Preliminary Group Classification of nonlinear wave equation
utt +ut=f(x, ux)uxx +g(x, ux)
TSHIDISO MASEBE
Orchid no. 0000-0002-3792-5213
Tshwane University of Technology
Maths,Science& Business Education Department
No 2 Aubrey Matlala Road, Soshanguve H
SOUTH AFRICA
Abstract: The paper discusses the non-linear wave equations whose coefficients are dependent on first order spatial
derivatives. We construct the principal Lie algebra, the equivalence Lie algebra, and the extensions by one of the
principal Lie algebra. We further construct the optimal system of one-dimensional subalgebras for first three
extended five-dimensional Lie algebras. These are finally used to determine invariant solutions of some examples.
Key–Words: Principal Lie Algebra, Equivalence Lie algebra, Invariant solution, One-dimensional optimal systems.
Received: September 20, 2021. Revised: September 26, 2022. Accepted: October 29, 2022. Published: December 1, 2022.
1 Introduction
Lie group analysis of differential equations is the area
of mathematics pioneered by Sophus Lie in the 19th
century (1849-1899). The first general solution of the
problem of classification was given by Sophus Lie
for an extensive class of partial differential equations,
[5]. Since then many researchers have done work on
various families of differential equations. The results
of their work have been captured in several outstand-
ing literary works, [1],[3],[5],[8],[9].The preliminary
group classification by Ibragimov, Torrisi and Valenti
[5] gave us up to thirty three equivalence classes of
submodels of the wave model of the form
utt =f(x, ux)uxx +g(x, ux).(1)
The present work examines a model which repre-
sents families of the nonlinear wave with dissipation,
namely
utt +ut=f(ux)uxx +g(ux).(2)
In this work we use the results of one-dimensional
optimal systems
(i) of the equivalence Lie algebra to obtain X5and
hence the classification of the family of equations
(2) above ,
(ii) of the extended principal Lie algebra of equation
(2) to calculate the invariant solutions of some
examples.
The method followed in the construction of the
one-dimensional optimal systems is found in the pa-
per by Ibragimov, Torrisi and Valenti, [4]. In this pa-
per while constructing the principal Lie algebra, we
also show how to determine the Lie point symmetries
of (2). We proceed to construct the equivalence Lie
algebra, and give the extensions by o ne of the princi-
pal algebra of equation (2),[1],[2],[4]. We also show
the method of determining invariant solutions,[6],[7].
The paper also illustrates the construction of one-
dimensional optimal systems of extended principal
Lie algebras L5. We conclude by calculating invari-
ant solutions of some one-dimensional subalgebras of
each extended algebra L5.
2 Principal Lie Algebra
The principal Lie algebra Lpof the non-linear
wave equation with dissipation namely utt +ut=
f(ux)uxx +g(ux),is determined as follows:
Let the generator of equation (2) be given by
X=ξ1(t, x, u)∂
∂t +ξ2(t, x, u)∂
∂x +η(t, x, u)∂
∂u
(3)
The second prolongation of (3) is given by
e
X2=X+ζt∂
∂ut
+ζx∂
∂ux
+ζtt ∂
∂utt
+ζxx ∂
∂uxx
,(4)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.90
Tshidiso Masebe
E-ISSN: 2224-2880
785
Volume 21, 2022
where
ζt=Dt(η)−utDt(ξ1)−uxDt(ξ2),
ζx=Dx(η)−utDx(ξ1)−uxDx(ξ2),
ζtt =Dt(ζt)−uttDt(ξ1)−utxDt(ξ2),
ζxx =Dx(ζx)−utxDx(ξ1)−uxxDx(ξ2),
(5)
[[5],[6],[7]].
The operators Dtand Dxdenote the total deriva-
tives with respect to t and x respectively as follows:
Dt=∂
∂t +ut∂
∂u +utx ∂
∂ux+utt ∂
∂ut+.....
