Applied Mathematics, 2011, 2, 666-675
doi:10.4236/am.2011.26088 Published Online June 2011 (http://www.SciRP.org/journal/am)
Computation of Some Geometric Properties for New
Nonlinear PDE Models
Nassar Hassan Abdel-All1, Mohammed Abdullah Abdullah Hamad1,
Mohammed Abdullah Abdel-Razek1, Amal Aboelwafa Khalil2
1Department of Mat hem at i cs, Faculty of Science, Assiut University, Assiut, Egypt
2Department of Mat hem at i cs, Faculty of Science, Sohag University, Sohag, Egypt
Received March 16, 2011; revised March 25, 2011; accepted March 28, 2011
Abstract
The purpose of the present work is to construct new geometrical models for motion of plane curve by Dar-
boux transformations. We get nonlinear partial differential equations (PDE). We have obtained the exact so-
lutions of the resulting equations using symmetry groups method. Also, the Gaussian and mean curvatures of
Monge form of the soliton surfaces have been calculated and discussed.
Keywords: Motion of Curve, Darboux Transformations, Gaussian and Mean Curvatures, Symmetry Groups
1. Introduction
Kinematics of moving curves in two dimension is for-
mulated in terms of intrinsic geometries. The velocity is
assumed to be local in the sense that it is a functional of
the curvature and its derivatives. Plane curves have
received a great deal of attention from mathematics,
physics, biology, dynamic system, image processing and
computer vision [1,2]. Evolution of plane curve can be
understand as a deformation of a plane curve in arbitrary
direction according to arbitrary a mount in a conti-
nuously process that has the time as a parameter. Phy-
sically this arbitrary a mount is a function of velocity, so
this process create a sequence of evolving planer curve
moving by a funcion of velocity, the type of the motion
(evolution) of this family of planer curves classified
depend on our choice of velocity function.
Let is the position vector of a curve C
moving in space and let

denote respectively
the unit tangent, principal normal and binormal vectors
vary along . If we introduce the darboux vector (see
[3]),

,rst
C


,,Tnb
=Tk b

 (1)
then the Serret-Frenet equations may be written as the
following [4]:
==,
==
==
s
s
s
TTkn
nnbk
bbn



 
 
where
s
is the arc length parameter along , the
curvature and
Ck

the torsion. In the present moving
curve context, the time
t
enters into the system (2) as a
sparameter. The general temporal evolution in which the
triad


,,bTn remains orthonormal adopts the form
(darboux formula) [3]
=,
=
=
t
t
t
Tnb
nT
bT
b
n









(3)
where

is the geodesic curvature,

is the normal
curvature and

is the geodesic torsion. Here, it is
required that the arc length and time derivatives com-
mute. This implies inextensibility of . Accordingly,
the compatibility conditions
C
=
s
tts
TT , =
s
t
nn
ts
and
=
s
t
bts b, applied to the systems (2) and (3) yield
=,
=
=.
st
s
st
k
k
k
,









(4)
If the velocity vector =t
r

of a moving curve
has the form
C
=Tnb,



 (5)
then imposition of the condition yields
stts rr =
=
sssss s
ttnn bb nb.

 

  (6)
,T (2)
Substitute about Serret-Frenet equations

N. H. ABDEL-ALL ET AL.667
=0,
=,
=.
s
s
s
k
k




 



(7)
The temporal evolution of the curvature and the
torsion
k

of the curve may now be expressed in
terms of the components of velocity
C
,

 and

by
substitution of (7) into (4) to obtain


=,
=,
ts
s
ts s
kk
k

s

 
 
 
 (8)
where

1
=.
ss
sk
k


 

 

