JOURNAL OF MATHEMATICAL PHYSICS 46, 073513 共2005兲
Comparing symmetries and conservation laws of
nonlinear telegraph equations
G. W. Blumana兲 and Temuerchaolub兲
Department of Mathematics, University of British Columbia, Vancouver, BC,
Canada V6T 1Z2
共Received 8 February 2005; accepted 18 March 2005; published online 6 July 2005兲
A comparison is made between the symmetries and conservation laws admitted by
nonlinear telegraph 共NLT兲 systems. Such systems are not variational. Unlike the
situation for variational systems where all conservation laws arise from symmetries, there are many NLT systems that admit more conservation laws than symmetries. The results are summarized in a table which includes the numbers of symmetries and conservation laws for each NLT system. It is also indicated when
symmetries map conservation laws to other conservation laws. © 2005 American
Institute of Physics. 关DOI: 10.1063/1.1915292兴
I. INTRODUCTION
In this paper, we consider the problem of comparing the multipliers of conservation laws and
the point symmetries of a given nonlinear system of partial differential equations 共PDEs兲 that is
not variational, i.e., whose associated Fréchet derivative is not self-adjoint. As a protypical example, we consider nonlinear telegraph 共NLT兲 systems of the form
H1关u, v兴 = vt − F共u兲ux − G共u兲 = 0,
H2关u, v兴 = ut − vx = 0.
共1兲
One physical example related to system 共1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.1 Another physical example related to system 共1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. 2 and 3兲.
A point symmetry
x* = x + ␰ˆ 共x,t,u, v兲␧ + O共␧2兲,
t* = t + ␶ˆ 共x,t,u, v兲␧ + O共␧2兲,
u* = u + ␩ˆ 共x,t,u, v兲␧ + O共␧2兲,
a兲
Electronic mail: bluman@math.ubc.ca
Permanent address: Inner Mongolia University of Technology, Hohhot, People’s Republic of China. Electronic mail:
tmchaolu@impu.edu.cn
b兲
0022-2488/2005/46共7兲/073513/14/$22.50
46, 073513-1
© 2005 American Institute of Physics
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073513-2
J. Math. Phys. 46, 073513 共2005兲
G. W. Bluman and T. Temuerchaolu
v* = v + ␨ˆ 共x,t,u, v兲␧ + O共␧2兲,
共2兲
with corresponding infinitesimal generator 共in evolutionary form兲
X = ␩⳵u + ␨⳵v
4–6
is admitted by system 共1兲
if and only if
X共1兲H1关u, v兴
X共1兲H2关u, v兴
where X
共1兲
冏
冏
H1关u,v兴=0,
H2关u,v兴=0
H1关u,v兴=0,
H2关u,v兴=0
冏
冏
= 0,
共3兲
= 0,
is the first extension of X and
␩ = ␩ˆ 共x,t,u, v兲 − ␰ˆ 共x,t,u, v兲ux − ␶ˆ 共x,t,u, v兲ut ,
␨ = ␨ˆ 共x,t,u, v兲 − ␰ˆ 共x,t,u, v兲vx − ␶ˆ 共x,t,u, v兲vt .
The Fréchet derivative associated with system 共1兲 is the linear operator
L关u兴 =
冋
Dt
− Dx
− F⬘共u兲ux − F共u兲Dx − G⬘共u兲
Dt
册
,
共4兲
and it yields the linearized system of 共1兲 given by
L关u兴
冏冋 册冏
⌽
⌿
H1关u,v兴=0,
共5兲
=0
H2关u,v兴=0
in terms of total derivative operators Dx and Dt. It is easy to show that a point symmetry of system
共3兲 is any solution 共⌽ , ⌿兲 = 共␩ , ␨兲 of the linearized system 共5兲.
On the other hand, a set of multipliers
␰ = ␰共x,t,U,V兲,
␾ = ␾共x,t,U,V兲,
共6兲
yields a conservation law of system 共1兲 if and only if
EU关␰共x,t,U,V兲H1关U,V兴 + ␾共x,t,U,V兲H2关U,V兴兴 ⬅ 0,
EV关␰共x,t,U,V兲H1关U,V兴 + ␾共x,t,U,V兲H2关U,V兴兴 ⬅ 0,
共7兲
for all differentiable functions U共x , t兲 and V共x , t兲, where
EU =
⳵
⳵
⳵
− Dx
− Dt
,
⳵Ux
⳵Ut
⳵U
EV =
⳵
⳵
⳵
− Dx
− Dt
⳵Vx
⳵Vt
⳵V
are Euler operators. One can show that a necessary condition for 兵␰共x , t , U , V兲 , ␾共x , t , U , V兲其 to be
a set of multipliers for a conservation law of system 共1兲 is that
L*关u兴
冏冋
␰共x,t,u, v兲
␾共x,t,u, v兲
册冏
H1关u,v兴=0,
= 0,
共8兲
H2关u,v兴=0
where in terms of the Fréchet derivative operator L关u兴, the adjoint operator L*关u兴 is the unique
operator having the property that
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073513-3
J. Math. Phys. 46, 073513 共2005兲
Symmetries and conservation laws of NLT equations
关␣, ␤兴L关u兴
冋册
冋册
⌽
␣
− 关⌽,⌿兴L*关u兴
⌿
␤
is a divergence expression for any differentiable functions ␣, ␤, ⌽, and ⌿.4,7–9 It is easy to show
that for the Fréchet derivative operator L关u兴 defined by 共4兲, one has
L*关u兴 =
冋
− Dt F共u兲Dx − G⬘共u兲
Dx
− Dt
册
.