Dx=∂
∂x +ux∂
∂u +utx ∂
∂ut+uxx ∂
∂ux+..... (6)
The determining equation of (2) is given by
e
X2(utt +ut−f(ux)uxx −g(ux)) |(2) (7)
= (ζtt +ζt−fζxx −fuxζxuxx −gζx)|(2)
= 0.
In cases of arbitrary f and g it follows that
ζxx =ζx= 0,and ζtt +ζt= 0.(8)
From the equation (8) we have that
ζtt +ζt=ηtt +ut2ηtu −ξ1
tt −2uxξ2
tu
+u2
tηuu −2ξ1
tu −uxξ2
uu−u3
tξ1
uu
−utx 2ξ1
t+ 2uxξ2
u+utξ2
u
+ (−ut−f(ux)uxx −g(ux))
ηu−2ξ1
t−3utξ1
u+ηt+utηu−ξ1
t
−u2
tξ1
u−uxξ2
t−utuxξ2
u= 0.
(9)
From equation (9) we obtain
ξ2
u=ξ1
t= 0.
ξ1
u=ηu= 0.
ξ2
t= 0.
ηtt +ηt= 0 ⇒η=c1+c2e−t.
(10)
Thus we have that
ξ1=c, ξ2=c, η =c1+c2e−t.(11)
Thus the principal Lie algebra Lpof the non-
linear wave equation with dissipation (2)
is spanned by the following generators
X1=∂
∂x, X2=∂
∂t, X3=∂
∂u, X4=e−t∂
∂u.
(12)
2.1 Equivalence Lie Algebra and extensions
of the principal Lie Algebra
The equivalence Lie Algebra, is the non-degerate
changes in the variables, x, t and uwhich carries
equation (2) into an equation of the same form. The
family of non-linear waves utt +ut=f(ux)uxx +
g(ux),can be written as a system of differential equa-
tions
utt +ut=f1uxx +f2
fk
x=fk
t=fk
u=fk
ut= 0 (13)
k = 1, 2. The equivalence Lie algebra element for the
system (13) is given by the generators
E=ξ∂
∂x +τ∂
∂t +η∂
∂u +µk∂
∂fk(14)
where ξ=ξ(x, t, u),τ=τ(x, t, u),η=
η(x, t, u),µk=µk(x, t, u, ux, ut, f1, f2).We now
introduce the following total derivatives
g
Dα=∂
∂α +fk
α∂
∂fk+fk
αt ∂
∂fk
t
+
fk
αx ∂
∂fk
x+fk
αu ∂
∂fk
u+fk
αut
∂
∂fk
ut
+...
for α∈ {x, t, u, ut}.
The extension of the equivalence algebra element
E, takes the form
e
E=E+ζt∂
∂ut+ζx∂
∂ux+ζxx ∂
∂uxx
+$k
t∂
∂fk
t
+$k
x∂
∂fk
x+$k
u∂
∂fk
u+$k
ut
∂
∂fk
ut
,(15)
where
ζi=Di(η)−utDi(τ)−uxDi(ξ)
ζij =Di(ζi)−ujtDi(τ)−ujxDi(ξ)
for i, j ∈ {x, t}and
$k
α=g
Dα(µk)−fk
tg
Dα(τ)−fk
xg
Dα(ξ)
−fk
ug
Dα(η)−fk
utg
Dα(ζt)−fk
uxg
Dα(ζx)
where α∈ {x, t, u, ut},k=1,2.
The invariance condition for the system of equa-
tions (15) is given by
e
E(utt +ut−f1uxx −f2)|(15) = 0 (16)
e
E(fk
α)=0for α∈ {x, t, u, ut}.(17)
We thus obtain
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.90
Tshidiso Masebe
E-ISSN: 2224-2880
786
Volume 21, 2022
ζtt +ζt−µ1uxx −f0ζxx −µ2= 0
and
$k
α= 0 for α∈ {x, t, u, ut}.
