JOURNAL OF MATHEMATICAL PHYSICS 46, 023505 共2005兲
Local and nonlocal symmetries for nonlinear telegraph
equation
G. W. Bluman,a兲 Temuerchaolu,b兲 and R. Sahadevanc兲
Department of Mathematics, University of British Columbia, Vancouver,
BC, Canada V6T 1Z2
共Received 18 August 2004; accepted 12 October 2004; published online 2 February 2005兲
In this paper, local and nonlocal symmetry classifications are considered for four
equivalent nonlinear telegraph equations. A complete potential symmetry classification of a scalar nonlinear telegraph equation is given through the point symmetry
classification of a related potential system. Six new classes of equations are shown
to admit potential symmetries. The relationships between local 共including contact兲
and nonlocal 共potential兲 symmetries of these equations are explored. A physical
example is considered for possible applications of the obtained potential
symmetries. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1841481兴
I. INTRODUCTION
In Refs. 1–4, an algorithmic procedure has been developed to find nonlocal symmetries
共potential symmetries兲 of partial differential equations 共PDEs兲 to extend the classes of symmetries
admitted by PDEs. Various researchers have found examples of PDEs that admit potential symmetries or extended the procedure to find potential systems.5
In recent years, there have been several investigations 共Refs. 6–8兲 to find symmetries for
nonlinear telegraph equations of the form
utt = 关F共u兲ux兴x + 关G共u兲兴x .
共1兲
PDE 共1兲 is equivalent to the potential system
vx = ut ,
vt = F共u兲ux + G共u兲.
共2兲
In particular, if 共u , v兲 = 共U共x , t兲 , V共x , t兲兲 solves 共2兲, then u = U共x , t兲 solves 共1兲. Conversely, for
any u = U共x , t兲 solving 共1兲, there exists a pair of functions 共u , v兲 = 共U共x , t兲 , V共x , t兲兲 solving 共2兲 with
V共x , t兲 unique to within a constant.
Similarly, the potential system 共2兲 is equivalent to the potential system
wt = v ,
wx = u,
共3兲
a兲
Electronic mail: bluman@math.ubc.ca
Permanent address: Inner Mongolia Polytechnic University, Hohhot, 010062, Peoples Republic of China. Electronic mail:
tmchaolu@impu.edu.cn
c兲
Permanent address: Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, Chennai
600 005, Tamil Nadu, India. Electronic mail: sahadevan@unom.ac.in and riasm@md3.vsnl.net.in
b兲
0022-2488/2005/46共2兲/023505/12/$22.50
46, 023505-1
© 2005 American Institute of Physics
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023505-2
Bluman, Temuerchaolu, and Sahadevan
J. Math. Phys. 46, 023505 共2005兲
vt = F共u兲ux + G共u兲,
and hence to the potential equation
wtt = F共wx兲wxx + G共wx兲.
共4兲
In particular, if u = U共x , t兲 solves 共1兲, then from the integrability conditions associated with 共3兲
there exists a triplet 共u , v , w兲 = 共U共x , t兲 , V共x , t兲 , W共x , t兲兲 solving system 共3兲 with w = W共x , t兲 solving
共4兲. Conversely, if w = W共x , t兲 solves 共4兲, then 共u , v兲 = 共Wx共x , t兲 , Wt共x , t兲兲 solves 共2兲 and u
= Wx共x , t兲 solves 共1兲.
Consequently, a symmetry of any one of the PDE systems 共1兲–共4兲 defines a symmetry of the
remaining three equivalent systems. Moreover, due to the relationship connecting these four
equivalent systems, it is possible for a local symmetry of any one of these systems to yield a
nonlocal symmetry of another one.9
Equations related to 共1兲 include the nonlinear heat conduction equation when F共u兲 = 0, and the
nonlinear inhomogeneous vibrating string equation when G共u兲 = 0.
The group properties of the nonlinear heat equation for both the scalar form 共1兲 and the
potential system 共2兲 are presented in Ref. 2. The point symmetry classification of the nonlinear
wave equation 共1兲 with G共u兲 = 0 and F共u兲 replaced by F共x , u兲 is discussed in Ref. 6. The complete
point symmetry classifications of equation 共1兲 and the equivalent potential equation 共4兲 are given
in Refs. 7 and 8, respectively. Some exact solutions of system 共2兲 are given in Ref. 10 for special
forms of F共u兲 and G共u兲.
Among the equivalent systems 共1兲–共4兲, it appears that the potential system 共2兲 arises most
directly in physical situations. One physical example directly related to system 共2兲 is represented
by the equations of telegraphy of a two-conductor transmission line with v as the current in the
conductors, u as the voltage between the conductors, G共u兲 as the leakage current per unit length,
F共u兲 as the differential capacitance, t as a spatial variable and x as time.11 Another physical
example related to system 共2兲 is the equation of motion of a hyperelastic homogeneous rod whose
cross-sectional area varies exponentially along the rod. Here u is the displacement gradient related
to the difference between a spatial Eulerian coordinate and a Lagrangian coordinate x, v is the
velocity of a particle displaced by this difference, G共u兲 is essentially the stress tensor, F共u兲
= ␭G⬘共u兲 for some constant ␭, and t is time 共see Refs. 12 and 13兲.