In the study of physical systems, the importance of conservation laws is well known, including connections with integrability, linearization and modern numerical methods. Conservation
laws are intrinsic properties of field equations since they must hold for any posed data. Familiarly,
conservation laws are derived from variational principles through Noether’s theorem.4,5 A given
system of PDEs 共linear or nonlinear兲 can be directly obtained from a variational principle if and
only if its Fréchet derivative is self-adjoint,4,10 i.e., L*关u兴 = L关u兴. Noether’s theorem yields a conservation law for any point symmetry 共2兲 that leaves invariant the action functional of the variational principle. This is equivalent to the Euler operator annihilating the scalar product of the
multipliers and the given system of PDEs’ functions in which solutions 共u , v兲 are replaced by
arbitrary differentiable functions 共U , V兲, i.e., system 共7兲. Consequently, any set of multipliers for a
conservation law of a system of PDEs having a variational principle must also be admitted
symmetries of the given system. Hence, when a system of PDEs has an associated self-adjoint
Fréchet derivative, its multipliers for conservation laws are a subset of its admitted symmetries.
It is well known that for a linear system of PDEs, any solution of its adjoint system yields a
set of multipliers for a conservation law since for any linear operator L, a divergence expression
is yielded by vLu − uL*v. Moreover, from this it follows that if a given nonlinear system of PDEs
can be mapped into a linear system by a point or contact transformation, then its infinite number
of admitted symmetries exhibit this linear system and its infinite number of multipliers for conservation laws exhibit the adjoint of this linear system.5,11,12
It is easy to show that for any F共u兲 and G共u兲, the NLT system 共1兲 is not self-adjoint. The extra
conditions beyond the necessary condition 共8兲 for ␰共x , t , U , V兲, ␾共x , t , U , V兲 to be multipliers for a
conservation law of system 共1兲 are given in Refs. 7–9. At first sight, one might think that a given
system 共1兲 admits more point symmetries 共2兲 which involve four unknowns 兵␰ˆ , ␶ˆ , ␩ˆ , ␨ˆ 其 than sets of
multipliers of the form 共6兲 which involve only two unknowns 兵␰ , ␾其. However, we will show that
for many NLT systems 共1兲, there are more admitted sets of multipliers of the form 共6兲 than
admitted point symmetries 共2兲.
The point symmetry and conservation law classifications of the NLT system 共1兲 have been
separately investigated in Refs. 13 and 14, respectively. In Ref. 15, it is shown how to obtain new
conservation laws from the action of an admitted symmetry on a known conservation law.
In Sec. II, we give the determining equations for point symmetries and multipliers admitted by
NLT systems 共1兲. We present the Symmetry and Conservation Law Classification Table for NLT
systems 共1兲 and show that when 共1兲 is not linearizable, there are many cases where it can admit,
nontrivially, one symmetry and four, three, two or zero conservation laws as well as cases where
it can admit, nontrivially, zero symmetries and four or two conservation laws. We also comment
on situations when a symmetry maps a conservation law into another conservation law共s兲. Further
comments are presented in Sec. III.
II. COMPARISON OF SYMMETRIES AND CONSERVATION LAWS FOR NLT SYSTEMS
By inspection, any NLT system 共1兲 obviously admits, as point symmetries, translations t → t
+ ␧1, v → v + ␧2, and x → x + ␧3, corresponding to admitted infinitesimal generators X1 = ut⳵u + vt⳵v,
X2 = ⳵v, and X3 = ux⳵u + vx⳵v, respectively, as well as a set of multipliers 共␰ , ␾兲 = 共0 , 1兲 since the
second PDE of any NLT system 共1兲 is written as a conservation law. Any additional admitted point
symmetries or sets of multipliers for conservation laws are considered to be nontrivial.
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073513-4
J. Math. Phys. 46, 073513 共2005兲
G. W. Bluman and T. Temuerchaolu
A point symmetry 共2兲 admitted by an NLT system 共1兲 is represented by any solution
兵␰ˆ 共x , t , u , v兲 , ␶ˆ 共x , t , u , v兲 , ␩ˆ 共x , t , u , v兲 , ␨ˆ 共x , t , u , v兲其 of the linear determining system of PDEs4–6
␰ˆ v − ␶ˆ u = 0,
␩ˆ u − ␺ˆ v + ␰x − ␶t = 0,
G共u兲␩ˆ v + ␩ˆ t − ␺ˆ x + G共u兲␶ˆ x = 0,
␰ˆ u − F共u兲␶ˆ v = 0,
␺ˆ u − G共u兲␶ˆ u − F共u兲␩ˆ v = 0,
G共u兲␰ˆ v + ␰ˆ t − F共u兲␶ˆ x = 0,
关␺ˆ v − ␶ˆ t − 2G共u兲␶ˆ v − ␩ˆ u + ␰ˆ x兴F共u兲 − F⬘共u兲␩ˆ = 0,
关␺ˆ v − ␶ˆ t − G共u兲␶ˆ v兴G共u兲 − F共u兲␩ˆ x − G⬘共u兲␩ˆ + ␺ˆ t = 0.