From the equations (17) we have
(µk)α= (ζx)α= 0, α ∈ {x, t, u, ut}
and k= 1,2,which implies that the µkare inde-
pendent of x, t, u, utand hence
µk=µk(ux, f1, f2), k = 1,2.
Furthermore (ζx)α= 0 yields
ξ=a1x+a2u+p(t)
τ=τ(t)
η=b1u+b2x+q(t)
(18)
where a1, a2;b1, b2are constants. The equations
(18), together with the invariance condition yield
ξ=a1x+a2
τ=a3
η=a4u+a5t+a6x+a7
µ1= 2a1f1
µ2=a5+a4f2.
(19)
For the model utt +ut=f(ux)uxx +g(ux),we
have
µ1= 2a1f
µ2=a5+a4g.
Therefore we obtain a 7-dimensional equivalence
algebra for the non-linear wave equation (2), which is
spanned by the following operators
E1=∂
∂x (20)
E2=∂
∂t
E3=∂
∂u
E4=x∂
∂u
E5=u∂
∂u +g∂
∂g ,
E6=t∂
∂u +∂
∂g ,
E7=x∂
∂x + 2f∂
∂f
The classification of the equation (2) is obtained by
extending the principal Lie algebra X1=∂
∂x ,
X2=∂
∂t , X3=∂
∂u ,
X4=e−t∂
∂u by X5in the section that follow.
3 One-Dimensional Optimal System
In order to determine X5and hence the classification
of equation (2) we give details of the determination of
the one-dimensional optimal systems L4below. Since
fand gdepend on ux,we prolong the equivalence
operators Ei(20), to the following operators
e
Ei=Ei+ζx∂
∂ux
,for i= 1,2, ......, 7.
Therefore we have
e
Ei=Ei,for i= 1,2,3
e
E4=x∂
∂u +∂
∂ux
,e
E5=u∂
∂u +g∂
∂g +ux
∂
∂ux
(21)
e
E6=E6, E7=x∂
∂x + 2f∂
∂f −ux
∂
∂ux
,
We form new operators Ziby projecting each
e
Ei(18), onto the (ux, f, g)-subspace of the
(x, t, u, ut, ux, f, g)−space. We have
pr(e
Ei)=0,for i= 1,2,3
Zi=pr(e
Ei+3),for i= 1,2,3,4.
Z1=pr(e
E5) = ∂
∂ux
Z2=g∂
∂g +ux
∂
∂ux
, Z3=∂
∂g ,
Z4= 2f∂
∂f −ux
∂
∂ux
,
We now consider the algebra L4, which is
spanned by Z1, Z2, Z3, Z4.We wish to determine the
optimal system of one-dimensional subalgebras of the
algebra L4.The non-zero structure constants of L4are
as follows:
[Z1, Z2] = Z1,[Z1,Z4] = −Z1,[Z2, Z3] = −Z3,
We now consider the algebra L4, which is spanned by
Z1, Z2, Z3, Z4.We wish to determine the optimal sys-
tem of one-dimensional subalgebras of the algebra L4.
The non-zero structure constants of L4are as follows:
[Z1, Z2] = Z1,[Z1,Z4] = −Z1,[Z2, Z3] = −Z3,
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.90
Tshidiso Masebe
E-ISSN: 2224-2880
787
Volume 21, 2022
The generators of the adjoint algebra LA
4are given by
A1=Z1
∂
∂Z2
−Z1
∂
∂Z4
A2=−Z1
∂
∂Z1
−Z3
∂
∂Z3
A3=Z3
∂
∂Z3
A4=Z1
∂
∂Z1
In order to obtain the elements of the adjoint group GA
or the group of inner automorphisms of the algebra
L4,we integrate the equations (19) to obtain a four
parameter Lie group:
A1:Z2=Z2+a1Z1, Z4=Z4−a1Z1
A2:Z1=a−1
2Z1, Z3=a−1
2Z3
A3:Z2=Z2+a3Z3,
A4:Z1=a4Z1
A matrix representation of an arbitrary element of
the adjoint group GAis of the form
M=
a−1
2a4a10−a1
0 1 0 0
0a−1
2a3a−1
20
0 0 0 1
.