In this paper we give the complete point symmetry classification of the potential system 共2兲
and compare our results with the complete point symmetry classification of the scalar equation 共1兲
included in Ref. 7 and the complete point symmetry classification of the potential equation 共4兲
given in Ref. 8. In particular, we will show the following.
共I兲 The point symmetry classifications of the scalar equations 共1兲 and 共4兲 are identical, i.e., for
any F共u兲 and G共u兲, a point symmetry admitted by 共1兲 induces a point symmetry admitted by 共4兲
and vice versa.
共II兲 For wide classes of F共u兲 and G共u兲, there exist point symmetries of the potential system 共2兲
which are nonlocal symmetries of the scalar equation 共1兲.
共III兲 Each point symmetry of the potential system 共2兲 which is a nonlocal symmetry of 共1兲
yields a contact symmetry of the potential equation 共4兲 that is not a point symmetry of 共4兲.
共IV兲 For all but one particular class of F共u兲 and G共u兲, a point symmetry of the scalar equation
共1兲 is a point symmetry of 共2兲.
In Sec. II, we give the set of determining equations for point symmetries of the potential
system 共2兲 and the complete potential symmetry classification of the scalar equation 共1兲 related to
共2兲. In Secs. III and IV, we present the complete point symmetry classifications of the scalar
equations 共1兲 and 共4兲 given in Refs. 7 and 8, respectively. In Sec. V, we compare the point
symmetry classifications of the systems 共1兲, 共2兲, and 共4兲 by proving theorems that yield statements
共I兲–共IV兲. The second physical example is considered in Sec. VI. Further comments are given in
Sec. VII.
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023505-3
J. Math. Phys. 46, 023505 共2005兲
Local and nonlocal symmetries
II. POTENTIAL SYMMETRY CLASSIFICATION OF THE SCALAR EQUATION „1…
Consider the potential system 共2兲.
The point symmetry
x * = x + ␧␰共x,t,u, v兲 + O共␧2兲,
t * = t + ␧␶共x,t,u, v兲 + O共␧2兲,
u * = u + ␧␩共x,t,u, v兲 + O共␧2兲,
v * = v + ␧␾共x,t,u, v兲 + O共␧2兲,
共5兲
is admitted by system 共2兲 if and only if it satisfies the determining equations,
X共1兲共vx − ut兲 = 0,
X共1兲共vt − F共u兲ux − G共u兲兲 = 0,
共6兲
for any 共u , v兲 that solves system 共2兲;
X = ␰共x,t,u, v兲
⳵
⳵
⳵
⳵
+ ␶共x,t,u, v兲 + ␩共x,t,u, v兲 + ␾共x,t,u, v兲
⳵x
⳵t
⳵u
⳵v
is the infinitesimal generator of the point symmetry 共5兲;
X共1兲 = X + ␩共1兲
⳵
⳵
⳵
⳵
+ ␩共2兲
+ ␾共1兲
+ ␾共2兲 ,
⳵ux
⳵ut
⳵vx
⳵vt
with
␩共1兲 =
D␩ D␰
D␶
−
ux −
u t,
Dx Dx
Dx
␩共2兲 =
D␩ D␰
D␶
−
ux −
ut ,
Dt Dt
Dt
␾共1兲 =
D␾ D␰
D␶
−
vx −
v t,
Dx Dx
Dx
␾共2兲 =
D␾ D␰
D␶
−
vx −
vt ,
Dt Dt
Dt
is the first extension 共prolongation兲 of X;
⳵
⳵
⳵
⳵
⳵
⳵
D
⳵
=
+ ux + vx + uxx
+ uxt
+ vxx
+ vxt ,
⳵u
⳵v
⳵ux
⳵ut
⳵vx
⳵vt
Dx ⳵x
⳵
⳵
⳵
⳵
⳵
⳵
D ⳵
= + ut + vt + uxt
+ utt
+ vxt
+ vtt
⳵u
⳵v
⳵ux
⳵ut
⳵vx
⳵vt
Dt ⳵t
are total derivative operators. Note that ␩共i兲 , ␾共i兲 are functions of x , t , u , v , ux , ut , vx , vt, i = 1 , 2.
The global one-parameter 共␧兲 Lie group of point symmetries associated with 共5兲 is obtained by
solving the initial value problem for the first order system of ordinary differential equations
共ODEs兲
dx*
= ␰共x * ,t * ,u * , v * 兲,
d␧
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023505-4
J. Math. Phys. 46, 023505 共2005兲
Bluman, Temuerchaolu, and Sahadevan
dt*
= ␶共x * ,t * ,u * , v * 兲,
d␧
du*
= ␩共x * ,t * ,u * , v * 兲,
d␧
dv *
= ␾共x * ,t * ,u * , v * 兲,
d␧
共7兲
with x * = x, t * = t, u * = u, v * = v at ␧ = 0.
A point symmetry 共5兲 yields a nonlocal symmetry of the scalar equation 共1兲 if and only if
␰2v + ␶2v + ␩2v ⫽ 0, i.e., if and only if the infinitesimals ␰ , ␶, and ␩ have an essential dependence on the
potential variable v. Such a nonlocal symmetry is called a potential symmetry of scalar equation
共1兲 related to the potential system 共2兲 共for details, see Ref. 3兲.