共9兲
The complete solution of the determining system 共9兲 is presented in Ref. 13.
A set of multipliers for a conservation law of an NLT system 共1兲 is represented by any solution
兵␰共x , t , U , V兲 , ␾共x , t , U , V兲其 of the linear determining system of PDEs7,9
␾V − ␰U = 0,
␾U − F共U兲␰V = 0,
␾x − ␰t − G共U兲␰V = 0,
F共U兲␰x − ␾t − G共U兲␰U − G⬘共U兲␰ = 0.
共10兲
The complete solution of the determining system 共10兲 and the corresponding conservation laws are
presented in Ref. 14.
Equivalence transformations10,11 simplify the symmetry and conservation law classifications
of NLT systems 共1兲. In particular, in Refs. 13 and 14, it is shown how to obtain the corresponding
conservation law for any 共F̄共u兲 , Ḡ共u兲兲 pair related to the conservation law for a given 共F共u兲 , G共u兲兲
pair through any similarity transformation
F̄共u兲 = ␥F共␣u + ␤兲,
Ḡ共u兲 = ␦G共␣u + ␤兲 + ␳ .
共11兲
In the following Symmetry and Conservation Law Classification Table 共Table I兲, we list and
compare the additional admitted nontrivial symmetries and nontrivial conservation laws for all
possible pairs 共F共u兲 , G共u兲兲, modulo any similarity transformation 共11兲. For each such pair, we
indicate the number of additional admitted point symmetries, the number of additional admitted
conservation laws, list all such admitted point symmetries in evolutionary form, show where to
find such admitted conservation laws in Ref. 14, and state pertinent comments. Most important, in
the comments column we indicate where an admitted symmetry can map a conservation law to
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073513-5
J. Math. Phys. 46, 073513 共2005兲
Symmetries and conservation laws of NLT equations
TABLE I. Symmetry and conservation law classification table.
Number of additional
point symmetries;
listing of symmetries
Number of additional
conservation laws;
see Ref. 14 for
conservation laws
Comments;
symmetry
mappings of
conservation
laws 共Ref. 15兲
G共u兲
F共u兲
0
Arbitrary
⬁
X = 关−A共u , v兲ux + B共u , v兲ut兴⳵u
+关−A共u , v兲vx + B共u , v兲vt兴⳵v
with
Au = −F共u兲Bv , Av = −Bu
⬁
Multipliers
␰ = a共U , V兲,
␾ = b共U , V兲
with
a U = b V,
F共U兲aV = bU
Linearizable
by a hodograph
transformation;
共a , b兲 system
is adjoint of
共A , B兲 system
u
1
⬁
X⬁ = B共x , t兲⳵u − A共x , t兲⳵v
with
Ax + Bt = 0,
At + Bx + B = 0
⬁
Multipliers
␰ = b共x , t兲,
␾ = a共x , t兲
with
a x = b t,
at = bx − b
Linear system;
共a , b兲 system
is adjoint of
共A , B兲 system
u
1
X = 共2u − 2xux − tut兲⳵u
+共3v − 2xvx − tvt兲⳵v
4
Table 4: Case 1
with ␤2 = 1,
␤1 = ␤3 = 0
t→t+␧
maps 共␰3 , ␾3兲
to additional
three 共␰i , ␾i兲,
i=1,2,4
u␣关␣ ⫽ 0 , 1兴
1
2
Table 1
t→t+␧
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
e␣u关␣ ⫽ 0兴
1
2
⬙
⬙
u 2 + ␣ 1u + ␣ 2
关␣21 ⫽ 4␣2兴
0
4
Table 4: Case 1
with ␤1 = 1,
␤2 = ␣1 , ␤3 = ␣2
In Ref. 14: in Table 1,
t → t + ␧ maps
共␰1 , ␾1兲 to 共␰2 , ␾2兲;
in Table 3,
共t , V兲 → 共−t , −V兲
maps a different
共 ␰ 1 , ␾ 1兲
to a different 共␰2 , ␾2兲
than for Table 1
