðÞ u

k :

8 < :η 1

=u r

r+

k−1

uu2 r

−1 ru

r,η 2

=v r

r−

n+4

n+2

ðÞvv2 r

−1 rv

r:

(

19u t

=1

rn−1

rn−1u−

4/n+

2ð

Þ ur

�� r+

a 1u+b 1u

n+4

ðÞ/

n+2

ðÞ +

p 1u

n+4

ðÞ/

n+2

ðÞ v

−2/n

+2ð

Þ ,

v t=

1rn

−1rn

−1v−

4/n+

2ð

Þ vr

�� r+

a 2v+b 2v

n+4

ðÞ/

n+2

ðÞ +

p 2v

n+4

ðÞ/

n+2

ðÞ u

−2/n

+2ð

Þ :

8 < :η 1

=u r

r−

n+4

n+2

ðÞuu2 r

−1 ru

r,

η 2=v r

r−

n+4

n+2

ðÞvv2 r

−1 rv

r:

8 < :20

u t=

1rn

−1rn

−1u2

3−n

ðÞ/

n−4

ðÞ u

r

�� r+

a 1u+p 1u

2n−7

ðÞ/

n−4

ðÞ v

3−n

ðÞ/

n−4

ðÞ ,

v t=

1rn

−1rn

−1v2

3−n

ðÞ/

n−4

ðÞ v

r

�� r+

a 2v+p 2v

2n−7

ðÞ/

n−4

ðÞ u

3−n

ðÞ/

n−4

ðÞ :

8 < :η 1

=u r

r−

2n−7

n−4

ðÞuu2 r

+n−3

ru r

+n−4

r2u,

η 2=v r

r−

2n−7

n−4

ðÞvv2 r

+n−3

rv r+

n−4

r2v:

8 < :

21u t

=1 rru

−1u r

�� r+

a 1u+

p 1u3

/2 ffiffi vp ,v t=

1 rrv

−1v r

�� r+

a 2v+

p 2v3

/2 ffiffi up :8 > < > :

η 1=u r

r−

3 2uu2 r

−1 ru

r+

s 1 r2u,

η 2=v r

r−

3 2vv2 r

−1 rv

r+

s 1 r2v:

(


conclude that Fðr, uÞ and Nðr, vÞ respectively satisfy thelinear PDE

Fru +2kuFr = 0,

Nrv −2lvNr = 0:

ð24Þ

It is easy to know that

F r, uð Þ = F1 uð Þ + F2 rð Þu−2k,N r, vð Þ =N1 vð Þ +N2 rð Þv−2l:

ð25Þ

The eighth one of system (16) becomes

ku−3k−1F22 rð Þ + −u−2kP′1 uð Þ − 2ku−2k−1P1 uð Þh

+ 2ku−k−1F1 uð ÞiF2 rð Þ − u−kF2′′ rð Þ −

n − 1r

u−kF2′ rð Þ+ F′1 uð ÞP1 uð Þ − P′1 uð ÞF1 uð Þ + kuk−1F21 uð Þ+ p1 u−kF1′ uð Þ − ku−3k−1F2 rð Þ + ku−k−1F1 uð Þ

h ivl+1

− p1 l + 1ð Þu−kN2 rð Þv−l − p1 l + 1ð Þu−kvlN1 vð Þ = 0:ð26Þ

Considering p1 ≠ 0 and l ≠ −1, we can set N1ðvÞ = c1v +c2v

−2l + c3v−l. As a consequence, we got

ku−3k−1F22 rð Þ + −u−2kP′1 uð Þ − 2ku−2k−1P1 uð Þh

+ 2ku−k−1F1 uð ÞiF2 rð Þ − u−kF2′′ rð Þ −

n − 1r

u−kF2′ rð Þ+ F′1 uð ÞP1 uð Þ − P′1 uð ÞF1 uð Þ + kuk−1F21 uð Þ− p1 l + 1ð Þc3u−k + p1 u−kF1′ uð Þ − ku−3k−1F2 rð Þ

h+ ku−k−1F1 uð Þ − l + 1ð Þc1u−k

iv1+l

− p1 l + 1ð Þu−k N2 rð Þ + c2½ �v−l = 0:ð27Þ

It is noted that the above polynomial about v include twoterms for the case of l = −1/2. Firstly, we consider the case ofk ≠ −1/2. The polynomial is zero will yield that N2ðrÞ = −c2.In addition, F2ðrÞ = c4 and