The determining equations 共6兲 simplify to
␰v − ␶u = 0,
␩u − ␾v + ␰x − ␶t = 0,
G共u兲␩v + ␩t − ␾x + G共u兲␶x = 0,
␰u − F共u兲␶v = 0,
␾u − G共u兲␶u − F共u兲␩v = 0,
G共u兲␰v + ␰t − F共u兲␶x = 0,
关␾v − ␶t − 2G共u兲␶v − ␩u + ␰x兴F共u兲 − F⬘共u兲␩ = 0,
关␾v − ␶t − G共u兲␶v兴G共u兲 − F共u兲␩x − G⬘共u兲␩ + ␾t = 0.
共8兲
We now consider the classification problem of finding all F共u兲, G共u兲 such that the potential
system 共2兲 yields a potential symmetry of 共1兲.
If
F共u兲 =
c
,
共au + b兲2
G共u兲 =
d
+f
au + b
共9兲
for arbitrary constants a , b , c , d , f, or if
F共u兲 is arbitrary,
G共u兲 = const,
then through the potential system 共2兲, the scalar equation admits an infinite number of potential
symmetries and the potential system 共2兲 is linearizable by a point transformation 共see Refs. 3 and
4兲.
Various symbolic manipulation algorithms exist to solve the set of determining equations 共8兲
共for example, see Refs. 14 and 15兲. Using the symmetry manipulation algorithm presented in Ref.
15, one can prove the following results.
Theorem 1: The scalar equation (1) admits a potential symmetry related to potential system
(2) if and only if the functions F共u兲 and G共u兲, with G⬘共u兲 ⫽ 0, satisfy the system of ODEs,
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023505-5
J. Math. Phys. 46, 023505 共2005兲
Local and nonlocal symmetries
共c4u + c5兲F⬘共u兲 − 2共c1 − c3 − c2G共u兲兲F共u兲 = 0,
共10兲
共c4u + c5兲G⬘共u兲 + c2G2共u兲 − 共c1 − 2c3 + c4兲G共u兲 − c6 = 0,
共11兲
for any fixed constants c1 , c2 , . . . , c6 with c2 ⫽ 0.
In the linearizable case 共9兲,
c1 = 0,
c6 =
c3共c4 − c3兲
.
c2
Theorem 2: For any F共u兲, G共u兲 satisfying the system of ODEs (10) and (11) with c2 ⫽ 0, the
potential system (2) admits the point symmetry (5) with
␰ = c 1x + c 2
冕
F共u兲du,
␶ = c 3t + c 2v ,
␩ = c 4u + c 5 ,
␾ = c6t + 共c1 − c3 + c4兲v ,
共12兲
and hence the scalar equation (1) admits the corresponding potential symmetry.
Now we find the functions F共u兲 and G共u兲 satisfying 共10兲, 共11兲 and the corresponding potential
symmetries 共12兲.
Note that the point transformation
x̄ = ax + b,
t̄ = ct + d,
ū = ␣u + ␤,
v̄ = ␥v + ␳t
for any constants a, b, c, d, ␣, ␤, ␥, and ␳ such that ac␣␥ ⫽ 0 and a␣ = c␥ is an equivalence
transformation for system 共2兲. Under this transformation, system 共2兲 becomes the equivalent
system
v̄x̄ = ūt̄,
v̄t̄ = F̄共ū兲ūx̄ + Ḡ共ū兲,
where
F̄共ū兲 =
冉 冊
a␥ ū − ␤
F
c␣
␣
and
冉 冊
␥ ū − ␤
␳
+ .
Ḡ共ū兲 = G
c
␣
c
We use such equivalence transformations to simplify the analysis. For example, if G共u兲 = a0共b0u
+ c0兲d0 + f 0, without loss of generality we can assume that G共u兲 = ud0.
Modulo translations and scalings in u and G, we obtain six distinct classes of ODEs for F共u兲
and G共u兲 where scalar equation 共1兲 admits potential symmetries. These six classes of ODEs and
their solutions 关modulo equivalence classes of F共u兲 and G共u兲兴 are presented in Table I. In Table II
for each class we display the corresponding infinitesimals 共␰ , ␶ , ␩ , ␾兲 and global group
共x * , t * , u * , v * 兲 obtained from solving the corresponding ODEs 共7兲.
All symmetries presented in Tables I and II are new for each of the equivalent systems 共1兲–共4兲.
Note that classes 1 and 6 are linearizable4 if ␤ = 0 and ␣ = 1 / 2.
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023505-6
J. Math. Phys. 46, 023505 共2005兲
Bluman, Temuerchaolu, and Sahadevan
TABLE I. Classes of F共u兲 and G共u兲 yielding potential symmetries of 共1兲.