(9)
Motion in a plane occurs if =0

and =0

. Then
Equation (8) becomes

==
tsss
s
kk kk.
ss

  (10)
From Equation (7), we have then Equation
(10) becomes
=dks 

2
=
tss s
kkk   
d.ks (11)
If we take then
 
d=d =ksFkFk 


 
dFkk
d=d= d
s
ks Fk= Fk
ks k
 , hence the equa-
tion (11) becomes
 
2
=.
ss
ts
ss
kk
kFkkFkkFk
kk




 (12)
2. Symmetry Group
Now, we want to present the most general Lie group of
point transformations, which apply on obtaining equa-
tions
Definition 1. We consider a scalar order
PDE represented by
m th


Δ,=0, where is natural number
m
sk m (13)
where

=i
s
s, is a vector for which the
components i
=1,,ip
s
are independent variables and


=
j
kk,
is a vector cosest of =1,j,q
j
k dependent vari-
ables, and

=
m
m
m
k
k
s

. The infinitesimal generator of
the one-parameter Lie group of transformations for equa-
tion (13) is

=1
=,
pi
i
i
lXs k
s




1=

q





,
s
k,

k
 (14)
where

,
i
s
k  , 


,

s
k are the infinitesimals, and
the th prolongation of the infinitesimal generator (14)
is (see [5-8])
m

=
m
lpr XX

1=

j
j
q




,m sk ,

j
k
 (15)
where

j
l 



,=
m
sk
j
D



1=

i
i
p
i
k 





i
.
,
1=

ij
i
pk


(16)
and is the total derivative operator defined by
D
k
k
k
k
s
=
, =, =1,,
jj
k
kj
s



D
i
ijj
j
j








p
(17)
A vector field
X
is an infinitesimal symmetry of the
system of differential Equations (13) if and only if it
satisfies the infinitesimal invariance condition


=0
=0
m
lpr  X (18)
3. Soliton Geometry
In this paper, we construct the soliton surfaces associated
with the single soliton solutions (similarity solution) of
the Equation (12). For this purpose, if


=,kkst is a
similarity solution of Equation (2), we have a solution
surface

given from the Monge patch




,kst=,fs,t. The tangent vectors
s
ft f, for the
soliton surface

are given by



=1,0, ,
=0,1, .
s
tt

s
f
k
f
k (19)
The normal unit vector field on the tangents

p
T is
given by
=.
f
s
f
Nt
s
t
f

 (20)
f
The and fundamental forms on 1st2nd

are de-
fined respectively by
22
1112 22
22
111222
=d,d=d2ddd ,
=d,d = d2ddd,
I
ff gsgstgt
I
IfNLsLstLt


(21)
where the tensor and are given by
ij
gij L
11 1222
11 1222
=,, =,, =,,
=,, =,, =,.
ss sttt
ss sttt
g
ffgff gff
LfNLfNLfN
(22)
The Gauss equations associated with

are
12
1111 11
12
1212 12
12
2222 22
=,
=,
=,
sss t
sts t
tts t
f
ffLN
f
ffLN
f
ffL


N
(23)

N. H. ABDEL-ALL ET AL. 668
while the Weingarten equations comprise
121222 1112 111112
122222 12121211 22
=
=,
,
s
s
ts
gLgLgL gL
Nf
gg
gLgLgL gL
Nf
gg




t
t
f
f
(24)
where
22
11 2212
==
st .
g
ff ggg
(25)
The functions i
j

 in (23) are the usual Christoffel
symbols given by the relations

,,,
1
=2
iil
j
jll jjl
gg g g



(26)
The compatibility conditions ts

=
ss st
f
f and
ts
 
=
st tt
f
f applied to the linear Gauss system (23)
produce the nonlinear Mainardi-Codazzi system
222 1112 111222
221211
111 2212 111222
221211
2=
2=
ts
st
LLL LL
ggg gg
LLL LL
ggg gg

 



 


0,
0,
(27)
or, equivalently,



121
111211 121212112211
121
122211 221222122212
=,
=,
ts
ts
LL LLL
LL LLL


2
2

(28)
The Gaussian and mean curvatures at the regular
points on the soliton surface are given by respectively
2
11 2212
12 2
11 2212
===,
LL L L
Kkk g
ggg g