F1 uð Þ = −c4u−2k +l + 1ð Þc1k + 1 u + s1u

−k, ð28Þ

is derived due to

u−kF1′ uð Þ − ku−3k−1F2 rð Þ + ku−k−1F1 uð Þ − l + 1ð Þc1u−k = 0:ð29Þ

Solving the third one and the eleventh one of system (16),we induce

P1 uð Þ = a1u + b1u−k + s1 +l + 1ð Þc1k + 1 u

1+k,

Q1 vð Þ = a2v + b2v−l + c3 + c1v1+l:ð30Þ

The eighth one and the sixteenth one of the system (16)become

ks1a1 + a1s1 − c1b1 + p1c3 − c1lb1 + p1lc3 = 0,

−c1b2 + c3la2 + p2s1k + p2s1 + a2c3 − b2lc1 = 0,ð31Þ

which finally identify the undetermined functions in diffusionsystem (2) and the admitted CLBS (1). The correspondingresults are listed as items 1-3 of Table 1. Similar discussionas above for the case of k = −1/2 will present the correspond-ing CLBS (1) and the governing system (2), which are listedas items 4-6 of Table 1. The results for k = −1, l ≠ −1 andk = −1, l = −1 are also listed in Table 1. We omit thetedious computational procedure for other cases of HðuÞand LðvÞ and just list the corresponding results inTable 1. It is noted that the software Maple is used forcalculations.

4. Exact Solutions of System (2)

Since the admitted CLBS (1) of system (2) rightly corre-sponds to the DC (3), the exact solutions of system (2)listed in Table 1 can be derived due to the compatibilityof the governing system (2) and the admitted DC (3)corresponding to the invariant surface of CLBS (1). Onefirst solves the two ODEs in (3) to determine the formof uðr, tÞ and vðr, tÞ, which are both functions about rwith t-dependent integration constants. Substituting theresulting uðr, tÞ and vðr, tÞ into the original system (2),we can finally determine the time evolution integrationconstants. Here, we just present several examples to illus-trate the reduction procedure.

Example 4. System

ut =1

rn−1rn−1ukur

� �r+ a1u−k −

1 + lð Þp1s21 + kð Þs1

u + s1 + p1u−kv1+l,

vt =1

rn−1rn−1vlvr

� �r+ a2v−l −

1 + kð Þp2s11 + lð Þs2

v + s2 + p2v−lu1+k,

8>>><>>>:

ð32Þ

admits of CLBS

η1 = urr +kuu2r +

n − 1r

ur + s1u−k,

η2 = vrr +lvv2r +

n − 1r

vr + s2v−l:

8>><>>: ð33Þ

The exact solutions of (32) are listed as below.


(i) For n ≠ 2,

u r, tð Þ = α tð Þr2−n + β tð Þ − 1 + kð Þs12n r2

� �1/ 1+kð Þ,

v r, tð Þ = ϕ tð Þr2−n + ψ tð Þ − 1 + lð Þs22n r2

� �1/ 1+lð Þ,

8>>>><>>>>:

ð34Þ

where αðtÞ, βðtÞ, ϕðtÞ, and ψðtÞ satisfy the four-dimensionaldynamical system

α′ = − p1s2 1 + lð Þs1

α + 1 + kð Þp1ϕ,

β′ = − p1s2 1 + lð Þs1

β + 1 + kð Þp1ψ + 1 + kð Þa1,

ϕ′ = − p2s1 1 + kð Þs2

ϕ + 1 + lð Þp2α,

ψ′ = − p2s1 1 + kð Þs2

ψ + 1 + lð Þp2β + 1 + lð Þa2:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð35Þ

The solutions of this linear system of PDEs are presentedas follows.

