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[1] Fleming, Wendell H. and Soner, H. Mete. Controlled Markov processes and viscosity solutions,
Applications of Mathematics, 25, New York: Springer-Verlag, 1993.
[2] Watanabe, Shinzo. “Analysis of Wiener functionals (Malliavin calculus) and its applications to heat
kernels”. Ann. Probab., Vol. 15, No. 1 (1987), 1--39.
[3] Whitham, G. B. Linear and Nonlinear waves, New York: Wiley, 1999.
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Journal of Natural Sciences and Mathematics, Qassim University, Vol. 1 (December 2007/Thu Al-Hijjah 1428H)
Journal of
Natural Sciences and Mathematics
Volume 1
(Thu Al-Hijjah 1428H)
(December 2007)
Qassim University Scientific Publications
RefereedJournal
Deposit No.:1429/ 2025.
Journal of Natural Sciences and Mathematics, Qassim University, Vol. 1 (December 2007/Thu Al-Hijjah 1428H)
Foreword I
Praise be to Allah, the Cherisher and Sustainer of the worlds.
May peace and blessings be upon the Seal of the Prophets, Muhammad.
May the peace and blessings of Allah be upon him, his family, and all his companions.
Scientific research has always been considered as a minaret for highly developed countries, a real sign for their way
of life, in addition to being the solver of all their problems and the indication of highly advanced universities and
their uniqueness, which are the core embodiments from which knowledge and research products emerge.
Contribution in the different fields of research is considered a great civilized step offered by the researcher for his
country in the first place, and for the world at large in the second. The researcher will be able to add a new horizon to
knowledge and to learning, or to add complementary or advancing ideas to already established scientific projects.
In this Specialized Journal, Qassim University is giving the researchers the chance to publish their scientific
contributions and inventions. This will accomplish its belief that empirical research with its theories, analysis, and
conclusions will enrich scientific progress which will be reflected on modern life, which is witnessing a rigorous
scientific competition.
Through this window of research the University hopes to keep and to contribute to the Progress of our country. It
will also enable us to achieve some of the objectives maintained by the Higher Education Body to meet the general
policy of the Kingdom. It, furthermore, offers to the researchers an embodiment of their intellectual achievements
in natural science research.
On the emergence of the first volume of the Scientific Journal, (The Journal of Natural Sciences and Mathematics),
I would like to invite researchers to enrich the forthcoming editions, with their projects, discussions, and scientific
comments.
Finally, let’s all contribute to the scientific and information world in a way that reflects our civilized progress. I pray
to Allah that our efforts be successful and fruitful.
(Professor) Khalid Alhomoudi
Rector
Qassim University
Journal of Natural Sciences and Mathematics, Qassim University, Vol. 1 (December 2007/Thu Al-Hijjah 1428H)
Foreword II
We, at Qassim University, are pleased to launch the first issue of the Journal of Natural Sciences and Mathematics
(JNM). It is hoped that the journal will go a long way in evolving a culture of scientific thought and outlook in this
part of the world while providing a platform to the scientific community for prompt and timely publication of their
researches.
It goes without saying that mathematics and the natural sciences are among the most important parts of our scientific
outlook and research and that no nation can hope to be successful unless it has shown itself capable of fundamental
research in these fields of human activity. A technology-driven society has to have a strong tradition in research and
scholarship in these intellectual endeavors of utmost importance.
This first issue of the journal is only a humble beginning in this direction and here we reiterate our commitment to
strive hard to improve its quality in the subsequent issues. We would appreciate the comments and suggestions from
our readers/subscribers that would help us in further improving its standards to make it a world class journal in the
field of natural sciences and mathematics.
(Professor) Saleh A. Al-Damegh
Vice President Graduate Studies & Research
Qassim University
vii
Journal of Natural Sciences and Mathematics, Qassim University, Vol. 1 (December 2007/Thu Al-Hijjah 1428H)
Editorial
It is my great pleasure to announce the publication of the first issue of the Journal of Natural Sciences and Mathematics,
JNM for short, which is published at Qassim University, Saudi Arabia. It is our fervent hope that the journal meets
the high standards expected of an international research journal. Considering the explosive pace accompanying the
creation of knowledge and the need for a quick and timely dissemination of latest developments, the journal has been
designed to feature a wide variety of research areas, both in mathematical as well as natural sciences. Accordingly,
the focus shall be on publishing high quality research and survey articles in these areas. In the present issue, we have
preferred to include research articles in mathematics and in physics whereas the articles in chemistry and biological
sciences are planned to be included in the subsequent issues.
In search of the main aim in bringing out this journal which is to ensure speedy the publication of quality research,
we shall welcome and look forward to comments and suggestions from any quarter that shall help in further
improvement of the journal.
Last but not the least, it is my privilege to express my deepest sense of gratitude to Mohammed Amin Sofi and to
all those who have been involved, one way or the other, in accomplishing this task. My special thanks are due to the
editorial staff including the editors/associate editors in bringing this volume to fruition.
AbdulRahman Al-Hussein
Editor-in-Chief
Qassim University
ix
Journal of Natural Sciences and Mathematics, Qassim University, Vol. 1 (December 2007/Thu Al-Hijjah 1428H)
Contents
Foreword I........………......................................................................................................……......…….……........ v
Foreword II ........……….....................................................................................................……......…….….…... vii
Editorial ........………...........................................................................................................……......….….…….....ix
Traveling Wave Solutions to (2+ 1) –Dimensional Nonlinear Evolution Equations
Wazwaz, Abdul-Majid ............................................................................................................................ 01- 13
Two Basic Inequalities for Shape Operator of Hypersurfaces
Tripathi, Mukut Mani..............................................................................................................................15- 21
Exact Localized Structures and Their Dynamic Properties to a (2+ 1) Dimensional Dispersive Long
Wave Equation
Peng,Yan-ze ..............................................................................................................................................23- 33
Eigenvalues of a Minimal Hypersurface in The Unit Sphere
Deshmukh, Sharief ..................................................................................................................................35- 40

xi
JNM: Journal of Natural Sciences and Mathematics
Vol. 1, pp. 1-13 (December 2007/Thu Al-Hijjah 1428H )
c
°
Qassim University Publications
Travelling Wave Solutions to (2+1)-Dimensional
Nonlinear Evolution Equations
Abdul-Majid Wazwaz
Department of Mathematics and Computer Science,
Saint Xavier University,
Chicago, IL 60655.
E-mail: wazwaz@sxu.edu
Abstract: In this work, we investigate four (2+1)-dimensional nonlinear evolution
equations namely, the Konopelchenko-Dubrovsky (KD), the Burgers, the KadomtsevPetviashvili (KP), and the Kadomtsev-Petviashvili-Burgers (KPB) equations. Our approach stems mainly from the tanh-coth method which was proved to be effective and
reliable. New travelling wave solutions of distinct physical structures, are formally derived. The work shows the power of the proposed scheme.
Keywords: Konopelchenko-Dubrovsky equation, Burgers equation, KadomtsevPetviashvili equation, Kadomtsev-Petviashvili-Burgers equation, tanh-coth method,
kinks, solitons.
Received for JNM on May 6, 2007.
1
Introduction
The study of higher-dimensional nonlinear evolution equations is of major significance
in solitary wave theory [1–6, 23]. Nonlinear wave phenomena appear in many scientific and engineering fields, such as fluid dynamics, plasma physics, solid state physics,
optical fibers, acoustics, and many others. These phenomena are usually described by
nonlinear partial differential equations that may contain effects of dispersion, dissipation, diffusion, convection and chemical reactions [14–17]. The study of one or more of
these effects is very important to highlight the physical features and structures of the
obtained solutions and its characteristics as well.
Many types of travelling wave equations are of particular interest to researchers.
Some of these types are: solitary waves which are localized travelling waves , asymptotically zero at large distances, the kink solutions which rise or descend from one
1
2
Abdul-Majid Wazwaz
asymptotic state to another, and the compactons which are solitons with compact support free of exponential wings. In other words, compactons are solitons with compact
spatial support such that each compacton is a soliton confined to a finite core. The delicate balance between nonlinearity and dispersion in the KdV equation, and in other
equations as well, gives rise to solitons. However, the interaction between genuine
nonlinearity and genuinely nonlinear dispersion gives rise to the so-called compactons.
This can be clearly seen in the K(n,n) equations and in other equations [18–22]. Moreover, peakons, and cuspons are other forms of solitons where solution exhibits specific
physical structure for each type.
A variety of powerful analytical and numerical methods are used in the literature
for handling nonlinear evolution equations. The inverse scattering method [1], Hirota’s
bilinear method, the tanh method [11–12, 16], the sine-cosine method [19–20], Backlund transformation method, Darboux transformation [8], projective Riccati equation
method, and many other methods were used to obtain more solutions of distinct physical structures, and to develop more progress in this regard.
In this work, four well-known models that are of particular interest in science will
be investigated. These models are given in partial differential equations in two spatial
dimensions and one temporal dimension.
We will first investigate the (2+1)-dimensional Konopelchenko-Dubrovsky (KD)
equation [9, 13, 21] that reads
ut − uxxx − 6buux + 32 a2 u2 ux − 3vy + 3aux v = 0,
uy = v x ,
(1.1)
where a and b are real parameters. Equation (1.1) is a new nonlinear integrable evolution equation on two spatial and one temporal dimensions. In [9], this equation was
investigated by the inverse scattering transform method. The F-expansion method is
also used in [15] to investigate the KD equation.
We next investigate the (2+1)-dimensional Burgers equation [3]
ut + uux + uuy = αuxx + βuyy ,
(1.2)
where α and β are constants that define the kinematic viscosities. Equation (1.2) is
a nonlinear partial differential equation that incorporates both convection terms uux
and uuy , and dissipation terms uxx and uyy . Burgers introduced the (1+1)-dimensional
equation to capture some of the features of turbulent fluid in a channel caused by the
interaction of the opposite effects of convection and diffusion. An equivalent form
of the Burgers equation (1.2) is derived from the generalized Painlevé integrability
classification in [14].
We will next investigate the (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation [7]
(ut + 3(u2 )x + uxxx )x + λuyy = 0.
(1.3)
Travelling wave solutions to (2+1)-dimensional nonlinear evolution equations
3
The KP equation is a very interesting one because it is completely integrable, as the
KdV equation, that gives N-soliton solutions, and because it is of physical significance.
The KP equation is a generalization of the KdV equation and is used to model shallow
water waves with weakly nonlinear restoring forces. This equation was discovered
by Kadomtsev and Petviashvili where they relaxed the restriction that the waves be
strictly one dimensional, namely the x-direction of the KdV equation
ut + 3(u2 )x + uxxx + = 0.