ODEs satisfied by
F共u兲 and G共u兲
G共u兲
F共u兲
1
uG⬘ − ␣共1 − G2兲 = 0
uF⬘ − 共␤ − 1 − 2␣G兲F = 0
共u2␣ − 1兲 / 共u2␣ + 1兲
共u2␣ + 1兲 / 共u2␣ − 1兲
4u2␣+␤−1 / 共u2␣ + 1兲2
−4u2␣+␤−1 / 共u2␣ − 1兲2
F共u兲 = 共u␤ / ␣兲G⬘共u兲
2
uG⬘ − ␣共1 + G2兲 = 0
uF⬘ + 共1 − ␤ − 2␣G兲F = 0
tan共␣ ln u兲
u␤−1 sec2共␣ ln u兲
’’
3
uG⬘ + G2 = 0
uF⬘ − 共␤ − 1 − 2G兲F = 0
共ln u兲−1
−u␤−1共ln u兲−2
’’
4
G⬘ − G2 − 1 = 0
F⬘ − 2共␤ + G兲F = 0
tan u
e2␤u sec2 u
F共u兲 = e2␤uG⬘共u兲
5
G⬘ + G2 − 1 = 0
F⬘ − 2共␤ − G兲F = 0
tanh u
coth u
e2␤u sech2 u
−e2␤u csch2 u
’’
6
G⬘ + G2 = 0
F⬘ − 2共␤ − G兲F = 0
u−1
−u−2e2␤u
’’
Class
Relationship between
F共u兲 and G共u兲
III. POINT SYMMETRY CLASSIFICATION OF THE SCALAR EQUATION „1…
In Ref. 7, Kingston and Sophocleous considered the classification problem of finding all F共u兲,
G共u兲 such that the scalar equation 共1兲 admits a point symmetry. The point symmetry
x * = x + ␧␰共x,t,u兲 + O共␧2兲,
TABLE II. Potential
= 兰u+␧e−2␤s共兰sF共x兲dx兲ds兴.
symmetries
Class
1
2
F共u兲 = 4u2␣+␤−1 / 共u2␣ + 1兲2
G共u兲 = 共u2␣ − 1兲 / 共u2␣ + 1兲 or
F共u兲 = −4u2␣+␤−1 / 共u2␣ − 1兲2
G共u兲 = 共u2␣ + 1兲 / 共u2␣ − 1兲
F共u兲 = u␤−1 sec2共␣ ln u兲
G共u兲 = tan共␣ ln u兲
3
F共u兲 = −u␤−1共ln u兲−2
G共u兲 = 共ln u兲−1
4
F共u兲 = e2␤u sec2 u
G共u兲 = tan u
5
F共u兲 = e2␤u sech2 u
G共u兲 = tanh u or
F共u兲 = −e2␤u csch2 u
G共u兲 = coth u
6
F共u兲 = −u−2e2␤u
G共u兲 = u−1
of
共1兲
for
each
class
Infinitesimals
␰,␶,␩,␾
␰ = 2共␤x + ␣ 兰 F共u兲du兲
␶ = 共␤ + 1兲t + 2␣v
␩ = 2u
␾ = 2␣t + 共␤ + 1兲v
␰ = 2共␤x − ␣ 兰 F共u兲du兲
␶ = 共␤ + 1兲t − 2␣v
␩ = 2u
␾ = 2␣t + 共␤ + 1兲v
␰ = 2共␤x + 兰 F共u兲du兲
␶ = 共␤ + 1兲t + 2v
␩ = 2u
␾ = 共␤ + 1兲v
␰ = 2␤x − 兰F共u兲du
␶ = ␤t − v
␩=1
␾ = t + ␤v
␰ = 2␤x + 兰F共u兲du
␶ = ␤t + v
␩=1
␾ = t + ␤v
␰ = 2␤x + 兰F共u兲du
␶ = ␤t + v
␩=1
␾ = ␤v
2␧
关⌫共␤ , F , ␧兲 = 兰ue s−共1+␤兲共兰sF共x兲dx兲ds,
⍀共␤ , F , ␧兲
Global group
x * = e2␤␧关x + ␣u␤共⌫共␤ , F , ␧兲 − ⌫共␤ , F , 0兲兲兴
t * = 共1 / 2兲e共␤+1兲␧关共t + v兲e2␣␧ + 共t − v兲e−2␣␧兴
u * = ue2␧
v * = 共1 / 2兲e共␤+1兲␧关共t + v兲e2␣␧ − 共t − v兲e−2␣␧兴
x * = e2␤␧关x − ␣u␤共⌫共␤ , F , ␧兲 − ⌫共␤ , F , 0兲兲兴
t * = e共␤+1兲␧关t cos 2␣␧ − v sin 2␣␧兴
u * = ue2␧
v * = e共␤+1兲␧关v cos 2␣␧ + t sin 2␣␧兴
x * = e2␤␧关x + u␤共⌫共␤ , F , ␧兲 − ⌫共␤ , F , 0兲兲兴
t * = e共␤+1兲␧共2v␧ + t兲
u * = ue2␧
v * = ve共␤+1兲␧
x * = e2␤␧关x − e2␤u共⍀共␤ , F , ␧兲 − ⍀共␤ , F , 0兲兲兴
t * = e␤␧共t cos ␧ − v sin ␧兲
u* =u+␧
v * = e␤␧共v cos ␧ + t sin ␧兲
x * = e2␤␧关x + e2␤u共⍀共␤ , F , ␧兲 − ⍀共␤ , F , 0兲兲兴
t * = 共1 / 2兲e␤␧共共t + v兲e␧ + 共t − v兲e−␧兲
u* =u+␧
v * = 共1 / 2兲e␤␧共共t + v兲e␧ − 共t − v兲e−␧兲
x * = e2␤␧关x + e2␤u共⍀共␤ , F , ␧兲 − ⍀共␤ , F , 0兲兲兴
t * = e␤␧共t + v␧兲
u* =u+␧
v * = v e ␤␧
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023505-7
J. Math. Phys. 46, 023505 共2005兲
Local and nonlocal symmetries
TABLE III. Classes of F共u兲 and G共u兲 yielding point symmetries of 共1兲.