(i) For p2 ≠ −p1s22ð1 + lÞ/s21ð1 + kÞ,

α tð Þ = c1 + c2 exp −p1s

22 1 + lð Þ + p2s21 1 + kð Þ

s1s2t

� �,

β tð Þ = − s1s2c3p1s

22 1 + lð Þ + p2s21 1 + kð Þ

exp

� − p1s22 1 + lð Þ + p2s21 1 + kð Þ

s1s2t

� �

+ s1 1 + kð Þ p1s2a2 1 + lð Þ + p2s1a1 1 + kð Þ½ �p1s

22 1 + lð Þ + p2s21 1 + kð Þ

t + c4,

ϕ tð Þ = − p2s1c2s2p1

exp − p1s22 1 + lð Þ + p2s21 1 + kð Þ

s1s2t

� �

+ s2 1 + lð Þc1s1 1 + kð Þ

,

ψ tð Þ = p2s21c3

p1 p1s22 1 + lð Þ + p2s21 1 + kð Þ

� � exp� − p1s

22 1 + lð Þ + p2s21 1 + kð Þ

s1s2t

� �

+ 1 + lð Þ p1s2a2 1 + lð Þ + p2s1a1 1 + kð Þ½ �s2p1s

22 1 + lð Þ + p2s21 1 + kð Þ

t

+ s2 1 + lð Þ a2s1 − a1s2ð Þp1s

22 1 + lð Þ + p2s21 1 + kð Þ

+ s2 1 + lð Þc4s1 1 + kð Þ

:

ð36Þ

(ii) For p2 = −p1s22ð1 + lÞ/s21ð1 + kÞ,

α tð Þ = c1t + c2,

β tð Þ = p1 1 + lð Þ 1 + kð Þ s1a2 − s2a1ð Þ2s1t2 + c3t + c4,

ϕ tð Þ = s2c1 1 + lð Þs1 1 + kð Þ

t + s2 1 + lð Þc2s1 1 + kð Þ

+ c1p1 1 + kð Þ

,

ψ tð Þ = p1s2 1 + lð Þ2 a2s1 − a1s2ð Þ2s21

t2

+ 1 + lð Þ 1 + kð Þ a2s1 − a1s2ð Þ + s2c3½ �s1 1 + kð Þ

t

+ s2p1c4 1 + lð Þ − s1a1 1 + kð Þ + s1c3s1p1 1 + kð Þ

:

ð37Þ

(iii) For n = 2,

u r, tð Þ = α tð Þ ln r + β tð Þ − 1 + kð Þs14 r2

� �1/ 1+kð Þ,

v r, tð Þ = ϕ tð Þ ln r + ψ tð Þ − 1 + lð Þs24 r2

� �1/ 1+lð Þ,

8>>>><>>>>:

ð38Þwhere αðtÞ, βðtÞ, ϕðtÞ, and ψðtÞ satisfy the linear sys-tem of ODEs (35).

Example 5. System

ut =1

rn−1rn−1u−1/2ur� �

r+ a1u + s1 −

a1s1s2

ffiffiffiffiffiuv

p,

vt =1

rn−1rn−1v−1/2vr� �

r+ a2v + s2 −

a2s2s1

ffiffiffiffiffiuv

p,

8>>><>>>:

ð39Þ

admits of CLBS

η1 = urr −12u u

2r +

n − 1r

ur + s1ffiffiffiu

p+ 4 n − 4ð Þ

r2u,

η2 = vrr −12v v

2r +

n − 1r

vr + s2ffiffiffiv

p+ 4 n − 4ð Þ

r2v:

8>><>>: ð40Þ

The exact solutions of this system are listed as follows.

(i) For n ≠ 2, 6,

u r, tð Þ = α tð Þr4−n + β tð Þr2

−s1

8 n − 2ð Þ r2

� �2,

v r, tð Þ = ϕ tð Þr4−n + ψ tð Þr2

−s2

8 n − 2ð Þ r2

� �2,

8>>>><>>>>:

ð41Þ


where αðtÞ, βðtÞ, ϕðtÞ, and ψðtÞ satisfy the four-dimensional dynamical system

α′ = 12 a1α −a1s12s2

ϕ,

β′ = 12 a1β −a1s12s2

ψ − 2n + 8,

ϕ′ = 12 a2ϕ −a2s22s1

α,

ψ′ = 12 a2ψ −a2s22s1

β − 2n + 8:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð42Þ

The solutions of this linear system of PDE are given asfollows.