(1.4)
We close our work by studying the (2+1)-dimensional Kadomtsev-PetviashviliBurgers (KPB) equation [10, 24]
(ut + 3(u2 )x + uxxx + αuxx )x + λuyy = 0,
(1.5)
where α and λ are constants. The KPB equation is a combination of the KP equation
and the Burgers equation. This equation mainly arises from nonlinear wave models
of fluid in an elastic tube, liquid with small bubbles and turbulence [4]. The KPB
equation describes for the dust acoustic waves in dusty plasmas [24] with non-adiabatic
dust charge fluctuation.
The objectives of this work are twofold. First, we seek to extend others works to
establish new exact solutions of distinct physical structures for the nonlinear equations
(1.1) – (1.3) and (1.5). Second, we aim to implement the tanh-coth strategy to achieve
the first goal and to show that the power of this method is its ease of use to determine
shock or solitary type of solutions.
The rest of this paper is organized as follows. In section 2, we give the main steps
of the tanh-coth method. In section 3, we apply the proposed method to the (2+1)dimensional Konopelchenko-Dubrovsky (KD) equation. In section 4, the tanh-coth
method is used to handle the (2+1)-dimensional Burgers equation. In section 5, we
use the proposed method to study the (2+1)-dimensional Kadomtsev-Petviashvili equation. In section 6, we follow the same discussion to the (2+1)-dimensional KadomtsevPetviashvili-Burgers equation. A short summary and discussion is given in final.
2
The tanh-coth method
A PDE
P (u, ut , ux , uxx , uxxx , · · · ) = 0,
(2.1)
can be converted to an ODE
0
00
000
Q(u, u , u , u · · · ) = 0,
(2.2)
upon using a wave variable ξ = (x − ct). Equation (2.2) is then integrated as long as
all terms contain derivatives where integration constants are considered zeros.
4
Abdul-Majid Wazwaz
The tanh method, introduced by Malfliet in [13–14] introduces an independent
variable
Y = tanh(µξ), ξ = x − ct,
(2.3)
is introduced that leads to the change of derivatives:
d
dξ
d2
dξ 2
d
= µ(1 − Y 2 ) dY
,
d
d2
= −2µ2 Y (1 − Y 2 ) dY
+ µ2 (1 − Y 2 )2 dY
2.
(2.4)
The tanh-coth method admits the use of the finite expansion
u(µξ) = S(Y ) =
M
X
ak Y k +
k=0
M
X
bk Y −k ,
(2.5)
k=1
where M is a positive integer, in most cases, that will be determined. Expansion (2.5)
reduces to the standard tanh method [13–14] for bk = 0, 1 ≤ k ≤ M . The parameter
M is usually obtained by balancing the linear terms of highest order in the resulting
equation with the highest order nonlinear terms. Substituting (2.5) into the ODE
(2.2) results in an algebraic system of equations in powers of Y that will lead to the
determination of the parameters ak , (k = 0..M ), µ, and c.
3
The Konopelchenko-Dubrovsky (KD) equation
The (2+1)-dimensional Konopelchenko-Dubrovsky (KD) equation [1] reads
ut − uxxx − 6buux + 32 a2 u2 ux − 3vy + 3aux v = 0,
uy = v x ,
(3.1)
where a and b are real parameters.
Using the wave variable ξ = x + y − ct carries the KD equation (3.1) into a system
of ODEs
0
000
0
0
0
0
−cu − u − 3b(u2 ) + 12 a2 (u3 ) − 3v + 3au v = 0,
(3.2)
0
0
u =v,
where by integrating the second equation we find
u = v.
(3.3)
Substituting (3.3) into the first equation of (3.2) and integrating the resulting equation
we obtain
a
1
00
(c + 3)u + 3(b − )u2 − a2 u3 + u = 0, a 6= 0.
(3.4)
2
2
00
Case I. We first consider the case where a 6= 2b. Balancing u with u3 in (3.4)
gives
M + 2 = 3M,
(3.5)
Travelling wave solutions to (2+1)-dimensional nonlinear evolution equations
5
so that
M = 1.
(3.6)
The tanh-coth method (2.5) admits the use of the finite expansion
b1
.
(3.7)
Y
Substituting (3.7) into (3.4), collecting the coefficients of Y we obtain the system of
algebraic equations for a0 , a1 , b1 , c, and µ. Solving this system gives the following sets
of solutions
(i) The first set:
u(ξ) = a0 + a1 Y +
2b − a
2b − a
2b − a
4(ab − a2 − b2 )
,
a
=
±
,
b
=
0,
µ
=
,
c
=
.
1
1
a2
a2
2a
a2
(ii) The second set:
a0 =
2b − a
2b − a
2b − a
4(ab − a2 − b2 )
,
a
=
0,
b
=
±
,
µ
=
,
c
=
.
1
1
a2
a2
2a
a2
(iii) The third set:
a0 =
a0 =
2b−a
,
a2
(3.8)
(3.9)
a1 = ± 2b−a
, b1 = ± 2b−a
,
2a2
2a2
(3.10)
2b−a
,
4a
µ=
c=
4(ab−a2 −b2 )
.
a2
(iv) The fourth set:
a0 =
2b−a
,
a2
a−2b
a−2b
a1 = ± √
i, b1 = ± √
i,
2a2
2a2
(3.11)
µ=
a−2b
√ i,
2 2a
c=
4(ab−a2 −b2 ) 2
,i
a2
= −1.
In view of these results we obtain the kink solution
µ
·
¸¶
2b − a
2b − a
1 ± tanh
(x + y − ct) ,
u1 (x, y, t) =
a2
2a
and the following travelling wave solutions
µ
·
¸¶
2b − a
2b − a
u2 (x, y, t) =
1 ± coth
(x + y − ct) ,
a2
2a
¡
£ 2b−a
¤
u3 (x, y, t) = 2b−a
2 ± tanh
2a2
£ 2b−a4a (x + y − ct)
¤¢
± coth 4a (x + y − ct) ,
and
u4 (x, y, t) =
2b−a
√
2a2
h
i
√ (x + y − ct)
2 ± tan 22b−a
2a
h
i¢
2b−a
√
± cot 2 2a (x + y − ct) ,
(3.12)
(3.13)
(3.14)
¡√
(3.15)
where c is given above in the three sets of solutions and a 6= 2b, a 6= 0. Recall that
v(x, y, t) = u(x, y, t).
6
Abdul-Majid Wazwaz
Case II. We next consider the case where a = 2b, therefore equation (3.4) reduces to
1
00
(c + 3)u − a2 u3 + u = 0, a 6= 0.
2
(3.16)
00
Balancing u with u3 in (3.16) gives M = 1. The tanh-coth method (2.5) admits the
use of the finite expansion
b1
u(ξ) = a0 + a1 Y + .
(3.17)
Y
Substituting (3.17) into (3.16), collecting the coefficients of Y we obtain the system of
algebraic equations for a0 , a1 , b1 and µ. Solving this system gives the following sets of
solutions:
(i) The first set:
r
c+3
1√
.
(3.18)
a0 = 0, a1 =
c + 3, b1 = 0, µ =
a
2
(ii) The second set:
1√
a0 = 0, a1 = 0, b1 =
c + 3, µ =
a
r
c+3
.
2
(3.19)
(iii) The third set:
r
1√
1√
1
a0 = 0, a1 =
c + 3, b1 =
c + 3, µ =
a
a
2
c+3
, c > −3.
2
(3.20)
(iv) The fourth set:
a0 = 0, a1 = − a1
p
−(c + 3), b1 =
µ=
− 21
1
a
p
−(c + 3),
p
−(c + 3), c < −3.
In view of these results we obtain the kink solution
#!
Ã
"r
1p
c+3
(x + y − ct)
,
u1 (x, y, t) =
2(c + 3) tanh
a
2
and the following travelling wave solutions
Ã
"r
#!
1p
c+3
2(c + 3) coth
(x + y − ct)
,
u2 (x, y, t) =
a
2
u3 (x, y, t) =
1
a
q
h q
i
tanh 12 c+3
(x
+
y
−
ct)
h q 2
i¢
+ coth 12 c+3
(x
+
y
−
ct)
,
2
c+3
2
(3.21)
(3.22)
(3.23)
¡
(3.24)
Travelling wave solutions to (2+1)-dimensional nonlinear evolution equations
7
and
µ
·
¸
·
¸¶
1
1
1
u4 (x, y, t) = γ tanh γ(x + y − ct) − coth γ(x + y − ct) ,
(3.25)
a
2
2
p
where c is left as a free parameter, γ = −(c + 3), and a 6= 0. Recall that v(x, y, t) =
u(x, y, t).
We recognize that the speed wave c plays a significant role in the last solutions. In
other words we also obtain the following solutions:
Ã
"r
#!
−(c + 3)
1p
u5 (x, y, t) =
−2(c + 3) tan
(x + y − ct)
,
(3.26)
a
2
à "r
#!
−(c + 3)
1p
u6 (x, y, t) =
−2(c + 3) cot
(x + y − ct)
,
a
2
q
h q
i
¡
1
1
c+3
c+3
u7 (x, y, t) = a − 2 tan 2 − 2 (x + y − ct)
h q
− cot
1
2
i¢
− c+3
(x
+
y
−
ct)
,
2
(3.27)
(3.28)
valid for c + 3 < 0, and
µ
·
¸
·
¸¶
1
1
1
(3.29)
u8 (x, y, t) = γ 1 tan γ 1 (x + y − ct) + cot γ 1 (x + y − ct) ,
a
2
2
√
valid for where c+3 > 0, and γ 1 = c + 3, and a 6= 0. Recall that v(x, y, t) = u(x, y, t).
4
The (2+1)-dimensional Burgers equation
The (2+1)-dimensional Burgers equation [14] is given by
ut + uux + uuy = αuxx + βuyy ,
(4.1)
where α and β are constants that define kinematic viscosities. This equation contains
convection terms and dissipation terms as well.
Proceeding as before, the wave variable ξ = x + y − ct transforms the Burgers
equation (4.1) into the ODE
0
−cu + u2 + (α + β)u = 0,
(4.2)
obtained after integrating once and setting the constant of integration to zero. Bal0
ancing u with u2 gives M = 1. The tanh-coth method (2.5) uses the finite expansion
u(ξ) = a0 + a1 Y +
b1
.
Y
(4.3)
8
Abdul-Majid Wazwaz
Substituting (4.3) into (4.2), collecting the coefficients of Y we obtain a system of
algebraic equations for a0 , a1 , b1 and µ. Solving this system leads to the following
results
(i) The first set:
c
c
c
a0 = , a1 = , b1 = 0, µ =
.
2
2
2(α + β)
(4.4)
c
c
c
a0 = , a1 = 0, b1 = , µ =
,
2
2
2(α + β)
(4.5)
(ii) The second set:
where c is left as a free parameter.