0
(29)

11 2212122211
12 2
11 2212
2 11
==
22
LgLg Lg
Hkk ggg

.
(30)
where

=det ij
g
g,


=det ij LL
k
and 12 are the
principal curvatures. The surface for which is
called parabalic surface, but if 1 and 2
constant or 1constant and 2, we have surface
semi round semi flat (cylinderial like surface).The
integrability conditions for the systems (2) and (3) are
equivalente to the Mainardi-Codazzi system of PDE (27).
This give a geometric interpretation for the surface
defined by the variables
, kk
K
=0
=0
kk
=0
=
=k
t
s
, to be a soliton surface
[9,10].
4. Applications
4.1. Case I:

=
F
kk
The Equation (12) becomes
3234
1=322=
ts sssssss
kkkkk kkkkk0. (31)
The infinitesimal point symmetry of Equations (31)
will be a vector field of the general form
=X
s
tk






(32)
on 3
=
M
R; our task is to determine which particular
coefficient functions ,

 and

are functions of the
variables ,
s
t and and will produce infinitesimal
symmetries. In order to apply condition (18), we must
compute the third order prolongation of which is
the vector field
k
,X
 3
p
r= J
J
,
J
X
s
tk k


 
 
 
 (33)
whose coefficients, in view of (31), are given by the
explicit formulae





=,
=,
=,
=.
ssstssst
ttststtt
ss sss tssssst
sss
s
sss tssssssst
Dkkkk
Dkkkk
Dkkkk
Dkkkk




  
 
 
 
(34)
The vector field
X
is an infinitesimal symmetry of
the Equation (31) if and only if


332
1
2
234
pr= 0=333
32
682=0.
ts
sss ss
ss sss
s sss
ss
ss
Xkkkkk
kkkk k
kkkk
k
 

 


 
(35)
Substituting the prolongation Formulae (34), and
equa- ting the coefficients of the independent derivative
mono- mials to zero, leads to the infinitesimal
determining equations which together with their
differential conse- quences reduce to the system
11
=, =, ====0
22
tstk k st
k  . (36)
The general solution of this system is readily found
31 323
11
=, =, =
22
cs cctcck   , (37)
where the coefficients i c are arbitrary constants.
Therefore, Equation (31) admits the three-dimensional
Lie algebra of infinitesimal symmetries, spanned by the
three vector fields
123
11
=, =, =
22
XXXstk .
s
tstk


 


(38)
The combination of space and time translations


12
X
X
=ysct
lead to a reduction of (31) to an ordinary
differential equation (ODE) by the transformation

and


=kwy where c is the speed of the
travelling wave. That is

N. H. ABDEL-ALL ET AL. 669
4
=0.
4.2. Case II:

1
=Fkk
233
322wwwwwcwwwww
 
 (39)
Now, the solution of the Equation (39) is,

3
2
21
1d
2ln2 2
wyc
wwcwcwc


=
0,
(40)
In this case Equation (12), becomes
52 3
912=
tssss sss
kk kkkkkk 0, (45)
where 12 and 3 are the integration constants, if
we consider it equal zero, hence the solution of Equation
(39) becomes
,cc cLie point symmetry for this equation is given by
12
34
=, =,
1
=,=
3
XX
st
Xs kXtk,
s
kt



22 2
2
22
==
11
cc
wcy csct


.
(41)
k






(46)
this solution is a similar solution to Equation (31), This
solution is in the Monge form



==,wwy wst
which define a regular surface as show in Figure 1
().
=1, 15, 0.12cs t
The combination 12
==
X
XcX c
s
t



gives rise
to travelling wave solutions a wave speed . The vector
field
c
X
has invariants and which
reduces (45) to the ODE
=ysct=kw
This surface is a soliton surface . From (29)
and (30), one can see that the Gaussian and mean
curvatures of the soliton surface ( ) are given by
respectively