All other F共u兲
0
2
Table 1
t→t+␧
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
u−2
⬁
⬁
Multipliers
␰ = e−xb共Û , V兲,
␾ = a共Û , V兲
with
ˆ
aV − e−UbUˆ = 0,
ˆ
aUˆ − e−UbV = 0
关Û = x + ln U兴
Linearizable;
共a , b兲 system
is adjoint of
共A , B兲 system
u−1
1
4
Table 5: Case 1
with ␤2 = 1,
␤1 = ␤3 = 0
V→V+␧
maps 共␰3 , ␾3兲
to additional
three 共␰i , ␾i兲,
i=1,2,4
u−1
X = 共2u − 2␣xux − ␣tut兲⳵u
+关共2 + ␣兲v − 2␣xvx − ␣tvt兴⳵v
X = 共2 − 2␣xux − ␣tut兲⳵u
+共␣v + 2t − 2␣xvt兲⳵v
X = 关u−1A共û , v兲ux兴
关 − B共û , v兲ut + A共û , v兲兴⳵u
+关u−1A共û , v兲vx兴
关 − B共û , v兲vt兴⳵u
with
Av + Bû = 0,
Aû + Bv − A = 0
关û = x + ln u兴
X = 共2xux + 3tut − 2u兲⳵u
+共2xux + 3tut − v兲⳵v
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073513-6
J. Math. Phys. 46, 073513 共2005兲
G. W. Bluman and T. Temuerchaolu
TABLE I. 共Continued.兲
G共u兲
F共u兲
u−1
共u + 1兲 u2
u␦
关␦ ⫽ 0 , ± 1兴
Number of additional
point symmetries;
listing of symmetries
Number of additional
conservation laws;
see Ref. 14 for
conservation laws
Comments;
symmetry
mappings of
conservation
laws 共Ref. 15兲
Ⲑ
1
X = 关2共x + ln u − 1 / u兲ux兴
关 + 共3t + 2v兲ut − 2共u + 1兲兴⳵u
+关2共x + ln u − 1 / u兲vx兴
关 + 共3t + 2v兲vt − v兴⳵v
4
Table 5: Case 1
with ␤3 = 0,
␤1 = ␤2 = 1
X maps 共␰3 , ␾3兲
to 共␰2 , ␾2兲;
V→V+␧
maps 共␰3 , ␾3兲
to additional
three 共␰i , ␾i兲,
i=1,2,4
共u ± 1兲␤ u2
关␤ ⫽ 0 , 1兴
Ⲑ
1
2
Table 1
V→V+␧
maps 共␰2 , ␾2兲
to 共␰1 , ␾1兲
u␣
关␣ ⫽ −1 , −2兴
1
X = 关2共␤x ± 兰F共u兲du兲ux兴
+共共␤ + 2兲t ± 2v兲ut
关 − 2共u ± 1兲兴⳵u
+关2共␤x ± 兰F共u兲du兲vx兴
+共共␤ + 2兲t ± 2v兲vt
关 − ␤v兴⳵v
X = 关2共2 + ␣兲xux兴
+关共4 + ␣兲tut − 2u兴⳵u
+关2共2 + ␣兲xux兴
关 + 共4 + ␣兲tut − 共2 + ␣兲v兴⳵u
2
⬙
⬙
u−2e␣u
关␣ ⫽ 0兴
1
X = 关共2␣x + 2 兰 F共u兲du兲ux兴
关 + 共␣t + 2v兲ut − 2兴⳵u
关共2␣x + 2 兰 F共u兲du兲vx兴
关 + 共␣t + 2v兲vt − ␣v兴⳵v
2
⬙
⬙
u−2 + ␣1u−1 + ␣2
关␣21 ⫽ 4␣2兴,
关␣2 ⫽ 0兴
0
4
Table 5: Case 1
with
␤1 = 1, ␤2 = ␣1,
␤3 = ␣2
In Ref. 14: in Table 1,
V → V + ␧ maps
共␰2 , ␾2兲 to 共␾1 , ␾1兲;
in Table 3,
共t , V兲 → 共−t , −V兲
maps a different
共 ␰ 1 , ␾ 1兲
to a different 共␰2 , ␾2兲
than for Table 1
All other F共u兲
0
2
Table 1
v→v+␧
maps 共␰2 , ␾2兲
to 共␰1 , ␾1兲
u␦−1
1
X = 关共␦ − 1兲tut + 2u兴⳵u
+关共␦ − 1兲tvt兴
关 + 共1 + ␦兲v兴⳵v
3
Table 3: Case 2
with
␯ = ␦, ␮ = 0
t→t+␧
maps 共␰1 , ␾1兲
to 共␰3 , ␾3兲;
V→V+␧
maps 共␰3 , ␾2兲
to 共␰3 , ␾3兲
u␤关␤ ⫽ ␦ − 1兴
1
X = 关2共␦ − ␤ − 1兲xux兴
关 + 共2␦ − ␤ − 2兲tut + 2u兴⳵u
+关2共␦ − ␤ − 1兲xux兴
+共2␦ − ␤ − 2兲tvt
关 + 共2 + ␤兲兴⳵u
0
u␦−1 + ␤
关␤ ⫽ 0兴
0
Table 3: Case 2
with
␯ = ␦ , ␮ = ␦ 2␤
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲k
2
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073513-7
J. Math. Phys. 46, 073513 共2005兲
Symmetries and conservation laws of NLT equations
TABLE I. 共Continued.兲
Number of additional
point symmetries;
listing of symmetries
G共u兲
F共u兲
u␦
关␦ ⫽ 0 , ± 1兴
u␦−1
+␬共u␦ + ␳兲2
关␬ ⫽ 0兴
0
ln u
u−1
1
u␣
关␣ ⫽ −1兴
1
u−1 + ␣
关␣ ⫽ 0兴
0
u−1
+␬共ln u + ␳兲2
关␬ ⫽ 0兴
0
1 / 共u ln2 u兲
1
u␤ / ln2 u
关␤ ⫽ −1兴
1
1 / 共u ln2 u兲 + ␣
关␣ ⫽ 0兴
0
1 / 共u ln2 u兲
+␬共ln−1 u + ␳兲2
关␬ ⫽ 0兴
0
eu
1
ln−1 u
eu
Number of additional