Class
G共u兲
F共u兲
A
B
eu
u␣+␤+1
e共␣+1兲u
C
u−1
u−2
D
E
ln u
u
u␣
e ␣u
F
u−3
u−4
u␣
Infinitesimals ␰ , ␶ , ␩
␰ = 2␣x , ␶ = 共␣ − 1兲t , ␩ = 2
␰ = 2␤x , ␶ = 共␣ + 2␤兲t , ␩ = −2u
Those in class B and
␰ = ex , ␶ = 0 , ␩ = −uex
␰ = 2共␣ + 1兲x , ␶ = 共␣ + 2兲t , ␩ = 2u
␰ = 2␣x , ␶ = ␣t , ␩ = 2
Those in class B and
␰ = 0 , ␶ = t2 , ␩ = ut
t * = t + ␧␶共x,t,u兲 + O共␧2兲,
u * = u + ␧␩共x,t,u兲 + O共␧2兲,
共13兲
is admitted by the scalar equation 共1兲 if and only if it satisfies the determining equation
X共2兲共utt − 共F共u兲ux兲x − G共u兲x兲 = 0
共14兲
for any u solving the scalar equation 共1兲;
X = ␰共x,t,u兲
⳵
⳵
⳵
+ ␶共x,t,u兲 + ␩共x,t,u兲 ,
⳵x
⳵t
⳵u
is the infinitesimal generator of the point symmetry 共13兲; X共2兲 is the second extension of X.
From 共14兲, the determining equations for ␰共x , t , u兲, ␶共x , t , u兲, and ␩共x , t , u兲 are given by
␰u = ␶x = ␶u = ␩uu = 0,
␰t − F共u兲␶x = 0,
2F共u兲共␰x − ␶t兲 − F⬘共u兲␩ = 0,
␩tt − F共u兲␩xx − G⬘共u兲␩x = 0,
2␩tu − ␶tt + F共u兲␶xx + G⬘共u兲␶x = 0,
2F共u兲␩xu + ␰tt − F共u兲␰xx + 2F⬘共u兲␩x − G⬘共u兲共␰x − 2␶t兲 + G⬙共u兲␩ = 0.
For arbitrary F共u兲 and G共u兲, clearly each of the equivalent systems 共1兲–共4兲 admits translations
in x共␰ = 1 , ␶ = 0兲 and t共␰ = 0 , ␶ = 1兲. For specific F共u兲 and G共u兲 with G⬘共u兲 ⫽ 0, the results are
summarized in Table III.
Note that the classes presented in Ref. 7 where F共u兲 = 1, G共u兲 = e␭u as well as F共u兲 = 1, G共u兲
= u␭+1 can be obtained, respectively, by appropriately scaling u in class A and setting ␣ = 0 in class
B.
IV. POINT SYMMETRY CLASSIFICATION OF THE SCALAR EQUATION „4…
In Ref. 8, the authors considered the classification problem of finding all F共u兲, G共u兲 such that
the scalar equation 共4兲 admits a point symmetry. The point symmetry
x * = x + ␧␰共x,t,w兲 + O共␧2兲,
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023505-8
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Bluman, Temuerchaolu, and Sahadevan
TABLE IV. Classes of F共u兲 and G共u兲 yielding point symmetries of 共4兲.
Class
G共u兲 = G共wx兲
F共u兲 = F共wx兲
A
B
eu
u␣+␤+1
e共␣+1兲u
C
u−1
u−2
D
ln u
u␣
E
F
u
u−3
e ␣u
u−4
u␣
Infinitesimals ␰ , ␶ , ␩
␰ = 2␣x , ␶ = 共␣ − 1兲t , ⍀ = 2共␣w + x兲
␰ = 2␤x , ␶ = 共␣ + 2␤兲t , ⍀ = 2共␤ − 1兲w
Those in class B and
␰ = ex , ␶ = 0 , ⍀ = 0
␰ = 2共␣ + 1兲x , ␶ = 共␣ + 2兲t , ⍀ = 2共␣ + 2兲w + t2
␰ = 2␣x , ␶ = ␣t , ⍀ = 2␣w + t2 + 2x
Those in class B and
␰ = 0 , ␶ = t2 , ⍀ = tw
t * = t + ␧␶共x,t,w兲 + O共␧2兲,
w * = w + ␧⍀共x,t,w兲 + O共␧2兲,
共15兲
is admitted by the scalar equation 共4兲 if and only if it satisfies the determining equation
X共2兲共wtt − F共wx兲wxx − G共wx兲兲 = 0
for any w solving the scalar equation 共4兲;
X = ␰共x,t,w兲
⳵
⳵
⳵
+ ␶共x,t,w兲 + ⍀共x,t,w兲 ,
⳵x
⳵t
⳵w
is the infinitesimal generator of the point symmetry 共15兲; X共2兲 is the second extension of X.