(i) For a2 ≠ −a1,

α tð Þ = c1 + c2 exp12 a1 + a2ð Þt

� �,

β tð Þ = 2c3a1 + a2

exp 12 a1 + a2ð Þt� �

+ 2 4 − nð Þ a1s1 + a2s2ð Þa1 + a2ð Þs2

t + c4,

ϕ tð Þ = − a2s2c2a1s1

exp 12 a1 + a2ð Þt� �

+ s2c1s1

,

ψ tð Þ = − 2a2s2c3a1s1 a1 + a2ð Þ

exp 12 a1 + a2ð Þt� �

+ 2 4 − nð Þa1 a1s1 + a2s2ð Þa1s1 a1 + a2ð Þ

t

+ a1 + a2ð Þs2c4 + 4 4 − nð Þ s2 − s1ð Þa1 + a2ð Þs1

:

ð43Þ

(ii) For a2 = −a1,

α tð Þ = c1t + c2,

β tð Þ = a1 4 − nð Þ s2 − s1ð Þ2s2t2 + c3t + c4,

ϕ tð Þ = s2c1s1

t + a1c2 − 2c1a1s1

,

ψ tð Þ = a1 4 − nð Þ s2 − s1ð Þ2s1t2

+ 2 n − 4ð Þ s2 − s1ð Þ + a1s2c3a1s1

t

+ 2 a1c4 − 2c3ð Þs2 + 4 4 − nð Þs2a1s1

:

ð44Þ

For n = 2,

u r, tð Þ = α tð Þr2 + β tð Þr2

+ s1r2 4lnr − 1ð Þ

32

� �2,

v r, tð Þ = ϕ tð Þr2 + ψ tð Þr2

+ s2r2 4 ln r − 1ð Þ

32

� �2,

8>>>><>>>>:

ð45Þ

where αðtÞ, βðtÞ, ϕðtÞ, and ψðtÞ satisfy the linear systemof (42).

For n = 6,

u r, tð Þ = α tð Þ ln rð Þr2

+ β tð Þr2

+ s132 r2

� �2,

v r, tð Þ = ϕ tð Þ ln rð Þr2

+ ψ tð Þr2

+ s232 r2

� �2,

8>>>><>>>>:

ð46Þ

where αðtÞ, βðtÞ, ϕðtÞ, and ψðtÞ satisfy the linear systemof (42).

Example 6. System

ut =1

rn−1rn−1u−1ur� �

r+ a1u + b1u ln uð Þ + p1uv1+l,

vt =1

rn−1rn−1vlvr

� �r+ a2v + b2v−l + p2v−l ln uð Þ,

8>><>>:

ð47Þ

admits of CLBS

η1 = urr −1uu2r +

n − 1r

ur ,

η2 = vrr +lvv2r +

n − 1r

vr:

8>><>>: ð48Þ

The exact solutions of this system are listed as follows.For n ≠ 2,

u r, tð Þ = exp α tð Þr2−n + β tð Þ� �,v r, tð Þ = ϕ tð Þr2−n + ψ tð Þ� �1/ 1+lð Þ,

8<: ð49Þ


α′ = b1α + p1ϕ,β′ = b1β + p1ψ + a1,ϕ′ = a2 1 + lð Þϕ + p2 1 + lð Þα,ψ′ = a2 1 + lð Þψ + p2 1 + lð Þβ + 1 + lð Þb2:

8>>>>><>>>>>:

ð50Þ

The linear system is solvable. However, we do not list thecorresponding results here because of the complexity form ofthe solutions.


For n = 2,

u r, tð Þ = rα tð Þ exp β tð Þ½ �,v r, tð Þ = ϕ tð Þ ln r + ψ tð Þ½ �1/ 1+lð Þ,

(ð51Þ

where αðtÞ, βðtÞ, ϕðtÞ, and ψðtÞ satisfy the linear system ofODEs (50).