The first set gives the kink solution
µ
·
¸¶
c
c
u1 (x, y, t) =
1 + tanh
(x + y − ct) ,
2
2(α + β)
for fixed y. However the second set gives the travelling wave solution
µ
·
¸¶
c
c
u2 (x, y, t) =
1 + coth
(x + y − ct)
2
2(α + β)
5
(4.6)
(4.7)
The Kadomtsev-Petviashvili (KP) equation
The (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation is given by
(ut + 6uux + uxxx )x + λuyy = 0,
(5.1)
where λ is a constant.
Using the wave variable ξ = x + y − ct carries the KP equation (5.1) into an ODE
00
(λ − c)u + 3u2 + u = 0,
(5.2)
obtained upon integrating twice and setting coefficients of integration to zeros.
00
Balancingu with u2 gives M = 2. The tanh-coth method (2.5) introduces the finite
expansion
b1
b2
u(ξ) = a0 + a1 Y 2 + +a2 Y 2 + + 2 .
(5.3)
Y
Y
Substituting (5.3) into (5.2), collecting the coefficients of Y we obtain a system of
algebraic equations for a0 , a1 , a2 , b1 , b2 , and µ. Solving this system gives a1 = b1 = 0
and the following sets of solutions
(i) The first set:
a0 =
λ−c
λ−c
1√
, a2 = −
, b2 = 0, µ =
λ − c, λ > c.
6
2
2
(5.4)
Travelling wave solutions to (2+1)-dimensional nonlinear evolution equations
9
(ii) The second set:
a0 =
c−λ
c−λ
1√
, a2 = −
, b2 = 0, µ =
c − λ, λ < c.
2
2
2
(5.5)
λ−c
λ−c
1√
, a2 = 0, b2 = −
,µ=
λ − c, λ > c.
6
2
2
(5.6)
(iii) The third set:
a0 =
(iv) The fourth set:
c−λ
c−λ
1√
, a2 = 0, b2 = −
,µ=
c − λ, λ < c,
2
2
2
where c is left as a free parameter.
In view of the first and the second sets we obtain the soliton solutions
µ
¸¶
·
√
λ−c
2 1
u1 (x, y, t) =
1 − 3 tanh
λ − c(x + y − ct) , λ > c,
6
2
a0 =
and
c−λ
u2 (x, y, t) =
2
µ
·
2
sech
¸¶
1√
c − λ(x + y − ct) , λ < c,
2
whereas the third and the fourth sets give following travelling wave solutions
µ
¸¶
·
√
λ−c
2 1
u3 (x, y, t) =
1 − 3 coth
λ − c(x + y − ct) , λ > c,
6
2
and
c−λ
u4 (x, y, t) = −
2
µ
·
¸¶
√
2 1
csch
c − λ(x + y − ct) , λ < c.
2
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
The relation between λ and c plays a major role in changing the structure of the
obtained solutions. This in turn gives the trigonometric solutions
¸¶
µ
·
√
λ−c
2 1
u5 (x, y, t) =
c − λ(x + y − ct) , λ < c,
(5.12)
1 + 3 tan
6
2
¸¶
µ
·
√
c−λ
2 1
u6 (x, y, t) =
λ − c(x + y − ct) , λ > c,
sec
2
2
¸¶
µ
·
√
λ−c
2 1
u7 (x, y, t) =
c − λ(x + y − ct) , λ < c,
1 + 3 cot
6
2
and
c−λ
u8 (x, y, t) =
2
¸¶
µ
·
√
2 1
λ − c(x + y − ct) , λ > c.
csc
2
(5.13)
(5.14)
(5.15)
10
6
Abdul-Majid Wazwaz
The KP-Burgers (KPB) equation
The (2+1)-dimensional Kadomtsev-Petviashvili-Burgers (KPB) equation [4, 24] is
given by
(ut + 6uux + uxxx + αuxx )x + λuyy = 0,
(6.1)
where α and λ are constants. The main difference between the KP equation and the
KPB equation is the dissipation term uxx that is added to the KP equation.
Proceeding as before, the wave variable ξ = x + y − ct carries the KPB equation
(6.1) into the ODE
0
00
(λ − c)u + 3u2 + αu + u = 0,
(6.2)
00
Balancingu with u2 gives M = 2. The tanh-coth method (2.5) allows the use of the
finite expansion
b1
b2
u(ξ) = a0 + a1 Y + a2 Y 2 + + 2 .
(6.3)
Y
Y
Substituting (6.3) into (6.2), collecting the coefficients of Y we obtain a system of
algebraic equations for a0 , a1 , a2 , b1 , b2 , c, and µ. Solving this system gives the following
sets of solutions
(i) The first set:
1 2
a0 = − 50
α , a1 =
b1 = b2 = 0, µ =
1 2
α ,
25
1
α,
10
1 2
a2 = − 50
α ,
c=λ−
(6.4)
6 2
α .
25
(ii) The second set:
a0 =
3 2
α ,
50
a1 =
b1 = b2 = 0, µ =
1 2
α ,
25
1
± 10
α,
1 2
a2 = − 50
α ,
c=λ+
(6.5)
6 2
α .
25
(iii) The third set:
1 2
a0 = − 50
α , a1 = 0, a2 = 0, b1 =
b2 =
1 2
α ,
− 50
µ=
1
α,
10
c=λ−
1 2
α ,
25
(6.6)
6 2
α .
25
(iv) The fourth set:
a0 =
b2 =
3 2
α ,
50
a1 = 0, a2 = 0, b1 =
1 2
− 50
α ,
µ=
1
± 10
α,
c=λ+
1 2
α ,
25
(6.7)
6 2
α .
25
where c is left as a free parameter.
In view of the first and the second sets we obtain the kink solutions
¸¶2
µ
·
1 2
6 2
1
u1 (x, y, t) = − α 1 − tanh
,
α(x + y − (λ − α ) t)
50
10
25
(6.8)
11
Travelling wave solutions to (2+1)-dimensional nonlinear evolution equations
and
u2 (x, y, t) =
£1
¤¢
6 2
3 − tanh
α(x
+
y
−
(λ
+
α
)
t)
25
£ 1 10
¤¢
6 2
× 1 + tanh 10
α(x + y − (λ + 25
α ) t) ,
1 2
α
50 ¡
¡
(6.9)
for fixed y, whereas the third and the fourth sets give following travelling wave solutions
µ
·
¸¶2
1 2
6 2
1
u3 (x, y, t) = − α 1 − coth
α(x + y − (λ − α ) t)
,
(6.10)
50
10
25
and
7
u4 (x, y, t) =
£1
¤¢
6 2
3 − coth
α(x
+
y
−
(λ
+
α
)
t)
25
¤¢
£ 1 10
6 2
α(x + y − (λ + 25
α ) t) ,
× 1 + coth 10
1 2
α
50 ¡
¡
(6.11)
Concluding remarks
Four nonlinear evolution equations, each is derived in two spatial and one temporal
dimensions, were investigated. Our analysis depends mainly on the tanh-coth method.
The two goals of this work, which are the determination of travelling wave solutions
to these models and the demonstration of the power of the proposed method, were
achieved. Several exact solutions, with distinct physical structures, were formally derived. The performance of the method has been shown to be reliable and effective. The
tanh-coth method can be used for a wider applicability due to its ease to use and due
to its reliability. The applied method will be used in further works to establish more
entirely new solutions for other kinds of nonlinear evolution equations.
References
[1] Ablowitz, M. J. and Clarkson, P. A. Solitons, Nonlinear Evolution equations and
Inverse Scattering, Cambridge: Cambridge University Press, 1991.
[2] Baldwin, D., Goktas U., Hereman, W., Hong, L., Martino, R. S. and Miller, J. C.
“Symbolic computation of exact solutions in hyperbolic and elliptic functions for
nonlinear PDEs”. Journal of Symbolic Computation, 37 (2004), 669-705.
[3] Burgers, J.M. it The nonlinear diffusion equation, Dordtrecht: Reiedl, 1974.
[4] Fan, E., Zhang, J. and Hon, B. “A new complex line soliton for the two-dimensional
KdV-Burgers equation”. Phys. Lett. A, 291 (2001), 376–380.
[5] Goktas, U. and Hereman, W. “Symbolic computation of conserved densities for
systems of nonlinear evolution equations”. Journal of Symbolic Computation, 24
(1997), 591-621.
[6] Janaki, M. S., Dasgupta, B., Som, B. K. and Gupta, M. R. “K-P Burgers equation
for the decay of solitary magnetosonic waves propagating obliquely in a warm
collisional plasma”. J. Phys. Soc., 60 (1991), 2977–2984.
12
Abdul-Majid Wazwaz
[7] Kadomtsev, B. B. and Petviashvili, V. I. “On the stability of solitary waves in
weakly dispersive media”. Sov. Phys. Dokl., 15 (1970), 539–541.
[8] Kichenassamy, S. and Olver, P. “Existence and nonexistence of solitary wave solutions to higher-order model evolution equations”. SIAM J. Math. Anal., 23, 5
(1992), 1141-1166.
[9] Konopelchenko, B. G. and Dubrovsky, V. G. “Some new integrable nonlinear
evolution equations in 2 + 1 dimensions”. Phys. Lett. A, 102, No. 1,2 (1984),
15–17.
[10] Ma, Z. Y., Wu, X. F. and Zhu, J. M. “Multisoliton excitations for the KadomtsevPetviashvili equation and the coupled Burgers equation”. Chaos, Solitons and
Fractals, 31 (2007), 648–657.
[11] Malfliet, W. and Hereman, W. “The tanh method:I. Exact solutions of nonlinear
evolution and wave equations”. Physica Scripta, 54 (1996), 563-568.
[12] Malfliet, W. and Hereman, W. “The tanh method:II. Perturbation technique for
conservative systems”. Physica Scripta, 54 (1996), 569-575.
[13] Song, L. N. and Zhang, H. Q. “New exact solutions for Konopelchenko-Dubrovsky
equation using an extended Riccati equation rational expansion method”. Commun. Theor. Phys., 45, 5 (2006), 769–776.
[14] Wang, B.-D., Song, L.-N. and Zhang, H.-Q. “A new extended elliptic equation
rational expansion method and its application to (2 + 1)-dimensional Burgers
equation”. Chaos, Solitons and Fractals, 33, 5 (2007), 1546-1551.
[15] Wang, D. and Zhang, H.-Q. “Further improved F-expansion method and new exact
solutions of Konopelchenko-Dubrovsky equation”. Chaos, Solitons and Fractals,
25 (2005), 601–610.
[16] Wazwaz, A. M. “The tanh method for travelling wave solutions of nonlinear equations”. Appl. Math. Comput., 154, 3 (2004), 713-723.