If we let Z∈L4be given by
Z=e1Z1+e2Z2+e3Z3+e4Z4
Z=¯e= (e1, e2, e3, e4),
then e=Me defines an equivalence relation in L4and
hence subdivides this algebra into equivalence classes.
The components of Zmap as follows under M:
e1=a−1
2a4e1+a1(e2−e4)
e2=e2
e3=a−1
2a3e2+a−1
2e3
e4=e4
Therefore the optimal system of one-dimensional sub-
spaces of L4,obtained through the adjoint group GA,
are as follows:
Therefore the optimal system of one-dimensional
subspaces of L4,obtained through the adjoint group
GA,are as follows:
Z Generator Restrictions
Z(1) αZ2+Z4α6= 1
Z(2) αZ2+βZ3+Z4α6=β
Z(3) Z1+Z2+Z4
Z(4) Z1+Z2+αZ3+Z4
Z(5) Z3
Z(6) Z3+Z4
Z(7) Z1+Z3
Consider
Z(1) =αZ2+Z4,
with α6= 1 .
Z(1) =α(g∂
∂g +ux
∂
∂ux
)+2f∂
∂f −ux
∂
∂ux
=αg ∂
∂g + 2f∂
∂f + (α−1)ux
∂
∂ux
.
From the characteristic equation
dg
αg =df
2f=dux
(α−1)ux
,
we obtain
f=u
2
α−1
xand g=u
α
α−1
x.
To obtain the extending vector X5, we let
e
Z=αE5+E7
=α(u∂
∂u +g∂
∂g ) + x∂
∂x + 2f∂
∂f .
Let X5be the projection of e
Zonto the (x, t, u)−
space, i.e
X5=x∂
∂x +αu ∂
∂u.
For the vectors Z(i), i = 2,3,· · · ,7,we proceed in
a similar manner in order to determine the functions
f, g and the extension vector X5.The classification
for equation (2) is given in the following table:
Z(i)f(ux)g(ux)X5
Z(1) u
2
α−1
xu
2
α−1
xx∂
∂x +αu ∂
∂u
Z(2) u
2
α−1
xα−1u
2
α−1
−β
xx∂
∂x + (αu +βt)∂
∂u
Z(3) e2uxC x ∂
∂x + (u+x)∂
∂u
Z(4) e2uxαuxx∂
∂x + (u+x+αt)∂
∂u
Z(5)
Z(6) u−2
x−ln x x ∂
∂x +ut ∂
∂u
Z(7) C ux(t+x)∂
∂u (22)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.90
Tshidiso Masebe
E-ISSN: 2224-2880
788
Volume 21, 2022
In what follows we will give the classification for
equation (2) for the listed generators X5.
1.If X5=x∂
∂x + (x+u)∂
∂u then
f=e2ux,and g=c
2.If X5=x∂
∂x + (x+u+αt)∂
∂u then
f=e2ux,and g=αux
3.If X5= (x+t)∂
∂u then
f=c, and g=ux
4.If X5=x∂
∂x +t∂
∂u then
f=u−2
x,and g=−ln ux
5.If X5=x∂
∂x +αu ∂
∂u then
f=u
2
α−1
x,and g=u
2
α−1
xfor α6= 1
6.If X5=x∂
∂x + (αu +βt)∂
∂u then
f=u
2
α−1
x,and g=α−1(u
2
α−1
x−β)for α6=β
Each extension will give us a five-dimensional Lie al-
gebra L5. From the above we will concentrate on the
first four whose equations are given by the following
utt +ut=e2uxuxx +c. (23)
utt +ut=e2uxuxx +αux(24)
utt +ut=cuxx +ux.(25)
utt +ut=u−2
xuxx + ln ux.(26)
From the latter we have five-dimensional Lie al-
gebras for each of the equations (23) to (26). We will
only construct optimal systems of one-dimensional
Lie subalgebras for the first three equations. We will
then calculate the invariant solutions using some of
these one-dimensional subalgebras.