11

25
912wwwww cwww
 

3
=0. (47)


32
1
2
4
12
41363
=0, =,
32
ttss
KH ts
 
 
2

(42)
Now, solving the equation with the Lie symmetry
spanned by
1
2
=,
2
=.
3
Yy
Yy w
y
w







(48)
where

 


22
1
87246862
2
52324
424 246
2246
=12
=18366441 7
83783107
660708933
499157.
ttss
ttssssst s
tsstss s
tsstssss
tsss

 

 

(43)
If we take the vector field we obtain solution
2
Y
2
3
=wy 

, and substituting in Equation (47) we get w
1
3
2
=9c 




and the solution

1
3
2
3
2
9
=c k
s
ct





(49)
The symmetry generator 3
11
=22
X
st k
s
tk
 


leads to invariants 2 =t y
s
and . These the in- =wsk
variants transform Equation (31) to the following ODE,



3222 3
23 2324
23
83624
243616 4
21=0
''
ywwywyww w

y
wwywwyw yww
ww
  

 

(44)
Remark 1. For regularity the parameters of the
soliton surface must be satisfied
s
ct, i.e., for =
s
ct
we have singularit y

 cuspidedge as shown in Figure 3.
The Gaussian and mean curvatures respectively are
(shown in Equation (50))
If we take the vector field 13
X
X we here the in-
riants and =yt1 =1 k
The numerical solution of Equation (44) is shown in
Figure 2 (intial condition and
).
 
1=1,1=2ww


1=3w
w
s
, that is then
=0w=w
constant and

  
7
33
102 21111
32 23
3333333
=0,
270 6
= .
818 68 68124324381
K
st
H
st ttstststsstsst


 


(50)

N. H. ABDEL-ALL ET AL. 670
Figure 1. Soliton surfaces of (41).
=
1
a
k,
s
 (51)
thus we have Figure 4.
The vector field leads to the invariants
4
X
s
y=
and the tarnsformation
1
3
=ktw reduces (45) to an ODE
in the form
62
27336= 0,wwwwwww
 

3
(52)
this equation can be solved numericaly (intial condition
and
 
0=2, 0=2ww



0=3w ) as shown in Fig-
ure 5.
4.3. Case III:

=2
1
Fk k
In this case Equation (12) takes the form
62 34
22440 =
tssss ssss
kkkkkkkk kk 0, (53)
Lie point symmetry of this equation is given by
123
=,=,=XX Xstk,
s
tst
 

 
k

(54)
The combination 12 ==XcX Xc
s
t




gives rise
to travelling wave solutions with wave speed c. The
vector field
X
has invariants and
which reduces (53) to
=ysct=wk
62 34
224 40=cwwwwww wwww

 
solving Equation (55) hence
3
242
21
6d
241824 9
wyc
wcwcwcw


=
0, (56)
If we take the integration constants to be zero hence
the solution takes the form

22
22
==
88
wcy csct

 
,
(57)
For regularity the parameters of the soliton surface
must be satisfied 1 8
s
ct c at 1, i.e., for =cc
1
=8
s
ct c we have singularity (cuspidaledge) as
shown in Figure 6.
0. (55) Figure 2. Numerical solution of (44).