[17] Wazwaz, A. M. Partial Differential Equations: Methods and Applications, The
Netherlands: Balkema Publishers, 2002.
[18] Wazwaz, A. M. “Compactons in a class of nonlinear dispersive equations”. Mathematical and Computer Modelling, 37, No. 3,4 (2003) 333-341.
[19] Wazwaz, A. M. “Distinct variants of the KdV equation with compact and noncompact structures”. Appl. Math. Comput., 150 (2004), 365-377.
Travelling wave solutions to (2+1)-dimensional nonlinear evolution equations
13
[20] Wazwaz, A. M. “Variants of the generalized KdV equation with compact and noncompact structures”. Computers and Mathematics with Applications, 47 (2004),
583-591.
[21] Wazwaz, A. M. “New kinks and solitons solutions to the (2+1)-dimensional
Konopelchenko-Dubrovsky equation”. Mathematical and Computer Modelling, 45
(2007), 473–479.
[22] Wazwaz, A. M. “Travelling wave solutions of generalized forms of Burgers,
Burgers-KdV and Burgers-Huxley equations”. Appl. Math. Comput., 169, (2005),
639–656.
[23] Whitham, G. B. Linear and Nonlinear waves, New York: Wiley, 1999.
[24] Xue, J.-K. “Kadomtsev-Petviashvili (KP) Burgers equation in a dusty plasmas
with non-adiabatic dust charge fluctuation”. Eur. Phys. J. D., 26 (2003), 211–
214.
JNM: Journal of Natural Sciences and Mathematics
Vol. 1, pp. 15-21 (December 2007/Thu Al-Hijjah 1428H )
c
°
Qassim University Publications
Two Basic Inequalities for Shape Operator of
Hypersurfaces
Mukut Mani Tripathi
Department of Mathematics,
Banaras Hindu University, Varanasi, 221 005, India.
E-mail: mmtripathi66@yahoo.com
Abstract: Two basic inequalities for shape operator of Riemannian hypersurfaces are
established.
Keywords: Ricci curvature, k-Ricci curvature, scalar curvature, shape operator.
2000 Mathematics Subject Classification: 53C40.
Received for JNM on May 6, 2007.
1
Introduction
In [1] B.-Y. Chen recalled that one of the basic interests of submanifold theory is to
establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold. Many famous results in differential geometry can
be regarded as results in this respect. The main extrinsic invariant is the squared mean
curvature and the main intrinsic invariants include the classical curvature invariants
namely the scalar curvature, the sectional curvature and the Ricci curvature. In the
literature, we find several work done in establishing basic inequalities involving the
squared mean curvature and one of the classical curvature invariants namely the scalar
curvature, the sectional curvature and the Ricci curvature for different kind of submanifolds of real space forms and complex space forms. First results in these directions
were given by B.-Y. Chen in [1], [2] and [3]. Motivated by a result of B.-Y. Chen [3],
a basic inequality, involving the Ricci curvature and the squared mean curvature of
the submanifold of any Riemannian manifolds, was proved recently [5]. The goal was
achieved by use of the concept of k-Ricci curvature [3]. Then some basic inequalities
were obtained [4] for a submanifold of any Riemannian manifold involving the squared
mean curvature and one of the intrinsic invariants namely the scalar curvature and the
sectional curvature of the submanifold.
15
16
Mukut Mani Tripathi
In this paper we apply these results to find two basic inequalities for shape operator
of Riemannian hypersurfaces. In section 2, we recall the definitions of Ricci curvature,
k-Ricci curvature and scalar curvature. Then we list required known results for later
use. In section 3, we give main results.
2
Some known results
Let M be an n-dimensional Riemannian manifold equipped with a Riemannian metric
g. Let Πk be a k-plane section of Tp M and X a unit vector in Πk . We choose an
orthonormal basis {e1 , . . . , ek } of Πk such that e1 = X. The Ricci curvature RicΠk of
Πk at X is defined by [3]
RicΠk (X) = K12 + K13 + · · · + K1k ,
(2.1)
where Kij is the sectional curvature of the plane section spanned by ei and ej at
p ∈ M . RicΠk (X) is called a k-Ricci curvature. The scalar curvature τ (Πk ) of the
k-plane section Πk is given by [3]
X
Kij ,
(2.2)
τ (Πk ) =
1≤i<j≤k
where {e1 , . . . , ek } is any orthonormal basis of the k-plane section Πk . Geometrically,
τ (Πk ) is the scalar curvature of the image expp (Πk ) of Πk at p under the exponential
map at p. The Ricci curvature of X now becomes Ric (X) = RicTp M (X). The scalar
curvature τ (p) of M at p is identical with the scalar curvature of the tangent space
Tp M of M at p, that is, τ (p) = τ (Tp M ). If Π is a plane section, τ (Π) is simply the
sectional curvature K(Π) of Π.
Now, let M be an n-dimensional Riemannian submanifold of an m-dimensional
Riemannian manifold (M̃ , g). The induced metric on M will also be denoted by g.
The equation of Gauss is given by
R(X, Y, Z, W ) = R̃(X, Y, Z, W ) + g (σ(X, W ), σ(Y, Z)) − g (σ(X, Z), σ(Y, W )) (2.3)
for all X, Y, Z, W ∈ T M , where R̃ and R are the Riemann curvature tensors of M̃ and
M respectively; and σ is the second fundamental form. The mean curvature vector
H(p) is given by nH(p) = Tr (σ). The submanifold M is totally geodesic in M̃ if σ = 0,
and minimal if H = 0. If σ (X, Y ) = g (X, Y ) H for all X, Y ∈ T M , then M is totally
umbilical. Let {e1 , . . . , en } be an orthonormal basis of the tangent space Tp M and er
belongs to an orthonormal basis {en+1 , . . . , em } of the normal space Tp⊥ M . We put
σ rij
= hσ (ei , ej ) , er i
and
2
kσk =
n
X
i,j=1
hσ (ei , ej ) , σ (ei , ej )i .
17
Two basic inequalities for shape operator of hypersurfaces
Let Kij and K̃ij denote the sectional curvature of the plane section spanned by ei and
ej at p in the submanifold M and in the ambient manifold M̃ respectively. In view of
the equation of Gauss (2.3), we have
Kij = K̃ij +
m
X
¡
¢
σ rii σ rjj − (σ rij )2 .
(2.4)
r=n+1
From (2.4) it follows that
2τ (p) = 2τ̃ (Tp M ) + n2 kHk2 − kσk2 ,
(2.5)
P
where τ̃ (Tp M ) = 1≤i<j≤n K̃ij denotes the scalar curvature of the n-plane section Tp M
in the ambient manifold M̃ .
We recall the following three results, in which basic inequalities involving intrinsic
invariants namely the Ricci curvature, the scalar curvature and the sectional curvature
and the extrinsic invariant namely the squared mean curvature of any submanifold of
a Riemannian manifold are established.
Theorem 2.1 ([5, Theorem 3.1]) Let M be an n-dimensional submanifold of a Riemannian manifold. For a unit vector X ∈ Tp M , it follows that
1 2
f (Tp M ) (X) ,
n kHk2 ≥ Ric (X) − Ric
4
(2.6)
f (Tp M ) (X) is the n-Ricci curvature of Tp M at X with respect to the ambient
where Ric
manifold M̃ . The equality case of (2.6) is satisfied by the unit vector X ∈ Tp M if and
only if
n
σ (X, X) = H (p) and σ (X, Y ) = 0
(2.7)
2
for all Y ∈ Tp M such that g(X, Y ) = 0. The equality case of (2.6) holds for all unit
vectors X ∈ Tp M and for all p ∈ M if and only if either M is totally geodesic or M is
a totally umbilical surface.
Proof. Here we give a very simple proof, which is taken from [4]. We put
n
2σ 0 (X, Y ) = σ(X, Y ) − g(X, Y )H ,
X, Y ∈ Tp M.
2
Then for a unit vector X ∈ Tp M , we obtain
0 ≤
n
X
g(σ 0 (X, ei ), σ 0 (X, ei ))
i=1
=
n
X
i=1
g (σ(X, ei ), σ(X, ei )) − ng (H, σ(X, X)) +
n2
kHk2 .
4
According to the Gauss equation (2.3) and the above inequality, we can easily get our
theorem. ¥
18
Mukut Mani Tripathi
Theorem 2.2 ([4, Theorem 4.2]) For an n-dimensional submanifold M in a Riemannian manifold, at each point p ∈ M , we have
n (n − 1)
kHk2 ≥ τ (p) − τ̃ (Tp M ) ,
2
(2.8)
P
where τ̃ (Tp M ) = 1≤i<j≤n K̃ij denotes the scalar curvature of the n-plane section Tp M
in the ambient manifold M̃ . The equality holds at each point p ∈ M if and only if M
is totally umbilical.
Theorem 2.3 [4, Theorem 5.2] Let M be an n-dimensional (n ≥ 3) submanifold of an
m-dimensional Riemannian manifold M̃ . Then, for each point p ∈ M and each plane
section Π ⊂ Tp M , we have
τ − K (Π) ≤
n2 (n − 2)
kHk2 + τ̃ (Tp M ) − K̃ (Π) .
2 (n − 1)
(2.9)
The equality in (2.9) holds at p ∈ M if and only if there exist an orthonormal basis
{e1 , . . . , en } of Tp M and an orthonormal basis {en+1 , . . . , em } of Tp⊥ M such that (a)
Π = Span {e1 , e2 } and (b) the forms of shape operators Ar ≡ Aer , r = n + 1, . . . , m,
become


a 0
0
,
An+1 =  0 b
(2.10)
0
0 0 (a + b) In−2


cr dr
0
Ar =  dr −cr
r ∈ {n + 2, . . . , m} .
(2.11)
0 ,
0
0 0n−2
3
Main results
Let M be a orientable hypersurface in an m-dimensional Riemannian manifold
The induced Riemannian metric on M will also be denoted by g, and ∇ denotes
Riemannian connection of (M, g). Let N be a local unit normal field of M and A
shape operator of M with respect to N . Then Gauss and Weingarten formulae
given respectively by
˜ X Y = ∇X Y + g(AX, Y )N,
∇
and
M̃ .
the
the
are
˜ X N = −AX.
∇
The equation of Gauss is given by
R(X, Y )Z = R̃(X, Y )Z + g(AY, Z)AX − g(AX, Z)AY.
We shall need the following:
(3.1)
19
Two basic inequalities for shape operator of hypersurfaces
Lemma 3.1 Let M be hypersurface in an m-dimensional Riemannian manifold M̃ .