4 Invariant Solutions
Consider the equation
utt +ut=e2uxuxx +c, (27)
whose set of generators is given by X1=∂
∂x , X2=
∂
∂t , X3=∂
∂u , X4=e−t∂
∂u ,X5=x∂
∂x + (u+
x)∂
∂u .
We will use the one dimensional subalgebra X=
X1+ (1 + ρ)X3i.e.
X=∂
∂x + (1 + ρ)∂
∂u.(28)
The characteristic equation of the above generator
(28) is given by
dt
0=du
k=dx
1where k= 1 + ρ. (29)
From equation (29) the invariants are given by
I1=u−kx ;I2=t. (30)
If we define I1=φ(I2)for some function φ, then
u(t, x) = kx +φ(t).(31)
The substitution of (31) into equation (27) asserts
that
ut=φ0(t)
utt =φ00 (t)
ux=k
uxx = 0
hence
utt +ut−e2uxuxx −c=φ00 (t)+φ0(t)−c= 0.(32)
The equation (32) simplifies to
φ00 (t) + φ0(t) = c, (33)
which is a second order ODE whose solution is given
by
φ(t) = c1+c2e−t+ct −c. (34)
Thus the invariant solution of (27) is given by
u(t, x) = kx +c1+c2e−t+ct −c, (35)
where k= 1 + ρ.
Consider the equation
utt +ut=e2uxuxx +αux(36)
which has the following set of generators X1=
∂
∂x , X2=∂
∂t , X3=∂
∂u , X4=e−t∂
∂u ,X5=
x∂
∂x + (u+x+αt)∂
∂u .
We will use the one dimensional subalgebra X=
X1+X4i.e.
X=∂
∂x +e−t∂
∂u.(37)
The characteristic equation of the above generator
(37) is given by
dt
0=du
e−t=dx
1(38)
From equation (38) the invariants are given by
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.90
Tshidiso Masebe
E-ISSN: 2224-2880
789
Volume 21, 2022
I1=u−xe−t;I2=t. (39)
If we define I1=φ(I2)for some function φ, then
u(t, x) = xe−t+φ(t).(40)
The substitution of (40) into equation (36) asserts
that
ut=−xe−t+φ0(t)
utt =xe−t+φ00 (t)
ux=e−t
uxx = 0,
hence
utt +ut−e2uxuxx −αux=φ00 (t)+φ0(t)−αe−t= 0.
(41)
The equation (41) simplifies to
φ00 (t) + φ0(t) = αe−t,(42)
which is a non-linear second order ODE whose
solution is given by
φ(t) = c1+c2e−t+αe−t−αte−t.
The invariant solution of utt +ut=e2uxuxx +
αuxis given by
u(t, x) = xe−t+c1+c2e−t+αe−t−αte−t.(43)
Consider the equation
utt +ut=cuxx +ux(44)
whose set of generators is given by X1=∂
∂x , X2=
∂
∂t , X3=∂
∂u , X4=e−t∂
∂u ,X5= (x+t)∂
∂u .
We will use the one dimensional subalgebras
X=αX1+X5and X=βX2+X5i.e. X=
α∂
∂x + (x+t)∂
∂u , and X=β∂
∂t + (x+t)∂
∂u respec-
tively to calculate the invariant solutions of (44).
Consider the one dimensional subalgebra
X=α∂
∂x + (x+t)∂
∂u.(45)
The characteristic equation of () is given by
dx
α=du
x+t=dt
0.(46)
From equation () the invariants are given by I1=
αu −1
2(x+t)2,I2=t.