N. H. ABDEL-ALL ET AL.671
Figure 3. Soliton surfaces of (49).
Figure 4. Solution of (51).
Gaussian and mean curvatures are



2222
2
2
21
48 242
=0, =
8
ttss tsts
KH
st
  
 
(58)
where



 
1
6224
2
42
2422
=8
=818896246
641889615
4415
st
tts s tssts
ttst tsts
sst s


 

Figure 5. Numerical solution of (52).
Figure 6. Soliton surface of (57).
The symmetry generator 3=
X
stk
s
tk




leads
to the invariants and . After some
detailed and tedious calculations, (53) becomes ODE
=yst=wsk




3222 3
62 2
32 43
21824
72 36
4014= 0
'
'
ywwywyww w
wywwyw
yw yww w

  



(60)
5
22
(59)

N. H. ABDEL-ALL ET AL. 672
The numerical solution of Equation (60) is shown in
Figure 7 (intial condition and
).
 
1=1,1=2ww


1=3
''
w
4.4. Case IV:

=1
1
Fk k
In this case Equation (12) becomes
765434
333233
32 2
5
464
282122
912
3=0,
tttt tsss
4
s
sss ssss
s ssssss ss
sss s
kkkkkkkk kkkk
kkkkkkk kkkk
kkk kkkkk
kk kkk

 


(61)
Lie point symmetry of this equation is given by
12
=, =XX
s
t



(62)
The combination 12 ==XcX Xc
s
t




gives rise
to travelling wave solutions with wave speed c. The
vector field
X
has invariants and =ysct=wk
Figure 7. Numerical solution of (60).
which reduces (62) to
765434
3332334
32 25
464
28212 2
9123
cwwcwwcwwcwwcwwww
wwwwwww wwww
www wwwwwwwwww=0

 
  

 
 
 
 
(63)
solving the Equation (63) we get

3
1d
1wyc ww  
=
0 (64)
where
 
 
 
 
 

22
12 2
222
12
2
2
22
21
=342 ln2ln1
2ln 122ln2
4ln22 4ln
4ln14ln 12ln
2ln2 22ln1
2ln1 2,
wcc wwcww
ww cwwwcw
ww cwwcww
cw wwww
cwcc w
cw c

 
 




(65)
If we take ,
1=cc 2=1c

and
3=0c

2
1d=
1122
wy
wwcw cw  
0,
(66)
then

1
2
1
2
124
=tan 2441
164
.
tan 24145
w
yww
w
ww















(67)
Hence, we have a soliton surface given by the implicit
equation

1
2
1
2
124
=tan 2441
164
,
tan 24145
w
sct ww
w
ww















(68)
Gaussian and mean curvatures of implicit surface are



 

22
3
22 3456
2
=0,
11141812
=
21 1211124
K
cw ww ww
H
cw www w

 
3
(69)
This surface is illustrated as in Figure 8.
4.5. Case V:




=ln
F
kk
In this case Equation (12) becomes

N. H. ABDEL-ALL ET AL.673
Figure 8. Soliton surface of (68).

4234 4
66 ln
tssss sssss
kk kkkkkkkkkkk =0 (70)
Lie point symmetry for this equation is given by

123
=,=, =XX Xtstk,
s
tst
 

 k

(71)
The combination 12 ==XcX Xc
s
t




gives rise
to travelling wave solutions with wave speed c. The
vector field
X
has invariants and
wkich reduces (70) to
=ysct=wk


2
4244
6
6ln
ww www
cwwwww w
 

 =0.
(72)
We have solved the Equation (72) with intial condition
( and ) numerically,
which represented in Figures 9(a), (b). The Figure 9(a)
represents the numerical solution at the forword wave,
while Figure 9(b) at the backword wave.
 
0=2, 0=2ww


0=3w
4.6. Case VI:


=ek Fk 
In this case Equation (12) becomes
42 2332
343
32
32 =0,
ktss sssss
sss sss
ekkkkskkkkk kk
kk kkkkkk

 (73)
Lie point symmetry for this equation is given by
(a)
(b)
Figure 9. Numerical solution of (72).
12
=, =XX,
s
t



 (74)
the travelling wave solution is obtained by

N. H. ABDEL-ALL ET AL. 674
=Xc
s
t



by which (73) becomes the o.d.e (with
the new independent variable , being the
speed of the wave)
=ysctc


22
22422 4 3
33
22 =
w
wwww ww
wwcewwwwww w
 




0,


(75)
by numerical (intial condition and
, range of to 5) Figures 10(a), (b), This
solution represents a curve on the soliton surface
and .
 