Let Π be a plane section in Tp M and X be a unit vector in Tp M . Then
K (Π) = K̃ (Π) + det(A|Π ),
(3.2)
¡
¢
f (Tp M ) (X) + Tr (A) g (AX, X) − g A2 X, X ,
Ric (X) = Ric
¡ ¢
2τ (p) = 2τ̃ (Tp M ) + (Tr (A))2 − Tr A2 ,
(3.3)
(3.4)
f (Tp M ) (X) is the (m − 1)-Ricci curvature of Tp M at X with respect to the
where Ric
ambient manifold M̃ and τ̃ (Tp M ) be the scalar curvature of the (m − 1)-plane section
Tp M in the ambient manifold M̃ .
Proof. Let M be a hypersurface in an m-dimensional Riemannian manifold M̃ . Now,
let {e1 , . . . , em−1 } be an orthonormal basis of the tangent space Tp M . Then from (3.1)
we get
Kij = K̃ij + g(Aei , ei )g(Aej , ej ) − g(Aei , ej )g(Aej , ei ),
(3.5)
where Kij and K̃ij are the sectional curvature of the plane section spanned by ei and
ej at p in the hypersurface M and in the ambient manifold M̃ respectively. Then (3.2)
is implied by (3.5). From (3.5) for a fixed i ∈ {1, . . . , m − 1} we have
m−1
X
Kij =
m−1
X
K̃ij + g(Aei , ei )
=
m−1
X
K̃ij + g(Aei , ei )
Ãm−1
X
Ãm−1
X
m−1
X
g(Aei , ej )g(Aei , ej )
j6=i
!
g(Aej , ej ) − g(Aei , ei )
j=1
j6=i
−
g(Aej , ej ) −
j6=i
j6=i
j6=i
m−1
X
!
g(Aei , ej )g(Aei , ej ) − g(Aei , ei )g(Aei , ei )
j=1
=
m−1
X
K̃ij + g(Aei , ei )
j6=i
m−1
X
g(Aej , ej ) −
j=1
m−1
X
g(Aei , ej )g(Aei , ej ),
j=1
which gives
f (Tp M ) (ei ) + Tr (A) g(Aei , ei ) − g (Aei , Aei ) .
Ric (ei ) = Ric
(3.6)
From (3.6) for a unit vector X ∈ Tp M , we immediately get (3.3). Also from (3.6) we
get
m−1
X
Ric (ei ) =
i=1
m−1
X
i=1
f (Tp M ) (ei ) + Tr (A)
Ric
m−1
X
i=1
g (Aei , ei ) −
m−1
X
¢
¡
g A2 ei , ei ,
i=1
which gives (3.4). ¥
Now, we prove two basic inequalities for shape operator of a hypersurface in the
following two Theorems:
20
Mukut Mani Tripathi
Theorem 3.2 Let M be a hypersurface of an m-dimensional Riemannian manifold.
Then for each unit vector X ∈ Tp M , it follows that
¡
¢
1
(Tr (A))2 ≥ Tr (A) g (AX, X) − g A2 X, X .
4
(3.7)
The equality case of (3.7) is true for all unit vectors in Tp M , p ∈ M if and only if
either M is totally geodesic or M is a totally umbilical surface.
Proof. From (m − 1) H =
Pm−1
i=1
g (Aei , ei ) N , we get
(m − 1)2 kHk2 = (Tr (A))2 .
(3.8)
Now, using (3.3) and (3.8) in (2.6), we find the inequality (3.7). Rest of the proof is
straightforward. ¥
Theorem 3.3 Let M be a hypersurface of an m-dimensional (m > 3) Riemannian
manifold. Then, for each point p ∈ M and each plane section Π ⊂ Tp M , we have
1
1 ¡ ¢
(Tr (A))2 ≤ Tr A2 + det(A|Π ).
2(m − 2)
2
(3.9)
The equality in (3.9) holds at p ∈ M if and only if there exists an orthonormal basis
{e1 , . . . , em−1 } of Tp M such that Π = Span {e1 , e2 } and the shape operator A becomes
of the form


a 0
0
.
A= 0 b
(3.10)
0
0 0 (a + b) Im−3
Proof. If M is a hypersurface of an m-dimensional (m > 3) Riemannian manifold,
then from (3.2) and (3.4) we get
τ (p) − K (Π) = τ̃ (Tp M ) − K̃ (Π) − det(A|Π ) +
1
1 ¡ ¢
(Tr (A))2 − Tr A2 .
2
2
Using the Schwarz inequality (m − 1) Tr. A2 ≥ (Tr. A)2 and the above equation in (2.9)
we get (3.9). ¥
References
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Arch. Math., 60 (1993), 568-578.
[2] Chen, B.-Y. “Mean curvature and shape operator of isometric immersions in real
space form”. Glasgow Math. J., 38, No. 1 (1996), 87-97.
Two basic inequalities for shape operator of hypersurfaces
21
[3] Chen, B.-Y., “Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions”. Glasgow Math. J., 41 (1999), 33-41.
[4] Hong, S. P., Matsumoto, K. and Tripathi, M. M. “Certain basic inequalities for
submanifolds of locally conformal Kaehlerspace forms”. SUT J. Math., 41, No. 1
(2005), 75-94.
[5] Hong, S. P. and Tripathi, M. M. “On Ricci curvature of submanifolds”. Internat.
J. Pure Appl. Math. Sci., 2, No. 2 (2005), 227-245.
JNM: Journal of Natural Sciences and Mathematics
Vol. 1, pp. 23-33 (December 2007/Thu Al-Hijjah 1428H )
c
°
Qassim University Publications
Exact Localized Structures and Their Dynamic
Properties to A (2+1) Dimensional Dispersive
Long Wave Equation
Yan-ze Peng1
School of Mathematics and Statistics,
Wuhan University, Wuhan, 430072, P. R. China.
E-mail: yanzepeng@163.com
Abstract: The nonlinear transformation method is developed to solve a (2+1) dimensional dispersive long wave equation. A general solution involving an arbitrary
low dimensional function is then obtained for the equation in question. New types of
localized coherent structures, including periodic dromion-like, multi-dromion-like and
multi-solitoff-like, are obtained by choosing appropriately the arbitrary function. Their
dynamic properties are numerically studied and illustrated by graphs. The creation phenomenon of multi-dromion-like structures, a new nonlinear phenomenon, is reported. A
simple review of the method is also given.
Keywords: A (2+1) dimensional dispersive long wave equation, exact solutions, the
nonlinear transformation method.
2000 Mathematics Subject Classification: 35Q35, 35B20, 37K45.
Received for JNM on May 6, 2007.
1
Introduction
In nonlinear science, searching for exact solutions, especially localized structures, of
nonlinear partial differential equations (PDEs) is one of the most important tasks.
There are a huge variety of methods available for constructing exact solutions of nonlinear partial differential equations (PDEs). Some of the most important methods are
the inverse scattering transformation [1], the bilinear method [2], symmetry reductions
[3-4], Bäcklund and Darboux transformations [5], the singular manifold method [6-8]
1
This work is supported by the research fund for the doctoral program of higher education of
China: No. 20070486094.
23
24
Yan-ze Peng
and so on. However, there is not an unified method to deal with a large number of
nonlinear PDEs. So, it is also important to seek new methods for solving these nonlinear equations. In this paper, the nonlinear transformation method is developed to
solve the (2+1) dimensional dispersive long wave equation
ut + uxx − 2uw = 0,
vt − vxx + 2vw = 0,
(uv)x − wy = 0.
(1.1)
This system is obtained in [9] as a reduction of self-dual Yang-Mills equations while its
complex version appears in [10]. Some exact solutions and localized coherent structures
are studied [11-12] for the complex version of equation (1.1) under the name of Melnikov
equation. Some basic dromion structures has been obtained [13] for equation (1.1) by
using the bilinear method [2].
The organization of paper is as follows. In section 2, the nonlinear transformation is
introduced to obtain a general functional separation solution containing an arbitrary
function of the variable y alone. Some new types of localized coherent structures
are obtained and their dynamic properties are numerically studied and illustrated by
graphs in section 3. The conclusions and discussions are given in last section.
2
A general solution to equation (1.1)
Introducing the nonlinear transformation
u=
ϕx
,
ϕ
v=
ϕy
,
ϕ
(2.1)
w = −(ln ϕ)xx ,
then, substituting it into equation (1.1) and equating the coefficients of like powers of
ϕ to zero, we have
ϕx (ϕt + ϕxx ) = 0,
−ϕy (ϕt + ϕxx ) + 2ϕx ϕxy = 0,
(2.2)
ϕx ϕxy = 0, (ϕt + ϕxx )x = 0,
ϕty − ϕxxy = 0, ϕxxy = 0,
which has a general functional separation solution
ϕ = f (x, t) + g(y),
(2.3)
where f (x, t) satisfies the equation
ft + fxx = 0,
(2.4)
Exact localized structures and their dynamic properties to a (2+1) dimensional . . .
25
and g(y) is an arbitrary function of indicated variable. Therefore, we obtain a general
functional separation solution of equation (1.1)
fx
gy
, v=
,
f +g
f +g
fxx
fx2
−
,
w=
2
(f + g)
f +g
u=
(2.5)
where f (x, t) satisfies equation (2.4) and g(y) is an arbitrary function of variable y. It
is pointed out that equation (1.1) possesses some special types of localized structures
for the following combined field
U ≡ uv =
fx g y
,
(f + g)2
(2.6)
or the potential field U ≡ −uy ≡ −[ln(f + g)]xy rather than the physical field u or v
itself.
3
Some localized coherent structures of equation (1.1)
In what follows, our attention are payed to localized coherent structures of equation
(1.1). Since equation (2.4) is linear, it has a general solution
f =1+
n
X
2
eki x−ki t ,
(3.1)
i=1
where ki are arbitrary real constants and n is a positive integer. For simplicity, we take
n = 2 in this paper. Several interesting cases are considered for the arbitrary function
g(y).
Case 1. g = esn(ly|m) .
It follows from equation (2.6) that
2
2
U = −[ln(1 + ek1 x−k1 t + ek2 x−k2 t + esn(ly|m) )]xy ,
(3.2)
where sn is Jacobi elliptic function [14-16]. Its time evolution process is shown in
Figure 1 with parameter values k1 = 1, k2 = −1, l = 1, m = 0.2 and t = −5, 0, 5. By
the way, these parameter values are valid throughout the paper, unless stated otherwise.
From the figures, we can see that equation (3.2) is a periodic dromion-like structure
(because it is periodic in y direction) and a new type of nonlinear phenomenon has
taken place. When t < 0, the amplitude of the localized structure is increasing with
time, and arrives at its maximum at t = 0. After this, the periodic dromion-like
structure propagates steadily. As m → 1, snξ → tanhξ, equation (3.2) is reduced to
2
2
U = −[ln(1 + ek1 x−k1 t + ek2 x−k2 t + etanh(ly) )]xy ,
(3.3)
26
Yan-ze Peng
whose time evolution is illustrated in Figure 2. The same nonlinear phenomenon of
two-dromion-like structure is found for equation (3.3).