Modulo equivalence transformations, the results presented in Ref. 8 are summarized in Table
IV.
V. DISCUSSION OF THE SYMMETRY CLASSIFICATIONS
In this section we prove the statements 共I兲–共IV兲 presented in the Introduction through proofs
of the following four theorems.
Theorem 3: The point symmetry classifications of the scalar equations (1) and (4) are identical, i.e., for any F共u兲 and G共u兲 a point symmetry admitted by (1) induces a point symmetry
admitted by (4) and vice versa.
Proof: The infinitesimals of a point symmetry of the scalar equation 共1兲 are of the form
␰ = ␰共x兲,
␶ = ␶共t兲,
␩ = 共f共t兲 − ␰⬘共x兲兲u + b
for some functions ␰共x兲, ␶共t兲, f共t兲 and constant b. A corresponding point symmetry of the potential
scalar equation 共4兲 共with u = wx兲 must have ␰ = ␰共x兲, ␶ = ␶共t兲, ⍀ = ⍀共x , t , w兲 with ⍀共x , t , w兲 satisfying
⍀x共1兲 = ␩
in terms of its first extended infinitesimal ⍀x共1兲, i.e.,
⍀x共1兲 =
D⍀ D␰
D␶
−
wx −
wt = ⍀x + ⍀wwx − ␰⬘共x兲wx = 共f共t兲 − ␰⬘共x兲兲wx + b.
Dx Dx
Dx
共16兲
Hence from Eq. 共16兲, we must have
⍀x = b,
⍀w = f共t兲.
Then it is necessary that
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023505-9
J. Math. Phys. 46, 023505 共2005兲
Local and nonlocal symmetries
⍀ = bx + f共t兲w + g共t兲,
共17兲
for some g共t兲 in order that a point symmetry of the scalar equation 共1兲 yields a point symmetry of
the scalar equation 共4兲. From Tables III and IV, it is easy to check that condition 共17兲 is satisfied
for each point symmetry of the scalar equation 共4兲.
Conversely, the infinitesimals of a point symmetry of 共4兲 are of the form
␰ = ␰共x兲,
␶ = ␶共t兲,
⍀ = f共t兲w + ␤共x,t兲
for some functions ␰共x兲, ␶共t兲, f共t兲, and ␤共x , t兲. A corresponding point symmetry of the scalar
equation 共1兲 with u = wx must have
␰ = ␰共x兲,
␶ = ␶共t兲,
␩ = ␩共x,t,u兲.
In terms of the first extended infinitesimal ⍀x共1兲, the infinitesimal ␩共x , t , u兲 must satisfy
␩ = ⍀x共1兲 =
D⍀ D␰
D␶
−
wx −
wt = ⍀x + ⍀wwx − ␰⬘共x兲wx = ␤x + 共f共t兲 − ␰⬘共x兲兲wx .
Dx Dx
Dx
共18兲
From Tables III and IV, it is easy to check that condition 共18兲 is satisfied for each point symmetry
of 共1兲.
䊏
Theorem 4: For each of the six classes of F共u兲 and G共u兲 listed in Table I, there exist point
symmetries of the potential system (2) which are nonlocal symmetries of the scalar equation (1).
Proof: Since each point symmetry in Table II has ␶v ⫽ 0, it follows that for all classes of F共u兲
and G共u兲 listed in Table I 共also repeated in Table II兲 there exist point symmetries of the potential
system 共2兲 which are nonlocal 共potential兲 symmetries of the scalar equation 共1兲.
䊏
Note that only the class F共u兲 = u−2, G共u兲 = u−1 is common for Tables I and III. This class is
linearizable since it admits potential symmetries leading to the linearization of 共2兲 by a point
transformation.4
Theorem 5: Each point symmetry of the potential system (2) which is a nonlocal symmetry of
the scalar equation (1) yields a contact symmetry of the potential equation (4) that is not a point
symmetry of (4).
Proof: A contact symmetry of a PDE with dependent variable w and independent variables x
and t is defined by
x * = x + ␧␰共x,t,w,wx,wt兲 + O共␧2兲,
t * = t + ␧␶共x,t,w,wx,wt兲 + O共␧2兲,
w * = w + ␧⍀共x,t,w,wx,wt兲 + O共␧2兲,
共19兲
if and only if
⳵⍀
⳵␰
⳵␶
=
wx +
w t,
⳵wx ⳵wx
⳵wx
⳵⍀ ⳵␰
⳵␶
=
wx +
wt .
⳵wt ⳵wt
⳵wt
共20兲
Let characteristic function W = ⍀ − ␰wx − ␶wt. A contact symmetry 共19兲 is a point symmetry if and
only if
⳵ 2W
⳵ 2W
⳵ 2W
=
=
= 0.
⳵共wx兲2 ⳵共wt兲2 ⳵wx⳵wt
共21兲
For details, see Chap. 5 of Ref. 3.