0=2, 0=2ww


0=3w

=wws
0
=
ct ysct
4.7. Case VII:


=
F
kk
In this case Equation (12) becomes
7
23
2
84181512= 0,
tssss ssss
kk kkkkkkkk 
4
(76)
Lie point symmetry for this equation is given by
123
22
=,=, =
33
XXXstk,
s
tst
 

 k

(77)
the travelling wave solution is obtained by
=Xc
s
t



by which (76) becomes the ODE (with
the new independent variable , being the
speed of the wave)
=ysctc
7
22
2
418 1512=
'
wwwwwcwww w

 
 



4
0, (78)
solving the Equation (78) we get
3
79
43
22
21
d=
22
wwyc
wcwwcwcw
0,






(79)
If we take the integration constants to be zero
Equation (79) becomes

3
22
9
24
2
2
=,
32
wcwcw
y
cwwcw











(80)
we have a soliton surface given by the implicit equation

3
22
9
24
2
2
=,
32
wcwcw
sct
cwwcw





 




(81)
Gaussian and mean curvatures of implicit surface are



5
22
3
77
=0,
174
=,
K
cc w
H
2
24
22
14 12 2cwcww




(82)
  




(a)
(b)
Figure 10. Numerical solution of (75).

N. H. ABDEL-ALL ET AL. 675
Figure 11. Soliton surface of (81).
this surface is illustrated in Figure 11.
5. Conclusions
We have discussed motion of curves in a plane and
analysed nonlinear equations and related generalisations
like vector using symmetry methods. These lead to exact
solutions like travelling wave, soliton and other simi-
larity solutions. Gaussian curvature equal zero and mean
curvature don’t equal zero lead to be surfaces cylinder of
[1] B. Kimia, A. Tannenbaum and S. Zucker, “On the Evolu-
tion of Curves via a Function of Curvature. I. The Clas-
sical Case,” Journal Mathematical Analysis and Applica-
tion, Vol. 163, No. 2, 1992, pp. 438-458.
doi:10.1016/0022-247X(92)90260-K
these equations.
6. References
[2] B. Kimia, A. Tannenbaum and S. Zucker, “Shapes, Shochs,
and Deformations I: The Components of Two-Dimen-
sional Shape and the Reaction-Diffusion Space,” Interna-
tional Journal of Computer Vision, Vol. 15, No. 3, 1995,
pp. 189-224. doi:10.1007/BF01451741
[3] C. Rogers and W. K. Schief, “Backlund and Darboux
Transformations Geometry and Modern Application in
Soliton Theory,” Cambridge University press, Cambridge,
2002. doi:10.1017/CBO9780511606359
[4] M. do Carmo, “Differerntial Geometry of Curves and
Surfaces,” Prentice-Hall, Upper Saddle River, 1976.
[5] L. P. Eisenhart, “A Treatise on the Differential Geometry
of Curves and Surfaces,” Ginn, Boston, 1909.
[6] G. W. Bluman and S. Kumei, “Symmetries and Differen-
tial Equations,” Springer, New York, 1989.
[7] P. J. Olver, “Applications of Lie Groups to Differential
Equations,” Springer, New York, 1986.
[8] P. Winternitz, “Lie Groups and Solutions of Nonlinear
Partial Differential Equations, Integrable Systems, Quan-
tum Groups, and Qunatum Field Theories,” Kluwer Aca-
demic, Dordrecht, 1992.
[9] C. H. Gu, “Soliton Theory and Its Applications,”
Springer-Verlag, Berlin, 1995.
[10] C. H. Gu and H. S. Hu, “On the Determination of Nonlin-
ear PDE Admitting Integrable System,” Scientia Sinica,
Series A, 1986, pp. 704-719.