Case 2. g = tanh(sn(ly|m)) + 1.
In this case, the localized structure of equation (1.1) reads
2
2
U = −[ln(2 + ek1 x−k1 t + ek2 x−k2 t + tanh(sn(ly|m)))]xy ,
(3.4)
whose evolution figures are similar to those of equation (3.2), and thus omitted.
Case 3. g = sn(tanhly|m) + 1.
From equation (2.6), one has
2
2
U = −[ln(2 + ek1 x−k1 t + ek2 x−k2 t + sn(tanhly|m))]xy ,
(3.5)
which has the same evolution property as that of equation (3.3).
Case 4. g = tanh3 (ly) + 1.
In this case, we obtain a four-dromion-like structure
2
2
U = −[ln(2 + ek1 x−k1 t + ek2 x−k2 t + tanh3 (ly))]xy .
(3.6)
Figure 3 illustrates the creation phenomenon of dromion-like structure (3.6).
2
Case 5. g = ely .
The localized structure of equation (1.1) is
2
2
2
U = −[ln(1 + ek1 x−k1 t + ek2 x−k2 t + ely )]xy ,
(3.7)
which is a four-solitoff-like structure and has no creation phenomenon. And its typical
spatial structure is illustrated in Figure 4. However, if we take l = −1 and other
parameters are unchanged, equation (3.7) is another four-dromion-like structure, and
its creation phenomenon is shown in Figure 5.
4
Conclusion and discussion
We have used the nonlinear transformation method to obtain a general functional separation solution containing an arbitrary function for a (2+1) dimensional dispersive
long wave equation (1.1). Some new types of localized coherent structures, which are
quite different from the basic dromion structure [13], are given by selecting appropriately the arbitrary function, and their dynamic properties are numerically studied and
illustrated by graphs. And we find that the solution structures have relation to the
parameter values. For example, equation (3.7) is a four-solitoff-like structure as l = 1
while a four-dromion-like structure as l = −1. Moreover, the four solitoff-like can
steadily propagate but the four dromion-like can not. It is worth mentioning that all
the localized structures obtained in this paper have no interaction. It is an interesting
problem how to obtain localized structures with interaction by choosing appropriately
Exact localized structures and their dynamic properties to a (2+1) dimensional . . .
27
the functions f (x, t) and/or g(y) appearing in equation (2.6).
Another natural question is how to use this method to solve other nonlinear PDEs.
Here we propose a knack. For a given nonlinear PDE, usually one should take the
transformation of the form
∂ l+m+n lnϕ
u= l m n ,
(4.1)
∂ t∂ x∂ y
where l, m and n are integer and determined by balancing the linear term of the
highest order derivative with nonlinear term in the equation of interest. For equation
(1.1) we obtain thus the transformation (2.1), which can not be got by means of the
singular manifold method [6-8].
References
[1] Gardner, C. S., Green, J. M., Kruskal, M. D. and Miura, R. M. “Method for
solving the Korteweg-de Vries equation”. Phys. Rev. Lett., 19 (1967), 1095-1097.
[2] Hirota, R. “Exact solution of the Korteweg-de Vries equation for Multiple collisions
of Solitons”. Phys. Rev. Lett., 27 (1971), 1192-1194.
[3] Olver, P. J. Application of Lie Group to Differential Equation, New York: Springer,
1986.
[4] Clarkson, P. A. and Kruskal, M. D. “New Similarity reductions of the Boussinesq
equation”. J. Math. Phys., 30 (1989), 2201-2213.
[5] Matveev, V. B. and Salle, M. A. Darboux Transformations and Solitons, Berlin:
Springer, 1991.
[6] Weiss, J., Tabor, M. and Carnevale, G. “The Painlevé Property for Partial differential equations”. J. Math. Phys., 24 (1983), 522-526.
[7] Weiss, J. “The Painlevé property for partial differential equations. II: Bäcklund
transformation, Lax pairs, and the Schwarzian derivative”. J. Math. Phys., 24
(1983), 1405-1413.
[8] Cariello, F. and Tabor, M. “Similarity reductions from extended Painlevé expansions for nonintegrable evolution equations”. Physica D, 53 (1991), 59-70.
[9] Chakravarty, S., Kent, S. L. and Newman, E. T. “Some reductions of the self-dual
Yang-Mills equations to integrable systems in 2+1 dimensions”. J. Math. Phys.,
36 (1995), 763-772.
[10] Maccari, A. “The Kadomtsev¨CPetviashvili equation as a source of integrable
model equations”. J. Math. Phys., 37 (1996), 6207-6212.
28
Yan-ze Peng
[11] Peng, Y. Z. “Exact Periodic Wave Solutions to the Melnikov Equation”. Z. Naturforsch. A, 60 (2005), 321-327.
[12] Peng, Y. Z. “Localized Coherent Structures and their Interactions for the Melnikov
Equation”. Z. Naturforsch. A, 61 (2006), 253-257.
[13] Radha, R. and Lakshmanan, M. “Exotic coherent structures in the (2+1) dimensional long dispersive wave equation”. J. Math. Phys., 38 (1997), 292-297.
[14] Abramowitz, M. and Stegun, I. A. Handbook of Mathematial Functions, New York:
Dover, 1972.
[15] Prasolov, V. and Solovyev, Y. Elliptic Functions and Elliptic Integrals, Providence:
American Mathematical Society, 1997.
[16] Chamdrasekharan, K. Elliptic Functions, Berlin: Springer-Verlag, 1985.
Exact localized structures and their dynamic properties to a (2+1) dimensional . . .
t=-5
0.002
0.001
U
0
-0.001
-0.002
5
0
y
-5
0
-5
x
5
t=0
0.1
0.05
U
0
-0.05
-0.1
5
0
y
-5
0
-5
x
5
t=5
0.1
U
0
5
-0.1
0
-10
y
-5
0
x
-5
5
10
Figure 1: The time evolution graphs of equation (3.2)
29
30
Yan-ze Peng
t=-5
0.002
0.001
U
0
-0.001
-0.002
4
2
0 y
-2
-4
-2
0
2
x
4
-4
t=0
0.1
0.05
U
0
-0.05
-0.1
4
2
0 y
-2
-4
-2
0
2
x
-4
4
t=5
0.1
U
0
4
2
-0.1
0 y
-2
-10
-5
0
x
5
-4
10
Figure 2: The time evolution graphs of equation (3.3)
Exact localized structures and their dynamic properties to a (2+1) dimensional . . .
t=-5
0.001
U
0
4
2
-0.001
0 y
-2
-5
0
x
-4
5
t=0
0.05
U
0
-0.05
4
2
0 y
-2
-5
0
x
-4
5
t=5
0.1
0.05
U
0
-0.05
-0.1
-10
4
2
0 y
-2
-5
0
x
5
-4
10
Figure 3: The time evolution graphs of equation (3.6)
31
32
Yan-ze Peng
1
U
4
0
2
-1
0
-5
0
x
y
-2
5
-4
Figure 4: The structure graph of equation (3.7) with l = 1
Exact localized structures and their dynamic properties to a (2+1) dimensional . . .
t=-5
0.001
U
0
4
2
-0.001
0 y
-2
-5
0
x
5
-4
t=0
0.05
U
0
-0.05
4
2
0 y
-2
-5
0
x
5
-4
t=5
0.1
U
0
4
2
-0.1
0
-10
0
x
y
-2
10
-4
Figure 5: The time evolution graphs of equation (3.7) with l = −1
33
JNM: Journal of Natural Sciences and Mathematics
Vol. 1, pp. 35-40 (December 2007/Thu Al-Hijjah 1428H )
c
°
Qassim University Publications
Eigenvalues of A Minimal Hypersurface in The
Unit Sphere
Sharief Deshmukh1
Department of Mathematics,
College of Science, King Saud University,
P.O. Box 2455, Riyadh 11451, Saudi Arabia.
E-Mail: shariefd@yahoo.com
Abstract: For a compact and connected minimal hypersurface M in the unit sphere
S n+1 (1), it is shown that if the Ricci curvature is bounded below by
λ1
n (n
− 1), then
n
λ1 = n and M isometric to S (1), λ1 is the first nonzero eigenvalue of the Laplacian
operator of M . We also obtain an estimate of the volume of the minimal hypersurface
M of constant scalar curvature in the unit sphere in terms of a constant vector field on
the Euclidean space Rn+2 .
2000 Mathematics Subject Classification: 53C20, 53C40.
Received for JNM on August 10, 2007.
1
Introduction
One of the interesting questions about the compact minimal hypersurfaces in the unit
sphere S n+1 (1) is to show that the first non zero eigenvalue λ1 of it’s Laplacian operator
is λ1 = n (cf. [9]). It is known that λ1 ≤ n and for totally geodesic case λ1 = n
(cf. [5], [7-8]). For embedded compact minimal hypersurfaces in S n+1 (1), Choi and
Wang [3], have shown that λ1 ≥ n2 (see also [1]). However for immersed hypersurfaces
in S n+1 (1) we do not know any such result even for hypersurfaces in S 3 (1). For a
minimal hypersurface M of the unit sphere S n+1 (1) its Ricci curvature tensor satisfies
Ric ≤ n − 1 and that λn1 (n − 1) ≤ (n − 1). A natural quest is to study the minimal
hypersurfaces of S n+1 (1), those satisfy λn1 (n − 1) ≤ Ric ≤ (n − 1). In this paper we
prove the following:
1
This work is supported by the College of Science Research Center at King Saud University.
35
36
Sharief Deshmukh
Theorem 1.1 Let M be a compact and connected orientable minimal hypersurface of
the unit sphere S n+1 (1). If the Ricci curvature of M satisfies Ric ≥ λn1 (n − 1), then
λ1 = n and M is isometric to the unit sphere S n (1).
A result similar to above can be found in [4], where it is proved that if the Ricci
curvature of a compact minimal hypersurface M satisfies Ric ≥ (n−2),
then
³q
´ M is either
¡p
¢
m
n−m
totally geodesic or the Clifford hypersurface S m
× S n−m
for a positive
n
n
integer m < n and n ≥ 4. Other interesting question is to obtain an upper bound for
the volume of a compact minimal hypersurface M in the unit sphere S n+1 (1). Since
S n+1 (1) is a hypersurface of Rn+2 we can treat a minimal hypersurface M of S n+1 (1)
as submanifold of the Eulcidean space Rn+2 . We use a constant vector field V on Rn+2
and the tangential component V T of it’s restriction to M to obtain an upper bound for
the volume of a non-totally geodesic minimal hypersurface of constant scalar curvature
M in S n+1 (1) (For totally geodesic hypersurface its volume is known). Indeed we prove
the following:
Theorem 1.2 Let M be a compact and connected orientable minimal non-totally
geodesic hypersurface of constant scalar curvature in the unit sphere S n+1 (1). Then
for a nonzero constant vector field V on the Euclidean space Rn+2 the volume V ol(M )
of M satisfies
¡ −1
¢Z
° T °2
n (n + 1) kAk2 + (n − 1)(1 − c)
°V ° dV
V ol(M ) ≤
kAk2 kV k2
M
1
where A is the shape operator, c = n−1
inf Ric, is the infimum of the Ricci curvature
of the hypersurface M and V T is the tangential component of V to M .