conservation laws;
see Ref. 14 for
conservation laws
Comments;
symmetry
mappings of
conservation
laws 共Ref. 15兲
2
Table 3: Case 1
with
␥ = ␦ , ␣ = ␦ 2␬ ,
␤=␳
⬙
X = 共tut − 2u兲⳵u
+共tvt − v − 2t兲⳵v
3
Table 3: Case 2
with
␯ = 1, ␮ = 0
X maps 共␰2 , ␾2兲
to 共␰1 , ␾1兲;
t→t+␧
maps 共␰1 , ␾1兲
to 共␰3 , ␾3兲
X = 关2共␣ + 1兲xux + 共␣ + 2兲tut兴
关 − 2u兴⳵u
+关2共␣ + 1兲xvx + 共␣ + 2兲tvt兴
关 − 共2t + 共2 + ␣兲v兲兴⳵v
0
2
Table 3: Case 2
with
␯ = 1, ␮ = ␣
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
2
Table 3: Case 1
with
␥ = 1, ␣ = ␬,
␤=␳
⬙
3
Table 3: Case 2
with
␯ = −1, ␮ = 0
X maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲;
t→t+␧
maps 共␰1 , ␾1兲
to 共␰3 , ␾3兲
2
Table 3: Case 2
with
␯ = −1, ␮ = ␣
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
2
Table 3: Case 1
with
␥ = −1, ␣ = ␬,
␤=␳
⬙
Table 6: Case 4
with
␤1 = ␤3 = 0,
␤2 = 1
V→V+␧
maps 共␰4 , ␾4兲
to 共␰1 , ␾1兲;
t→t+␧
maps 共␰4 , ␾4兲
to 共␰2 , ␾2兲;
X maps 共␰4 , ␾4兲
to 共␰3 , ␾3兲
X = 关−2 ln−1 uux兴
关 + 共t + 2v兲ut − 2u兴⳵u
+关−2 ln−1 uvx兴
关 + 共t + 2v兲vt − v兴⳵v
X = 2关共共␤ + 1兲x + 兰F共u兲du兲ux兴
关 + 共共␤ + 2兲t + 2v兲ut − 2u兴⳵u
+2关共共␤ + 1兲x + 兰F共u兲du兲vx兴
+共共␤ + 2兲t + 2v兲vt
关 − 共␤ + 2兲v兴⳵v
X = 共2 + tut兲⳵u
+共v + tvt兲⳵v
0
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073513-8
J. Math. Phys. 46, 073513 共2005兲
G. W. Bluman and T. Temuerchaolu
TABLE I. 共Continued.兲
Number of additional
point symmetries;
listing of symmetries
Number of additional
conservation laws;
see Ref. 14 for
conservation laws
G共u兲
F共u兲
eu
e ␣u
关␣ ⫽ 0 , 1兴
1
eu + ␣
关␣ ⫽ 0兴
0
4
Table 6: Case 4
with
␤1 = 0, ␤2 = 1,
␤3 = ␣
e2u + ␣1eu + ␣2
关␣21 ⫽ 4␣2兴
0
4
Table 6: Case 1
with
␤1 = 1, ␤2 = ␣1,
␤3 = ␣2 ⬎ 0
共eu + ␣兲2
关␣ ⫽ 0兴
0
X = 关2共␣ − 1兲xux兴
关 + 共␣ − 2兲tut − 2兴⳵u
+关2共␣ − 1兲xvx兴
关 + 共␣ − 2兲tvt − ␣v兴⳵v
Comments;
symmetry
mappings of
conservation
laws 共Ref. 15兲
0
2
t→t+␧
maps 共␰3 , ␾3兲
to 共␰1 , ␾1兲;
共t , V兲 → 共−t , −V兲
maps 共␰3 , ␾3兲
to 共␰4 , ␾4兲
and maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
␰1 = eA sin B,
␾1 = eA共r cos B兲
共 − eU sin B兲
with
A = a共x + eU兲
+ 1 Ⲑ 2 共a␣1
− 1兲U
+␣t − ␳V,
B = ␬t + ␥V − b共x兲
共 + eU
+ 1 Ⲑ 2 ␣ 1U 兲;
a,b,r,␣,␥,␳,␬
are given in
Table 6: Case 2
with ␤1 = 1,
␤ 2 = ␣ 1,
␤3 = ␣2 ⬍ 0
共t , V兲→
共t + c1 , V + c2兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲;
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰3 , ␾3兲;
共t , V兲→
共−t + c1 , −V + c2兲
maps 共␰1 , ␾1兲
to 共␰4 , ␾4兲
关c1 = ␣1␲ / 2冑兩␣2兩 ,
c2 = 冑兩␣2兩␲兴
Table 6: Case 5
with
␤1 = 1, ␤2 = ␣1,
␤3 = ␣2 = 0
t→t+␧
maps 共␰3 , ␾3兲
to 共␰1 , ␾1兲;
共t , V兲 → 共−t , −V兲
maps 共␰3 , ␾3兲
to 共␰4 , ␾4兲 and
maps
共␰1 , ␾1兲 to 共␰2 , ␾2兲
Table 6: Case 3
with
␤1 = 1, ␤2 = 2␣,
␤3 = ␣2
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
other conservation laws through the methods presented in Ref. 15. In particular, we show that in
many cases the obvious admitted point symmetries t → t + ␧1 and v → v + ␧2 are very useful to
obtain new conservation laws from a known conservation law.