A point symmetry of potential system 共2兲 which is a nonlocal symmetry of the scalar equation
共1兲 共see Table II兲 is of the form
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023505-10
J. Math. Phys. 46, 023505 共2005兲
Bluman, Temuerchaolu, and Sahadevan
␰ = ax + b
冕
u
F共s兲ds,
␩ = fu + g,
␶ = ct + dv ,
␾ = ht + kv ,
共22兲
for some constants a, b, c, d, f, g, h, and k with d ⫽ 0, a + f = c + k.
Solving 共20兲, after using the substitution 共22兲, we find that
⍀共x,t,w,wx,wt兲 = b
冕
wx
d
sF共s兲ds + w2t + A共x,t,w兲
2
共23兲
yields a contact symmetry for an arbitrary function A共x , t , w兲.
Now we find A共x , t , w兲 so that 共23兲 yields a contact symmetry of the potential equation 共4兲.
Since u = wx, v = wt, it follows that we must have
␩ = ⍀x共1兲 =
D⍀ D␰
D␶
−
wx −
wt ,
Dx Dx
Dx
␾ = ⍀t共1兲 =
D⍀ D␰
D␶
−
wx −
wt ,
Dt
Dt
Dt
共24兲
in terms of first extended infinitesimals ⍀x共1兲 and ⍀t共1兲. After solving the equations 共24兲, we find
that A共x , t , w兲 = gx + 共a + f兲w + 21 ht2 and hence
␰ = ax + b
⍀=b
冕
wx
冕
wx
F共s兲ds,
␶ = ct + dwt ,
1
d
sF共s兲ds + w2t + gx + 共a + f兲w + ht2 ,
2
2
共25兲
defines a contact symmetry of the scalar potential equation 共4兲. Using condition 共21兲, it is clear
that 共25兲 is not a point symmetry of 共4兲 since d ⫽ 0.
䊏
Theorem 6: A point symmetry of the scalar equation (1) yields a point symmetry of the
potential equation (2) if and only if the infinitesimals for F共u兲 and G共u兲 belong to classes A–E in
Table III.
Proof: A point symmetry 共␰共x , t , u兲 , ␶共x , t , u兲 , ␩共x , t , u兲兲 of the scalar equation 共1兲 yields a
point symmetry of 共2兲 if and only if the set of determining equations 共8兲 has a solution for
␾共x , t , u , v兲. A solution of 共8兲 exists if and only if the six integrability conditions involving the
second order mixed partial derivatives of ␾共x , t , u , v兲 are satisfied. Consequently, it is easy to show
that a point symmetry of 共1兲 yields a point symmetry of 共2兲 if and only if it satisfies the additional
conditions
␶tt = 0,
␩xu + ␰xx = 0,
␩tt = 0,
F⬘共u兲␩x − 2F共u兲␰xx = 0.
共26兲
The infinitesimals for classes A–E in Table III satisfy 共26兲 but those for class F do not since here
␶tt ⫽ 0.
䊏
As a consequence of Theorem 6, we see that the point symmetry X = t2共⳵ / ⳵t兲 + ut共⳵ / ⳵u兲 for
G共u兲 = u−3, F共u兲 = u−4 yields a nonlocal symmetry of the potential system 共2兲.
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023505-11
J. Math. Phys. 46, 023505 共2005兲
Local and nonlocal symmetries
TABLE V. Physical examples 关F共u兲 = G⬘共u兲兴 yielding potential symmetries of 共1兲.
Class
Infinitesimals
␰,␶,␩,␾
Global group
1.1
G共u兲 = 共u2␣ − 1兲 / 共u2␣ + 1兲
␰ = 2␣共共u2␣ − 1兲 / 共u2␣ + 1兲兲
␶ = t + 2␣v
␩ = 2u
␾ = 2␣t + v
1.2
G共u兲 = 共u2␣ + 1兲 / 共u2␣ − 1兲
␰ = 2␣共共u2␣ + 1兲 / 共u2␣ − 1兲兲
␶ = t + 2␣v
␩ = 2u
␾ = 2␣t + v
2
G共u兲 = tan共␣ ln u兲
3
G共u兲 = 共ln u兲−1
4
G共u兲 = tan u
5.1
G共u兲 = tanh u
␰ = −2␣ tan共␣ ln u兲
␶ = t − 2␣v
␩ = 2u
␾ = 2␣t + v
␰ = 2共ln u兲−1
␶ = t + 2v
␩ = 2u
␾=v
␰ = −tan u
␶ = −v
␩=1
␾=t
␰ = tanh u
␶=v
␩=1
␾=t
5.2
G共u兲 = coth u
␰ = coth u
␶=v
␩=1
␾=t
x * = x + ln兩cos共u + ␧兲 / cos u兩
t * = t cos ␧ − v sin ␧
u* =u+␧
v * = v cos ␧ + t sin ␧
x * = x + ln共共e2共u+␧兲 + 1兲 / 共e2u + 1兲兲 − ␧
1
t * = 2 共共t + v兲e␧ + 共t − v兲e−␧兲
u* =u+␧
1
v * = 2 共共t + v兲e␧ − 共t − v兲e−␧兲
x * = x + ln兩共e2共u+␧兲 − 1兲 / 共e2u − 1兲兩 − ␧
1
t * = 2 共共t + v兲e␧ + 共t − v兲e−␧兲
u* =u+␧
1
v * = 2 共共t + v兲e␧ − 共t − v兲e−␧兲
G共u兲 = u−1
␰ = u−1
␶=v
␩=1
␾=0
x * = x + ln兩1 + ␧ / u兩
t * = t + v␧
u* =u+␧
v* =v
6
x * = x + ln共共u2␣e4␣␧ + 1兲 / 共u2␣ + 1兲兲 − 2␣␧