If we let I1be a function of I2,
u(t, x) = 1
α((x+t)2
2+φ(t))where φ(t) = I1i.e I1=φ(I2).
(47)
The substitution of (47) into (44) asserts that
ut=1
αn(x+t)−φ0(t)o
utt =1
α(1 −φ00 (t))
ux=1
α(x+t)
uxx =1
α.
(48)
Hence utt +ut−cuxx −ux=
1
αn1−φ00 (t)+(x+t)−(x+t)−c−φ0(t)o=
0,
simplifies to
φ00 (t) + φ0(t)=1−c. (49)
Solving the equation (49) we obtain that
φ(t) = c1−c2e−t+ (1 −t) (1 −c).(50)
Therefore the invariant solution of (44) is given
by
u(t, x) = 1
α((x+t)2
2+c1−c2e−t+ (1 −t) (1 −c)).
(51)
5 Conclusion
The purpose of the project was to gain an insight
into the method of Group classification on a non lin-
ear wave equation with dissipation. From the present
project, the methods of determining the principal Lie
algebra, the equivalence Lie algebra have been gained.
However, the technique and methods of finding opti-
mal systems of one-dimensional subalgebras, the ex-
tension of the principal Lie algebra by one for a va-
riety of differential equations has been acquired. We
would like to explore them further and even for higher
dimensional subalgebras. Future projects would also
include extending on the current one to determine a
complete classification for the equation (2).
Acknowledgements: The author would like to ac-
knowledge the assistance of fellow colleagues in go-
ing through the manuscript and for their invaluable
inputs. The research was supported by the financial
assistance from Tshwane University of Technology.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.90
Tshidiso Masebe
E-ISSN: 2224-2880
790
Volume 21, 2022
References:
[1] Blumen,G.W. Kumei,S. 1989. Symmetries and
Differential Equations. New York, Springer-
Verlag.
[2] Blumen,G.W. Anco,S.C. 2002. Symmetries and
Integration methods for Differential Equations.
New York. Springer-Verlag.
[3] Boyko, V.M., Lokaziuk, O.V. and Popovych,
R.O.,2021. Realizations of Lie algebras on
the line and the new group classification of
(1+1)-dimensional generalized nonlinear Klein-
Gordon equations. Anal.Math.Phys. 11, p.127.
https://doi.org/10.1007/s13324-021-00550-z.
[4] Hydon, P.E 2000. Symmetry methods for Differ-
ential Equations. New York. Cambridge Univer-
sity Press .
[5] Ibragimov,N.H. & Torrisi,M. and A.
Valenti,.(1991). Preliminary Group-
Classification of the equations utt =
f(x, ux)uxx +g(x, ux).J. Math. Phys.32,
2988-2995. June
[6] Ibragimov, N.H.1999. Elementary Lie Group
Analysis and Ordinary Differential Equations.
London. J. Wiley & Sons Ltd.
[7] Ibragimov, N.H.2010. A Practical Course in
Differential Equations and Mathematical Mod-
elling. Beijing. J. World Scientific Publishing
Co. Pty. Ltd.
[8] Kumar, S., Kumar, D. and Wazwaz, AM.
2021.Lie symmetries, optimal system, group-
invariant solutions and dynamical behaviors of
solitary wave solutions for a (3+1)-dimensional
KdV-type equation. Eur. Phys. J. Plus 136,
p.531. https://doi.org/10.1140/epjp/s13360-021-
01528-3
[9] Vaneeva, O.O., Bihlo, A. and Popovych, R.O.,
2020. Generalization of the algebraic method of
group classification with application to nonlinear
wave and elliptic equations. Communications in
Nonlinear Science and Numerical Simulation,
91, p.105419.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.90
Tshidiso Masebe
E-ISSN: 2224-2880
791
Volume 21, 2022