As a corollary to this result we have
Corollary 1.3 Let M be a compact and connected orientable non-totally geodesic minimal hypersurface of constant scalar curvature in the unit sphere S n+1 (1). Then no
nonzero constant vector field on Rn+2 is normal to M .
2
Priliminaries
Let M be a compact minimal hypersurface of the unit sphere S n+1 (1) with unit normal
vector field N and shape operator A. Let N be the unit normal vector field of the sphere
S n+1 (1) in the Euclidean space Rn+2 . We denote the second fundamental form of M
as submanifold of Rn+2 , by h. Then we have
h(X, Y ) = g (AX, Y ) N − hX, Y i N
(2.1)
Eigenvalues of a minimal hypersurface in the unit sphere
37
X, Y ∈ X(M ), where X(M ) is the Lie-algebra of smooth vector fields on M , g is
the induced metric on M and h, i is the Euclidean metric on Rn+2 . For basic tools
on submanifolds we refer to [2]. Since M is minimal hypersurface of S n+1 (1), using
equation (2.1), we see that the mean curvature vector field H of M in Rn+2 is given
by H = −N . Let V be a nonzero constant vector field on Rn+2 . Restricting V to M
we can express it as V |M = ξ + N ∗ , where ξ ∈ X(M ) and N ∗ ∈ Γ(υ), where Γ(υ) is
the space of smooth sections of the normal bundle υ of M in Rn+2 . Then it is straight
forward to get that
∗
∇X ξ = B(X), ∇⊥
X N = −h(ξ, X), X ∈ X(M )
(2.2)
where ∇ is the Riemannian connection induced on M and ∇⊥ is the connection in
the normal bundle υ, and B = AN ∗ is the Weingarten map of the submanifold M in
Rn+2 with respect to the normal vector field N ∗ . We take a local orthonormal frame
{e1 , .., en } on M to compute the trace of the operator B as
trB =
n
P
hBei , ei i =
i=1
n
P
­
®
hh(ei , ei ), N ∗ i = −n N , N ∗ = −nϕ
(2.3)
i=1
­
®
where ϕ = N , N ∗ is the smooth function defined on M . Since the divergence of
ξ ∈ X(M ) is trB, we see that the Laplacian of the function ϕ is given by ∆ϕ = −nϕ
and its gradient ∇ϕ is given by ∇ϕ = ξ. Also for the smooth function h = hN, N ∗ i
as N ∗ = hN + ϕN , using (2.2) we get ∇h = −Aξ and its Laplacian is given by
∆h = −h kAk2 . Thus we have proved the following
Lemma 2.1 Let M be a compact orientable minimal hypersurface of the unit sphere
S n+1 (1). Then for a constant vector field V on Rn+2 , the functions ϕ, h satisfy
∇ϕ = ξ,
∆ϕ = −nϕ,
∇h = −Aξ,
∆h = −h kAk2
where A is the shape operator of the minimal hypersurface M .
Next, for a smooth function f on M we define an operator Bf : X(M ) → X(M )
by Bf (X) = ∇X ∇f where ∇f is the gradient of the function f . This operator is
symmetric and it is trivial to check that it satisfies
(∇X Bf )(Y ) − (∇Y Bf )(X) = R(X, Y )∇f , T rBf = ∆f
(2.4)
where R is the curvature tensor of the hypersurface M . We denote by Ric the Ricci
tensor of the minimal hypersurface M which is given by
Ric(X, Y ) = (n − 1)g(X, Y ) − g(AX, AY )
(2.5)
The Ricci operator Q of M is defined by g(QX, Y ) = Ric(X, Y ), X, Y ∈ X(M ). Next
we prove the following:
38
Sharief Deshmukh
Lemma 2.2 Let {e1 , .., en } be a local orthonormal frame on the minimal hypersurface
M of the unit sphere S n+1 (1). Then
n
P
i=1
(∇ei Bf ) (ei ) = ∇∆f + Q(∇f ).
Proof. Since Bf is a symmetric operator, using equation (2.4), we get
X(∆f ) =
n
X
Xg(ei , Bf ei ) =
i=1
=
X
X
g ((∇X Bf )(ei ), ei )
i
g ((∇ei Bf )(X) + R(X, ei )∇f, ei )
i
= g ((∇ei Bf )(ei ), X) − Ric(∇f, X)
which proves the Lemma.
Let the smooth function f be the eigenfunction of the Laplacian operator ∆ corresponding to eigenvalue λ, that is ∆f = −λf . We compute the divergence of the vector
field Bf (∇f ) as
¶
µ
n
n
P
P
2
ei g (∇f, Bf ei ) = kBf k + g ∇f, (∇eI Bf ) (ei )
divBf (∇f ) =
i=1
i=1
2
2
= kBf k − λ k∇f k + Ric(∇f, ∇f )
where we used the equation (2.4) and Lemma 2.2. Thus integrating this equation we
conclude the following:
Lemma 2.3 Let M be a compact orientable minimal hypersurface of the unit sphere
S n+1 (1). Then for the smooth function f satisfying ∆f = −λf , we have
Z
©
ª
Ric(∇f, ∇f ) + kBf k2 − λ k∇f k2 dV = 0
M
3
Proofs of the theorems
In this section we establish the proofs of our main results.
3.1
Proof of Theorem 1.1
Let f be a non constant smooth function on M with ∆f = −λ1 f , where λ1 is the first
nonzero eigenvalue of the Laplacian operator ∆. Then it follows that
R
R
k∇f k2 dV = λ1 f 2 dV
(3.1)
M
M
39
Eigenvalues of a minimal hypersurface in the unit sphere
From Lemma 2.3 we get
R ¡
kBf k2 −
+
λ1
n
R ¡ M
Ric(∇f, ∇f ) −
M
¢
k∇f k2 dV
λ1
(n
n
¢
− 1) k∇f k2 dV = 0
(3.2)
2
Using Schwartz inequality we get kBf k2 ≥ n1 (T rBf )2 = λn1 f 2 , with equality holding
if and only if Bf = − λn1 f I. Thus in view of equation (3.1) and the condition in the
theorem we conclude
¶
Z µ
λ1
2
2
k∇f k dV ≤ 0
kBf k −
n
M
Using Schwartz inequality we get Bf = − λn1 f I, that is
∇X ∇f = −
λ1
f X, X ∈ X(M )
n
which q
is Obata’s differential equation
for non constant f (cf. [6]). Thus M is isometric
q
n
n
n
to S ( λ1 ) which gives λ1 = n λ1 , that is λ1 = n and M is isometric to S n (1).
3.2
Proof of Theorem 1.2
Let V be a non zero constant vector field on Rn+2 . Note that the restriction of V to
M can be expressed as
V = ξ + hN + ϕN
where ξ = V T is the tangential component of V to M . We have
kV k2 = kξk2 + h2 + ϕ2
(3.3)
Now using the facts ∆ϕ = −nϕ and ∆h = − kAk h and Lemma 2.1, we find 12 ∆ϕ2 =
−nϕ2 + kξk2 and 12 ∆h2 = − kAk2 h + kAξk2 and integrating these equations we arrive
at
Z
Z
Z
Z
2
2
2
2
kξk dV = n ϕ dV ,
kAξk dV = kAk
h2 dV
(3.4)
M
M
M
M
2
where kAk is a constant as the hypersurface M is minimal and has constant scalar
curvature. Now integrating equation (3.3) and using the equation in (3.4) we get
Z
Z
2
2
kV k V ol(M ) = (n + 1) ϕ dV + h2 dV
(3.5)
M
M
Multiplying equation (3.2) by kAk2 and using equation (3.4) and the expression for the
Ricci curvature given in equation (2.5) we get
Z
Z
¡
¢
2
2
2
2
kAk kV k V ol(M ) = (n + 1) kAk
ϕ dV +
(n − 1) kξk2 − Ric(ξ, ξ) dV
M
M
40
Sharief Deshmukh
1
inf Ric for the compact hypersurface M . That gives Ric(ξ, ξ) ≥ c(n −
Let c = n−1
2
1) kξk that is (n − 1) kξk2 − Ric(ξ, ξ) ≤ (n − 1)(1 − c) kξk2 . Inserting this in above
equation we get
Z
¡ −1
¢
2
2
2
kAk kV k V ol(M ) ≤ n (n + 1) kAk + (n − 1)(1 − c)
kξk2 dV
M
where we used Lemma 2.1. Since M is non-totally geodesic we get
¡ −1
¢Z
° T °2
n (n + 1) kAk2 + (n − 1)(1 − c)
°V ° dV
V ol(M ) ≤
kAk2 kV k2
M
which proves the result.
References
[1] Barros, A. and Bessa, G. P. Estimates of the first eigenvalue of minimal hypersurfaces of S n+1 , 10th School on Differential Geometry Portuguese (Belo Horizonte,
1998), Mat. Contemp., 17 (1999), 71–75.
[2] Chen, B. Y. Total Mean Curvature and Submanifolds of Finite type, Singapore:
World Scientific Publishing Co., 1984.
[3] Choi, H. I. and Wang, A. N. “A first eigenvalue estimate for minimal hypersurfaces”. J. Diff. Geom., 18 (1983), 559–562.
[4] Ejiri, N. “Compact minimal submanifolds of a sphere with positive Ricci curvature”. J. Math. Soc. Japan, 31, No. 2 (1979), 251–256.
[5] Gardner, R. B. “New viewpoints in the geometry of submanifolds of RN ”. Bull.
Amer. Math. Soc., 83, No. 1 (1977), 1–35.
[6] Obata, M. “Certain conditions for a Riemannian manifold to be isometric with a
sphere”. J. Math. Soc. Japan, 14 (1962), 333–340.
[7] Pigola, S., Rigoli, M. and Setti, A. G. “Some applications of integral formulas in
Riemannian geometry and PDE’s”. Milan J. Math., 71 (2003), 219–281.
[8] Reilly, R. “On the first eigenvalue of the Laplacian for compact submanifolds of
Euclidean space”. Comment. Math. Helv., 52 (1977), 525–533.
[9] Yau, S. T. Seminar on Differential Geometry, Princeton University Press (1982),
669–706.