Example 7. System

ut =1

rn−1rn−1u−1ur� �

r+ a1u −

s1p1s3

u ln u + p1u ln v + s3,

vt =1

rn−1rn−1v−1vr� �

r+ a2v −

s3p2s1

v ln v + p2v ln u + s1,

8>>><>>>:

ð52Þ

admits of CLBS


n − 1r

ur + s3u,

η2 = vrr −1vv2r +

n − 1r

vr + s1v:

8>><>>: ð53Þ

For n ≠ 2,

u r, tð Þ = exp α tð Þr2−n + β tð Þ − s32n r2

h i,

v r, tð Þ = exp ϕ tð Þr2−n + ψ tð Þ − s12n r2

h i,

8><>: ð54Þ


α′ = − s1p1s3

α + p1ϕ,

β′ = − s1p1s3

β + p1ψ + a1,

ϕ′ = − s3p2s1

ϕ + p2α,

ψ′ = − s3p2s1

ψ + p2β + a2:

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð55Þ

The solutions of this linear system of ODEs are given asbelow.

(i) For p2 ≠ −s21p1/s23,

α tð Þ = s3c1s1

−s1p1c2s3p2

exp − s23p2 + s21p1

s1s3t

� �,

β tð Þ = s3 a2s1p1 + a1s3p2ð Þts21p1 + s23p2

−s1s3c3

s21p1 + s23p2exp − s

23p2 + s21p1

s1s3t

� �+ c4,

ϕ tð Þ = c1 + c2 exp −s23p2 + s21p1

s1s3t

� �,

ψ tð Þ = s1 a2s1p1 + a1s3p2ð Þts21p1 + s23p2

+ s23c3p2

s21p1 + s23p2� �

p1exp − s

23p2 + s21p1

s1s3t

� �

+ s1 a2s3 − a1s1ð Þs21p1 + s23p2� �

s3+ s1c4

s3:

ð56Þ

(ii) For p2 = −s21p1/s23,

α tð Þ = s3c1s1

t + s1c1 + s3p2c2p2s1

,

β tð Þ = − p1 a1s1 − a2s3ð Þ2s3t2 + c3t + c4,

ϕ tð Þ = c1t + c2,

ψ tð Þ = − s1p1 a1s1 − a2s3ð Þ2s23t2 + a2s3 − a1s1 + s1c3

s3t

+ c3 − a1ð Þs3 + s1p1c4s3p1

:

ð57Þ

For n = 2,

u r, tð Þ = exp β tð Þ − s34 r2

h irα tð Þ,

v r, tð Þ = exp ψ tð Þ − s14 r2

h irϕ tð Þ,

8><>: ð58Þ

where αðtÞ, βðtÞ, ϕðtÞ, and ψðtÞ satisfy the linear system ofODEs (68).

Example 8. System

ut =1

rn−1rn−1ukur

� �r+ a1u + b1u1−k + p1u1−kvl,

vt =1

rn−1rn−1vlvr

� �r+ a2v + b2v1−l + p2v1−luk,

8>><>>:

ð59Þ


admits of CLBS

η1 = urr +k − 1u

u2r −1rur ,

η2 = vrr +l − 1v

v2r −1rvr:

8>><>>: ð60Þ

The solutions are listed as

u r, tð Þ = α tð Þr2 + β tð Þ� �1/k,v r, tð Þ = ϕ tð Þr2 + ψ tð Þ� �1/l,

8<: ð61Þ


α′ = 2 nk + 2ð Þk

α2 + ka1α + kp1ϕ,

β′ = 2nαβ + ka1β + kp1ψ + kb1,

ϕ′ = 2 nl + 2ð Þl

ϕ2 + la2ϕ + lp2α,

ψ′ = 2nϕψ + la2ψ + lp2β + lb2:

ð62Þ

Example 9. System

admits of CLBS

η1 = urr −3n − 10n − 4ð Þu u

2r −

3 n − 3ð Þr

ur + s1u 3n−10ð Þ/ n−4ð Þ −n − 4ð Þ2r2

u,

η2 = vrr −3n − 10n − 4ð Þv v

2r −

3 n − 3ð Þr

vr + s1v 3n−10ð Þ/ n−4ð Þ −n − 4ð Þ2r2

v:

8>>>><>>>>:

ð64Þ

The solutions are listed as below.For n ≠ 2,

u r, tð Þ = n − 3ð Þs1n − 2ð Þ2 n − 4ð Þ r

2 + α tð Þr4−n + β tð Þr6−2n" # 4−nð Þ/2 n−3ð Þ

,

v r, tð Þ = n − 3ð Þs2n − 2ð Þ2 n − 4ð Þ r

2 + ϕ tð Þr4−n + ψ tð Þr6−2n" # 4−nð Þ/2 n−3ð Þ

,

8>>>>>><>>>>>>:

ð65Þ


α′ = 2 n − 3ð Þp1n − 4ð Þs1

s2α − s1ϕð Þ,

β′ = 2 n − 4ð Þs21 + n − 3ð Þs2p1

� �n − 4ð Þs1

β −n − 2ð Þ2 n − 4ð Þ2 n − 3ð Þ α

2 −2 n − 3ð Þp1

n − 4 ψ,

ϕ′ = 2 n − 3ð Þp2n − 4ð Þs2

s1ϕ − s2αð Þ,

ψ′ = 2 n − 4ð Þs22 + n − 3ð Þs1p2

� �n − 4ð Þs2

ψ −n − 2ð Þ2 n − 4ð Þ2 n − 3ð Þ ϕ

2 −2 n − 3ð Þp2

n − 4 β:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð66Þ

For n = 2,

u r, tð Þ = 2r2 s1 ln2r + α tð Þ ln r + β tð Þ� � ,

v r, tð Þ = 2r2 s2 ln2r + ϕ tð Þ ln r + ψ tð Þ� � ,

8>>><>>>:

ð67Þ


α′ = s2p1s1

α − p1ϕ,

β′ = s2p1 + 2s21

s1β −

12 α

2 − p1ψ,

ϕ′ = s1p2s2

ϕ − p2α,

ψ′ = s1p2 + 2s22

s2ψ −

12 ϕ

2 − p2β:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð68Þ

Example 10. System

ut =1

rn−1rn−1ukur

� �r+ a1u + b1u1−k + p1u1−kv−2/ n+2ð Þ,

vt =1

rn−1rn−1v−4/ n+2ð Þvr

� �r+ a2v + b2v n+4ð Þ/ n+2ð Þ + p2v n+4ð Þ/ n+2ð Þuk,

8>><>>:

ð69Þ

ut =1

rn−1rn−1u2 3−nð Þ/ n−4ð Þur

� �r+ p1s2 3 − nð Þ + s

21

� �n − 3ð Þs1

u + p1u 3n−10ð Þ/ n−4ð Þv2 3−nð Þ/ n−4ð Þ,

vt =1

rn−1rn−1v2 3−nð Þ/ n−4ð Þvr

� �r+ p2s1 3 − nð Þ + s

22

� �n − 3ð Þs2

v + p2v 3n−10ð Þ/ n−4ð Þu2 3−nð Þ/ n−4ð Þ,

8>>><>>>:

ð63Þ


admits of CLBS

η1 = urr +k − 1u

u2r −1rur ,

η2 = vrr −n + 4n + 2ð Þv v

2r −

1rvr:

8>><>>: ð70Þ

The solutions are listed as

u r, tð Þ = α tð Þr2 + β tð Þ� �1/k,v r, tð Þ = ϕ tð Þr2 + ψ tð Þ� �− n+2ð Þ/2,

8<: ð71Þ


α′ = 2 nk + 2ð Þk

α2 + ka1α + kp1ϕ,

β′ = 2nαβ + ka1β + kp1ψ + kb1,

ϕ′ = 2nϕ2ψ − 2a2n + 2 ϕ −

2p2n + 2 α,

ψ′ = 2nϕψ2 − 2a2n + 2ψ −

2p2n + 2β −

2b2n + 2 :

8>>>>>>>>>><>>>>>>>>>>:

ð72Þ

Example 11. System

ut =1r

ru−1ur� �

r+ a1u +

p1u2

v,

vt =1r

rv−1vr� �

r+ a2v +

p2v2

u,

8>><>>: ð73Þ

admits of CLBS


srur +

2 s + 1ð Þr2

u,

η2 = vrr −2vv2r +

srvr +

2 s + 1ð Þr2

v:

8>><>>: ð74Þ

The solutions are listed as below.For s ≠ −3,

u r, tð Þ = 3 + sα tð Þr− s+1ð Þ + β tð Þr2 ,

v r, tð Þ = 3 + sϕ tð Þr− s+1ð Þ + ψ tð Þr2 ,

8>>><>>>:

ð75Þ


α′ = s + 3ð Þαβ − p1ϕ − a1α,β′ = −a1β − p1ψ,ϕ′ = s + 3ð Þϕψ − a2ϕ − p2α,ψ′ = −a2ψ − p2β:

8>>>>><>>>>>:

ð76Þ

For s = −3,

u r, tð Þ = 1α tð Þr2 ln r + β tð Þr2 ,

v r, tð Þ = 1ϕ tð Þr2 ln r + ψ tð Þr2 ,

8>>><>>>:

ð77Þ


α′ = −p1ϕ − a1α,β′ = −a1β − p1ψ − α2,ϕ′ = −a2ϕ − p2α,ψ′ = −a2ψ − p2β − ϕ2:

8>>>>><>>>>>:

ð78Þ

Example 12. System

ut =1

rn−1rn−1u2 3−nð Þ/ n−4ð Þur

� �r+ a1u + p1u 2n−7ð Þ/ n−4ð Þv 3−nð Þ/ n−4ð Þ,

vt =1

rn−1rn−1v2 3−nð Þ/ n−4ð Þvr

� �r+ a2v + p2v 2n−7ð Þ/ n−4ð Þu 3−nð Þ/ n−4ð Þ,

8>><>>:

ð79Þ

admits of CLBS

η1 = urr −2n − 7n − 4ð Þu u

2r +

n − 3r

ur +n − 4r2

u,

η2 = vrr −2n − 7n − 4ð Þv v

2r +

n − 3r

vr +n − 4r2

v:

8>>><>>>:

ð80Þ

The solutions are listed as below.For n ≠ 2,

u r, tð Þ = α tð Þr3−n + β tð Þr� � 4−nð Þ/ n−3ð Þ,v r, tð Þ = ϕ tð Þr3−n + ψ tð Þr� � 4−nð Þ/ n−3ð Þ,

8<: ð81Þ


α′ = n − 2ð Þ2

n − 3 αβ2 −

n − 3ð Þa1n − 4 α −

n − 3ð Þp1n − 4 ϕ,

β′ = n − 2ð Þ2

n − 3 β3 −

n − 3ð Þa1n − 4 β −

n − 3ð Þp1n − 4 ψ,

ϕ′ = n − 2ð Þ2

n − 3 ϕψ2 −

n − 3ð Þa2n − 4 ϕ −

n − 3ð Þp2n − 4 α,

ψ′ = n − 2ð Þ2

n − 3 ψ3 −

n − 3ð Þa2n − 4 ψ −

n − 3ð Þp2n − 4 β:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð82Þ


For n = 2,

u r, tð Þ = 1α tð Þr ln r + β tð Þr½ �2 ,

v r, tð Þ = 1ϕ tð Þr ln r + ψ tð Þr½ �2 ,

8>>><>>>:

ð83Þ


α′ = −α3 − 12 a1 + p1ð Þα,

β′ = −α2β − 12 a1 + p1ð Þβ,

ϕ′ = −ϕ3 − 12 a2 + p2ð Þϕ,

ψ′ = −ϕ2ψ − 12 a2 + p2ð Þψ:

8>>>>>>>>>>><>>>>>>>>>>>:

ð84Þ

Example 13. System

ut =1r

ru−1ur� �

r+ a1u +

p1u3/2ffiffiffiv

p ,

vt =1r

rv−1vr� �

r+ a2v +

p2v3/2ffiffiffiu

p ,

8>>><>>>:

ð85Þ

admits of CLBS

η1 = urr −32u u

2r −

1rur +

s1r2u,

η2 = vrr −32v v

2r −

1rvr +

s1r2v:

8>><>>: ð86Þ

The solutions are listed as below.For s1 + 2 < 0,


α′ = 12 α3 + 12 αβ

2 −12 a1α −

12 p1ϕ,

β′ = 12β3 + 12 α

2β −12 a1β −

12 p1ψ,

ϕ′ = 12 ϕ3 + 12 ϕψ

2 −12 a2ϕ −

12 p2α,

ψ′ = 12ψ3 + 12 ϕ

2ψ −12 a2ψ −

12 p2β:

8>>>>>>>>>>><>>>>>>>>>>>:

ð88Þ

For s1 + 2 > 0,

u r, tð Þ = s1 + 2r2 α tð Þ sin 1/2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−2 s1 + 2ð Þp ln r� � + β tð Þ cos 1/2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−2 s1 + 2ð Þp ln r� �h i2 ,

v r, tð Þ = s1 + 2r2 ϕ tð Þ sin 1/2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−2 s1 + 2ð Þp ln r� � + ψ tð Þ cos 1/2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−2 s1 + 2ð Þp ln r� �h i2 ,

8>>>>>><>>>>>>:

ð87Þ

u r, tð Þ = s1 + 2r2 α tð Þ sinh 1/2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 s1 + 2ð Þp ln r� � + β tð Þ cosh 1/2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 s1 + 2ð Þp ln r� �h i2 ,

v r, tð Þ = s1 + 2r2 ϕ tð Þ sinh 1/2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 s1 + 2ð Þp ln r� � + ψ tð Þ cosh 1/2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 s1 + 2ð Þp ln r� �h i2 ,

8>>>>>><>>>>>>:

ð89Þ



α′ = − 12 α3 + 12 αβ

2 −12 a1α −

12 p1ϕ,

β′ = 12β3 −

12 α

2β −12 a1β −

12 p1ψ,

ϕ′ = − 12 ϕ3 + 12 ϕψ

2 −12 a2ϕ −

12 p2α,

ψ′ = 12ψ3 −

12 ϕ

2ψ −12 a2ψ −

12 p2β:

8>>>>>>>>>>><>>>>>>>>>>>:

ð90Þ

For s1 + 2 = 0,

u r, tð Þ = 1r2 α tð Þ ln r + β tð Þ½ �2 ,

v r, tð Þ = 1r2 ϕ tð Þ ln r + ψ tð Þ½ �2 ,

8>>><>>>:

ð91Þ


α′ = −α3 − 12 a1α −12 p1ϕ,

β′ = −α2β − 12 a1β −12 p1ψ,

ϕ′ = −ϕ3 − 12 a2ϕ −12 p2α,

ψ′ = −ϕ2ψ − 12 a2ψ −12 p2β:

8>>>>>>>>>>><>>>>>>>>>>>:

ð92Þ

5. Conclusions

The second-order CLBS (1) is used to study nonlinear radi-ally symmetric diffusion system (2). Exact solutions and sym-metry reductions are provided due to the compatibility of thegoverning system (2) and the admitted DC (3) correspondingto the invariant surface condition for CLBS (1). Those solu-tions extend the known ones such as instantaneous sourcesolutions of the porous medium equation with absorptionterm. In addition, these results cannot be obtained in theframework within the classical symmetry and the nonclassi-cal symmetry.

The topics of exact solutions and symmetry reductionsare interesting. Recently, Kumar and collaborators [53–60]have done many superior works about these topics fordifferent kinds of multidimensional evolution equations.The studies of CLBS for multidimensional evolution equa-tions will be involved in our future studies.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by Chinese National NaturalScience Foundation (Grant No. 11501175) and Key ScientificResearch Project of Colleges and Universities in Henan Prov-ince (Grant No. 20A110017).

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Second-Order Conditional Lie-Bäcklund Symmetry and Differential Constraint of Radially Symmetric Diffusion System1. Introduction2. Preliminaries3. CLBS (1) of System (2)4. Exact Solutions of System (2)5. ConclusionsData AvailabilityConflicts of InterestAcknowledgments

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