1
t * = 2 e␧关共t + v兲e2␣␧ + 共t − v兲e−2␣␧兴
u * = ue2␧
1 ␧
*
=
e
关共t
+
v
v兲e2␣␧ − 共t − v兲e−2␣␧兴
2
x * = x + ln兩共u2␣e4␣␧ − 1兲 / 共u2␣ − 1兲兩 − 2␣␧
1
t * = 2 e␧关共t + v兲e2␣␧ + 共t − v兲e−2␣␧兴
u * = ue2␧
1
v * = 2 e␧关共t + v兲e2␣␧ − 共t − v兲e−2␣␧兴
x * = x + ln兩cos共␣共ln u + 2␧兲兲 / cos共␣ ln u兲兩
t * = e␧关t cos 2␣␧ − v sin 2␣␧兴
u * = ue2␧
␧
v * = e 关v cos 2␣␧ + t sin 2␣␧兴
x * = x + ln兩1 + 2␧ / ln u兩
t * = e␧共2v␧ + t兲
u * = ue2␧
v * = ve␧
VI. A PHYSICAL EXAMPLE
We now specialize to the situation for the second physical example mentioned in the Introduction. Here F共u兲 = ␭G⬘共u兲. Consequently, in Tables I and II, we have ␤ = 0 and 兰F共u兲du
= G共u兲. In Table V, we give the corresponding global group for each of the six classes yielding
potential symmetries of the scalar equation 共1兲.
After translations and scalings of u and G, class 5 includes bounded monotonic stress tensor
functions G共u兲 = ␣ tanh共␤u + ␥兲 for arbitrary constants ␣, ␤, ␥, and ␦.
VII. FURTHER DISCUSSION
In this paper we found new symmetries for equivalent telegraph equations 共1兲–共4兲. These
symmetries can be used to find families of solutions from any given solution and to construct
invariant solutions from the invariants of the symmetries or through the direct method discussed in
Ref. 3.
In future papers, we will find conservation laws for 共1兲–共4兲 through techniques introduced in
Refs. 16–18. Systems 共1兲–共4兲 are not self-adjoint. Hence a symmetry of 共1兲–共4兲 does not yield a
conservation law through Noether’s theorem.
For the physical case F共u兲 = G⬘共u兲, the second equation in the potential system 共2兲 becomes
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023505-12
J. Math. Phys. 46, 023505 共2005兲
Bluman, Temuerchaolu, and Sahadevan
共exv兲t = 共exG共u兲兲x .
This leads to another equivalent potential system
v x = u t,
w x = e xv ,
wt = exG共u兲.
The problem of finding equivalent potential systems for a given PDE is considered in Refs. 4 and
5.
ACKNOWLEDGMENT
The authors acknowledge financial support from the National Sciences and Engineering Research Council of Canada.
G. Bluman and S. Kumei, J. Math. Phys. 28, 307 共1987兲.
G. Bluman, S. Kumei, and G. Reid, J. Math. Phys. 29, 806 共1988兲; 29, 2320共E兲 共1988兲.
3
G. Bluman and S. Kumei, Symmetries and Differential Equations, Appl. Math. Sci. No. 81 共Springer, New York, 1989兲.
4
G. Bluman and P. Doran-Wu, Acta Appl. Math. 41, 21 共1995兲.
5
R. O. Popovych and N. M. Ivanova, math-ph/0407008.
6
M. Torrisi and A. Valenti, Int. J. Non-Linear Mech. 20, 135 共1985兲.
7
G. Kingston and C. Sophocleous, Int. J. Non-Linear Mech. 36, 987 共2001兲.
8
M. Gandarias, M. Torrisi, and A. Valenti, Int. J. Non-Linear Mech. 39, 389 共2004兲.
9
G. Bluman, An Overview of Potential Symmetries, Lectures in Applied Mathematics 共American Mathematical Society,
Providence, RI, 1993兲, Vol. 29, pp. 97–109.
10
E. Varley and B. Seymour, Stud. Appl. Math. 72, 241 共1985兲.
11
I. G. Katayev, Electromagnetic Shock Waves 共Iliffe, London, 1966兲.
12
A. Jeffrey, Wave Motion 4, 173 共1982兲.
13
A. C. Eringen and E. S. Suhubi, Elastodynamics 1: Finite Motions 共Academic, New York, 1974兲.
14
W. Hereman, Euromath Bull 1, 45 共1994兲.
15
Temuerchaolu, Adv. Math. 32, 208 共2003兲.
16
S. Anco and G. Bluman, Phys. Rev. Lett. 78, 2869 共1997兲.
17
S. Anco and G. Bluman, Eur. J. Appl. Math. 13, 545 共2002兲.
18
S. Anco and G. Bluman, Eur. J. Appl. Math. 13, 567 共2002兲.
1
2
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