‫جملة العلوم الطبيعية والرياضيات‪ ،‬جامعة القصيم‪ ،‬املجلد ‪ (1‬ذو احلجة ‪1428‬هـ ‪ /‬ديسمرب ‪2007‬م )‪.‬‬
‫كلمة رئيس هيئة التحرير‬
‫احلمد هلل والصالة والسالم عىل رسول اهلل نبينا حممد وعىل آله وصحبه أمجعني‪ ،‬وبعد‪:‬‬
‫نحمده سبحانه او ً‬
‫ال وآخر ًا عىل أن مكننا من تقديم العدد األول من جملة العلوم الطبيعية والرياضيات والتي تعد‬
‫أحد فروع املجلة العلمية جلامعة القصيم‪ .‬تعنى هذه املجلة بتحكيم ونرش األبحاث القيمة بجميع فروع الرياضيات‪،‬‬
‫اإلحصاء‪ ،‬بحوث العمليات‪ ،‬الفيزياء‪ ،‬الكيمياء وعلوم األحياء‪.‬‬
‫إن أملنا من ذلك هو إشاعة ما نطلق عليه ثقافة البحث العلمي وتوطينها يف املجتمع ألهنا قد أصبحت أمر ًا ملح ًا‬
‫وخصوص ًا مع هذا التقدم امللحوظ يف العلم‪ .‬وهذا لن يتأتى إال بجعل االهتامم بالعلوم الرياضية والطبيعية التي تعد‬
‫مصدر ًا للعديد من العلوم األخرى من ضمن أهم األولويات‪ .‬وكمحاولة لتحقيق جزء من هذا اهلدف السامي سنركز‬
‫عملنا باملجلة عىل حماولة استقطاب ونرش األبحاث األصيلة يف تلك املجاالت العلمية‪ .‬كام سنحاول من خالل إصدار‬
‫عددين للمجلة سنوي ًا الرقي بمستواها وإيصاهلا إن شاء اهلل إىل مصاف املجالت العلمية املرموقة‪ .‬ولذا فإننا كهيئة‬
‫حترير ال نستغني عن أي مالحظات أو اقرتاحات قد ترد من قبل القارئ الكريم‪.‬‬
‫إن قيامنا هبذا الدور سيمكننا بحول اهلل من خلق مسامهة للجامعات السعودية يف إثراء املعرفة‪ ،‬وتفعيل التواصل‬
‫العلمي مع باحثني متميزين من أقطار شتى من العامل‪.‬‬
‫وإنني هبذه املناسبة أتقدم بالشكر اجلزيل لكل من ساهم يف إنجاز هذا العمل‪ ،‬وأخص بالذكر زمالئي أعضاء‬
‫هيئة التحرير‪ .‬كام أثمن كل النصائح التي وردت من قبل أعضاء هيئة التحرير املشاركني‪.‬‬
‫الدكتور‪ /‬عبدالرمحن بن سليامن احلسني‬
‫‬
‫جملة العلوم الطبيعية والرياضيات‪ ،‬جامعة القصيم‪ ،‬املجلد ‪ (1‬ذو احلجة ‪1428‬هـ ‪ /‬ديسمرب ‪2007‬م )‪.‬‬
‫كلمة وكيل الجامعة للدراسات العليا والبحث العلمي‬
‫احلمد هلل والصالة والسالم عىل رسول اهلل نبينا حممد وعىل آله وصحبه أمجعني‪ ،‬وبعد‪:‬‬
‫بتوفيق من اهلل عز وجل ثم بدعم من معايل مدير اجلامعة ( حفظه اهلل ) نفتخر ونعتز بافتتاح باكورة اإلنتاج العلمي‬
‫للعدد األول من جملة اجلامعة العلمية املحكمة (فرع العلوم الطبيعية والرياضيات)‪ .‬وأمتنى أن يساهم هذا اإلصدار‬
‫يف أثراء املكتبة البحثية‪ ،‬وأن يكون رافد ًا من روافد العلم واملعرفة يستفيد منه املختصون يف جمال العلوم الطبيعية‬
‫والرياضيات والذين أدعوهم إىل املشاركة يف نرش أبحاثهم العلمية يف إعداد املجلة القادمة‪.‬‬
‫أقدم شكري وتقديري لرئيس حترير املجلة وأعضائها وأيض ًا الزمالء بإدارة النرش العلمي والرتمجة باجلامعة عىل‬
‫ما بذلوه من جهود إلخراج هذا العدد املرشف‪.‬‬
‫سدد اهلل اخلطى وبارك يف اجلهود ‪،،،‬‬
‫األستاذ الدكتور‪ /‬صالح بن عبد اهلل الدامغ‬
‫‬
‫جملة العلوم الطبيعية والرياضيات‪ ،‬جامعة القصيم‪ ،‬املجلد ‪ (1‬ذو احلجة ‪1428‬هـ ‪ /‬ديسمرب ‪2007‬م )‪.‬‬
‫كلمة معالي مدير الجامعة‬
‫رب العاملني‪ ،‬والصالة والسالم عىل النبي األمني‪ ،‬عليه أفضل الصلوات والتسليم وعىل آله وأصحابه‬
‫احلمداهلل ّ‬
‫أمجعني‪ ،‬وبعد‪:‬‬
‫فإن البحث العلمي يعدّ منارة للدول املتقدمة‪ ،‬ومعل ًام حقيقي ًا حلياهتا‪ ،‬ونور ًا هتتدي به لكثري من مشكالهتا‪ ،‬وعالمة‬
‫وتفردها‪ ،‬بكوهنا املحاضن األوىل التي ينبثق منها شعاع العلم واإلنتاج البحثي‪.‬‬
‫لرقي اجلامعات ّ‬
‫وإن اإلسهام يف جماالت البحث ُيعد هنضة حضارية يقدمها الباحث لوطنه أو ً‬
‫ال‪ ،‬ثم للعامل بأرسه ثاني ًا‪ ،‬ويضيف‬
‫للعلم واملعرفه رؤى جديدة‪ ،‬أو أفكار متممة أو مطورة ملرشوعات علمية قائمة‪.‬‬
‫وإن جامعة القصيم من خالل هذه املجلة املتخصصة متنح الباحثني جما ً‬
‫ال واسع ًا لنرش إبداعاهتم العلمية إليامهنا‬
‫العميق بقيمة العلوم التجريبية بنظرياهتا وحتليالهتا واستنتاجاهتا وقيمتها يف دفع عجلة التقدم العلمي وإثرائه وإنعكاسه‬
‫عىل احلياة املعارصه التي تشهد تنافس ًا علمي ًا حمتدم ًا‪.‬‬
‫وتأمل اجلامعة من خالل هذه النافذة البحثيه أن تساهم يف التطور العلمي والعناية به يف هذا الوطن املعطاء‪ ،‬وأن‬
‫حتقق بعض ًا من أهداف التعليم العايل التي رسمته سياسة التعليم يف اململكة‪ ،‬وتقدم للباحثني أ ّيا كان مقامهم وعاء‬
‫يعنى بنرش ماتوصلت إليه عقوهلم يف جمال البحوث الطبيعية‪.‬‬
‫وإنني مع إطاللة العدد األول من هذه املجلة العلمية (جملة العلوم الطبيعية والرياضيات) أدعو الباحثني إلثراء‬
‫أعدادها ببحوثهم ومناقشاهتم وتعقيباهتم العلمية لنسهم يف ٍ‬
‫بناء علمي ومعريف يعكس تطورنا احلضاري‪ ،‬داعيا املوىل‬
‫أن يكلل اجلهود بالنجاح والتقدّ م‪.‬‬
‫األستاذ الدكتور‪ /‬خالد بن عبدالرمحن احلمودي‬
‫هـ‬
‫هيئة التحرير‬
‫عبد الرمحن بن سليامن احلسني (قسم الرياضيات‪ ،‬جامعة القصيم) (رئيس هيئة التحرير)‬
‫حمسن أبو مندور‬
‫‬
‫أمحد الصاوي‬
‫‬
‫‬
‫أمحد اجلراحيي‬
‫جمدي خليفه‬
‫هيئة التحرير املشاركني‬
‫فو آنه ( اسرتاليا )‬
‫‬
‫توماس بيورك ( السويد )‬
‫راجيف شيكيت ( اهلند )‬
‫رشيف دشموخ (السعودية )‬
‫‬
‫قريد كوب ( أملانيا )‬
‫ستيفن كييل ( اململكة املتحدة )‬
‫ولفقانق لنرت ( النمسا )‬
‫هول إدواردز ( اململكة املتحدة )‬
‫يان‪-‬زي بنق ( الصني )‬
‫ديفيد إيلوارثي ( اململكة املتحدة )‬
‫جورج رترش‪-‬أدو ( أمريكا )‬
‫منق جيو فنق ( هونق كونق )‬
‫جورج واتسون ( اسكتلندا )‬
‫‬
‫نيكول القروي ( فرنسا )‬
‫يوتاكا فوكودا ( اليابان )‬
‫‬
‫حتسني غزال ( السعوديه )‬
‫أمحد محزة ( مرص )‬
‫‬
‫‬
‫حممد إقبال ( اهلند )‬
‫جاهيوان هي ( الصني )‬
‫‬
‫‬
‫‬
‫رقم اإليداع ‪1429/2025‬هـ‬
‫رشدي راشد ( فرنسا )‬
‫مصطفى سويالك ( تركيا )‬
‫عبد املجيد وزوز ( أمريكا )‬
‫نورمان وو ( الصني )‬
‫ﳎﻠﺔ ﺍﻟﻌﻠﻮﻡ ﺍﻟﻄﺒﻴﻌﻴﺔ ﻭﺍﻟﺮﻳﺎﺿﻴﺎﺕ‪ ،‬ﺟﺎﻣﻌﺔ ﺍﻟﻘﺼﻴﻢ‪ ،‬ﺍﳌﺠﻠﺪ ‪ )١‬ﺫﻭ ﺍﳊﺠﺔ ‪١٤٢٨‬ﻫـ ‪ /‬ﺩﻳﺴﻤﱪ ‪٢٠٠٧‬ﻡ (‪.‬‬
‫‪‬‬
‫‪‬‬
‫ﺍﳌﺠﻠﺪ ﺍﻷﻭﻝ‬
‫‪‬‬
‫) ﺫﻭ ﺍﳊﺠﺔ ‪١٤٢٨‬ﻫـ(‬
‫)ﺩﻳﺴﻤﱪ‪٢٠٠٧‬ﻡ(‬
‫ﺍﳌﺠﻠﺔ ﺍﻟﻌﻠﻤﻴﺔ ﳉﺎﻣﻌﺔ ﺍﻟﻘﺼﻴﻢ‬
‫)ﳎﻠﺔ ﳏﻜﻤﺔ (‬