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073513-9
J. Math. Phys. 46, 073513 共2005兲
Symmetries and conservation laws of NLT equations
TABLE I. 共Continued.兲
G共u兲
Number of additional
point symmetries;
listing of symmetries
F共u兲
u␦ ± 1
u␦−1
u␦ ⫿ 1
共u␦ ⫿ 1兲2
关␦ ⫽ 0 , ± 1兴
1
关
1
X= ⫿2
u␦ ± 1
u
u␦ ⫿ 1 x
Number of additional
conservation laws;
see Ref. 14 for
conservation laws
3
兴
+ 共t + ␦v兲ut − 2u ⳵u
关
1
+ ⫿2
Table 3: Case 2
with
␯ = ⫿ 2␦, ␮ = 0
X maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲;
t → t + ␧,
maps 共␰1 , ␾1兲
to 共␰3 , ␾3兲;
V → V + ␧,
maps 共␰2 , ␾2兲
to 共␰3 , ␾3兲
u␦ ± 1
v
u␦ ⫿ 1 x
兴
+ 共t + ␦v兲vt−共␦t+ v兲 ⳵v
u␦+␤−1
共u␦ ⫿ 1兲2
关␤ ⫽ 0兴
1
+␣
共u␦ ⫿ 1兲2
u␦−1
0
2
Table 3: Case 2
with
␯ = ⫿ 2␦,
␮ = 共2␦兲2␣
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
u␦−1
0
2
Table 3: Case 1
with
␥ = ⫿ 2␦,
␣ = 共2␦兲2␬,
␤=␳
⬙
Table 3: Case 2
with
␯ = ␦, ␮ = 0
X maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲;
t → t + ␧,
maps 共␰1 , ␾1兲
to 共␰3 , ␾3兲;
V → V + ␧,
maps 共␰2 , ␾2兲
to 共␰3 , ␾3兲
X = 关共2␤x + ␦ 兰 F共u兲du兲ux兴
关 + 共共␤ + 1兲t + ␦v兲ut − 2u兴⳵u
+关共2␤x + ␦ 兰 F共u兲du兲vx兴
+共共␤ + 1兲t + ␦v兲vt
关 − 共␦t + 共␤ + 1兲v兲兴⳵v
0
关␣ ⫽ 0兴
+␬
tan共␦ ln u兲
关␦ ⫽ 0兴
Comments;
symmetry
mappings of
conservation
laws 共Ref. 15兲
共u␦ ⫿ 1兲2
u␦ ± 1
共
+␳
u␦ ⫿ 1
关␬ ⫽ 0兴
兲
2
u−1 sec2共␦ ln u兲
1
X = 关−2 tan共␦ ln u兲ux兴
关 + 共t − 2␦v兲ut − 2u兴⳵u
+关−2 tan共␦ ln u兲vx兴
+共t − 2␦v兲vt
关 − 共2␦t + v兲兴⳵v
3
u␤ sec2共␦ ln u兲
关␤ ⫽ −1兴
1
X = 关2共共␤ + 1兲x
− ␦ 兰 F共u兲du兲ux兴
关 + 共共␤ + 2兲t − 2␦u兲ut
− 2u兴⳵u
+关2共共␤ + 1兲x − ␦ 兰 F共u兲du兲vx兴
+共共␤ + 2兲t − 2␦v兲vt
关 + 共2␦t + 共␤ + 2兲v兲兴⳵v
0
共 sec2共␦ ln u兲 Ⲑ u 兲 + ␣
0
2
Table 3: Case 2
with
␯ = ␦ , ␮ = ␦ 2␣
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
共 sec2共␦ ln u兲 Ⲑ u 兲 + ␬关␳兴
0
2
Table 3: Case 1
with
␥ = ␦ , ␣ = ␦ 2␬ ,
␤=␳
⬙
关␣ ⫽ 0兴
关 + tan共␦ ln u兲兴2
关␬ ⫽ 0兴
III. FURTHER DISCUSSION
共1兲 As can be seen in the Symmetry and Conservation Law Classification Table 共Table I兲, for
each 共F共u兲 , G共u兲兲 pair where the NLT system 共1兲 admits nontrivial conservation laws, there exists
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073513-10
J. Math. Phys. 46, 073513 共2005兲
G. W. Bluman and T. Temuerchaolu
TABLE I. 共Continued.兲
Number of additional
point symmetries;
listing of symmetries
Number of additional
conservation laws;
see Ref. 14 for
conservation laws
G共u兲
F共u兲
tanh u
sech2 u
1
X = 共tanh uux + vut − 1兲⳵u
+共tanh uvx + vvt − t兲⳵v
4
e␤u sech2 u
关␤ ⫽ 0兴
1
X = 关2共␤x + 兰F共u兲du兲ux兴
关 + 共␤t + 2v兲ut − 2兴⳵u
+关2共␤x + 兰F共u兲du兲vx兴
关 + 共␤t + 2v兲vt − 共2t
+ ␤v兲兴⳵v
0
tanh u + ␦
0
4
Table 7: Case 3
with
␤1 = 1,
␤2 = ␤3 = 0
t→t+␧
maps 共␰4 , ␾4兲
to 共␰i , ␾i兲, i = 1 , 3;
V → V + ␧,
maps 共␰4 , ␾4兲
to 共␰i , ␾i兲, i = 2 , 3;
X maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
Table 7: Case 1
with ␤1 = 0,
␤2 = 1, ␤3 = ␦,
兩␦兩 ⬎ 1
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
␰1
= eA cosh U cos B,
共t , V兲→
共t + c1 , V + c2兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲;
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰3 , ␾3兲;
共t , V兲→
共−t + c1 , −V + c2兲
maps 共␰1 , ␾1兲
to 共␰4 , ␾4兲
关c1 = −␲ / 4冑1 − ␦ , 兴兴
关c2 = 共2␦
− 1兲␲ / 4冑1 − ␦兴
␾1 = 关 共1
− ␦兲1/2 1
Ⲑ
+ e2U 兴eA cosh U
⫻关sin B
− re2U cos B兴
with
1
A = a共x + 2 U兲
−共␬t + ␥V兲,
1
B = −b共x + 2 U兲
−共␣t + ␳V兲,
a,b,r,␣,␥,␳,␬
are given in
Table 7: Case 2
with ␤1 = 0,
␤2 = 1, ␤3 = ␦,
兩␦兩 ⬍ 1
tanh2 u + ␦
关␦ ⫽ −1 , 0兴
0
4
Comments;
symmetry
mappings of
conservation
laws 共Ref. 15兲
Table 7: Case 3
with ␤1 = 0,
␤2 = 1, ␤3 = ␦,
兩␦兩 = 1
t→t+␧
maps 共␰3 , ␾3兲
to 共␰1 , ␾1兲;
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
and maps 共␰3 , ␾3兲
to 共␰4 , ␾4兲
Table 7: Case 1
with
␤1 = −1, ␤2 = 0,
␤3 = 1 + ␦
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
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073513-11
J. Math. Phys. 46, 073513 共2005兲
Symmetries and conservation laws of NLT equations
TABLE I. 共Continued.兲
G共u兲
F共u兲
tanhu
tanh2 u
+␣1 tanh u + ␣2
关␣21 ⫽ 4␣2 , ␣1 ⫽ 0兴
Number of additional
point symmetries;
listing of symmetries
0
Comments;
symmetry
mappings of
conservation
laws 共Ref. 15兲
Number of additional
conservation laws;
see Ref. 14 for
conservation laws
4
Table 7: Case 1
with ␤1 = −1,
␤ 2 = ␣ 1, ␤ 3 = 1
+ ␣ 2,
兩1 + ␣2兩 ⬎ 兩␣1兩
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
Table 7: Case 2
with
␤1 = −1, ␤2 = ␣1
⬍ 0,
␤ 3 = 1 + ␣ 2,
兩1 + ␣2兩 ⬍ −␣1
共t , V兲→
共t + c1 , V + c2兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲;
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰3 , ␾3兲;
共t , V兲→
共−t + c1 , −V + c2兲
maps 共␰1 , ␾1兲
to 共␰4 , ␾4兲
共2 + ␣1兲␲
c1 =
,
4冑−共1 + ␣1 + ␣2兲
关
关
−
␰1
= eA cosh1+a u cos B,
␾1
冑␣1 − 共1 + ␣2兲 A
=
e
1 + e2U
⫻cosh1+a U关sin B兴
关 − re2U cos B兴
with
1
A = a共x − 2 U兲
−共␬t + ␥V兲,
B = −b共x
+ ln cosh U兲
1
+ 2 ␣1U兲 − 共␣t
+ ␳V兲;
a,b,r,␣,␥,␳,␬
are given in
Table 7: Case 2
with ␤1 = −1,
␤2 = ␣1 ⬎ 0,
␤ 3 = 1 + ␣ 2,
兩1 + ␣2兩 ⬍ ␣1
关
兴
兴
c2 =
共 ␣ 1 + 2 ␣ 2兲 ␲
4冑−共1 + ␣1 + ␣2兲
兴
共t , V兲→
共t + c1 , V + c2兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲;
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰3 , ␾3兲;
共t , V兲→
共−t , + c1 , −V + c2兲
maps 共␰1 , ␾1兲
to 共␰4 , ␾4兲
共2 − ␣1兲␲
c1 =
,
4冑␣1 − 共1 + ␣2兲
共2␣2 − ␣1兲␲
c2 =
4冑␣1 − 共1 + ␣2兲
关
关
兴
兴
an admitted group of point transformations 共discrete and/or continuous兲 that maps a conservation
law to one or more 共up to three兲 new conservation laws for the same 共F共u兲 , G共u兲兲 pair through use
of the work presented in Ref. 15.
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073513-12
J. Math. Phys. 46, 073513 共2005兲
G. W. Bluman and T. Temuerchaolu
TABLE I. 共Continued.兲
Number of additional
point symmetries;
listing of symmetries
Number of additional
conservation laws;
see Ref. 14 for
conservation laws
Comments;
symmetry
mappings of
conservation
laws 共Ref. 15兲
G共u兲
F共u兲
tanh u
tanh2 u
+ ␣1 tanh u + ␣2
关␣21 ⫽ 4␣2 , ␣1 ⫽ 0兴
0
4
Table 7: Case 3
with ␤1 = −1,
␤ 2 = ␣ 1, ␤ 3 = 1
+ ␣ 2,
兩 ␤ 3兩 = 兩 ␣ 1兩
t→t+␧
maps 共␰3 , ␾3兲
to 共␰1 , ␾1兲;
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
and maps 共␰3 , ␾3兲
to 共␰4 , ␾4兲
共tanh u + ␦兲2
关␦ ⫽ ± 1兴
0
2
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
1
0
2
Table 7: Case 5
with ␤1 = −1,
␤2 = 2␦,
␤3 = 1 + ␦2
Table 7: Case 5
with ␤3 = 1
␤1 = ␤2 = 0
e␤u cosh2 u
1
0
or
4
See comments
␥1 coth2 u
+␥2 coth u + ␥3
0
2
or
4
See comments
Symmetries and
conservation laws
obtained from the
the cases where
G共u兲 = tanh u
through
equivalence
transformation
共x , t , u , v兲→
␲
x,t,u+ i,v
2
sec2 u
1
X = 共1 + tan uux + vut兲⳵u
+共t + tan uvx + vvt兲⳵v
4
Table 8: Case 2
with ␤1 = 1,
␤2 = ␤3 = 0
V→V+␧
maps 共␰1 , ␾1兲
to 共␰i , ␾i兲, i = 2 , 3;
t→t+␧
maps 共␰1 , ␾1兲
to ␰i , ␾i, i = 1 , 3;
X maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
e␤u sec2 u
关␤ ⫽ 0兴
1
X = 关2共␤x − 兰F共u兲du兲兴ux
关 + 共␤t − 2v兲ut − 2兴⳵u
+关2共␤x − 兰F共u兲du兲兴vx
+共␤t − 2v兲vt
关 − 共2t + ␤v兲兴⳵v
0
tan u + ␦
0
Table 8: Case 1
with ␤1 = 0,
␤2 = 1, ␤3 = ␦
共t , V兲 → 共−t , −V兲
maps 共␰1 , ␾1兲
to 共␰2 , ␾2兲
coth u
tan u
See comments
4
⬙
共
兲
共2兲 It can happen that a nonvariational system of PDEs can become variational through a
differential substitution. For example, the Korteweg–de Vries equation
ut + uux + uxxx = 0
is not variational but becomes variational after the differential substitution u = ⌫x. One can show
that the differential substitution 共u , v兲 = 共⌫x , ⌬x兲, leads to an NLT system 共1兲 that is variational if
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073513-13
J. Math. Phys. 46, 073513 共2005兲
Symmetries and conservation laws of NLT equations
TABLE I. 共Continued.兲
Number of additional
point symmetries;
listing of symmetries
Number of additional
conservation laws;
see Ref. 14 for
conservation laws
Comments;
symmetry
mappings of
conservation
laws 共Ref. 15兲
G共u兲
F共u兲
tan u
tan2 u + ␦
关␦ ⫽ 0 , 1兴
0
4
Table 8: Case 2
with ␤1 = 1,
␤2 = 0, ␤3 = ␦ − 1
⬙
tan2 u
+␣1 tan u + ␣2
关␣21 ⫽ 4␣2 , ␣1 ⫽ 0兴
0
4
Table 8: Case 1
with ␤1 = 1,
␤ 2 = ␣ 1,
␤3 = ␣2 − 1
⬙
共tan u + ␦兲2
0
2
Table 8: Case 3
with ␤1 = 1,
␤2 = 2␦,
␤3 = ␦2 − 1
⬙
1
0
2
Table 8: Case 3
with ␤3 = 1,
␤1 = ␤2 = 0
⬙
and only if G共u兲 = const 关in this variational case, the NLT system 共1兲 is linearizable by a hodograph
transformation兴.
共3兲 In general, suppose a system of PDEs with two dependent variables 共u , v兲 and two independent variables 共x , t兲 is not variational but becomes variational through the differential substitution 共u , v兲 = 共⌫x , ⌬x兲, then an admitted point symmetry
␩ = ␩ˆ 共x,t,⌫,⌬兲 − ␰ˆ 共x,t,⌫,⌬兲⌫x − ␶ˆ 共x,t,⌫,⌬兲⌫t ,
␨ = ␩ˆ 共x,t,⌫,⌬兲 − ␰ˆ 共x,t,⌫,⌬兲⌬x − ␶ˆ 共x,t,⌫,⌬兲⌬t ,
of the variational system would yield multipliers of the form 兵␰共x , t , U , V兲 , ␾共x , t , U , V兲其 of the
given system, if and only if ␶ˆ 共x , t , ⌫ , ⌬兲 = 0 and ␩ˆ , ␨ˆ and ␰ˆ do not depend explicitly on ⌫ and ⌬
共otherwise, such a set of multipliers yields a set of nonlocal multipliers and nonlocal symmetries
of the given system兲. Conversely, suppose the given system admits a conservation law resulting
from a set of multipliers of the form 兵␰共x , t , U , V兲 , ␾共x , t , U , V兲其, then such a set of multipliers
yields a point symmetry admitted by the variational system if and only if ␰V = ␰UU = ␾U = ␾VV = 0
关otherwise, such a set of multipliers would yield a local 共but not point兲 symmetry admitted by the
variational system兴.
共4兲 In general, for a nonvariational system of PDEs, there is a direct connection between
conservation laws and symmetries if the system is linear or directly linearizable by a point or
contact transformation. For the other exhibited cases, in view of the previous two remarks it would
be interesting to investigate if there exist nonlocal symmetries directly connected to the conservation laws.
ACKNOWLEDGMENT
The authors acknowledge financial support from the National Sciences and Engineering Research Council of Canada and also the second author 共T.兲 is thankful for partial support from NSF
of China 共Grant No. 10461005兲.
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G. W. Bluman and T. Temuerchaolu
J. Math. Phys. 46, 073513 共2005兲
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