Exact solutions of the nonlinear loaded Benjamin-Ono
equation
1BAZAR BABAJANOV, 2FAKHRIDDIN ABDIKARIMOV
1Department of Applied Mathematics and Mathematical Physics,
Urgench State University, Urgench, 220100,
UZBEKISTAN
2Khorezm Mamun Academy, Khiva, 220900,
UZBEKISTAN
Abstract: In this paper, we investigate the non-linear loaded two-dimensional Benjamin-Ono equation
by the functional variable method. The advantage of this method is reliability and efficiency. Using this
method we obtained exact solitary and periodic wave solutions. The solving procedure is very simple and
the traveling wave solutions of this equation are demonstrated by hyperbolic and trigonometric functions.
The graphical representations of some obtained solutions are demonstrated to better understand their
physical features.
Key-Words: non-linear loaded Benjamin-Ono equation, solitary wave solutions, functional variable
method, periodic wave solutions, trigonometric function, hyperbolic function.
Received: August 15, 2021. Revised: July 14, 2022. Accepted: August 16, 2022. Published: September 20, 2022.
1 Introduction
Non-linear partial differential equations are im-
portant equations used in modeling many phe-
nomena in science and engineering applications.
One of the most important nonlinear evolution
equation is the Benjamin-Ono(BO) equation. In
1967, the general theoretical treatment of a new
class of long stationary waves with finite ampli-
tude was studied by Benjamin and Ono developed
Benjamin’s theory in which was taken a class of
non-linear evolution equations. The BO equation
describes internal waves between two stratified
homogenous fluids with different densities, where
one of the layers is infinitely deep[1, 2, 3, 4]. This
equation is expressed in the following basic form
utt +αu2xx +βuxxxx +γuyyyy = 0.(1)
where α,βand γare non zero constants.
The literature is rich in different studies to
find special solutions of the nonlinear BO equa-
tion, such as the existence of multi-soliton solu-
tion by several authors[1, 5], existence of peri-
odic solutions by Ambrose and Wilkening[6], and
certain discrete solutions by Tutiya[7]. We can
indicate several methods that can be used to ob-
tain exact special solutions to the non-linear BO
equation, such as, numerical method[8, 9], Hirota
bilinear method[10], extended truncated expan-
sion method[11], generalization of the homoclinic
breather method[12], tanh expansion method[13],
constructive method[14] and others[15, 16, 17, 18,
19, 20, 21] are used to derive accurate solutions.
In this paper, we consider the following the
non-linear loaded two-dimensional BO equation
utt −αu2xx −βuxxxx −γuyyyy+
+φ(t)u(0,0, t)uxx = 0,(2)
where u(x, y, t) is unknown function, x∈R,y∈
R,t≥0, α,βand γare any constants, and φ(t)
are the given real continuous functions.
We investigate the non-linear loaded two-
dimensional BO equation by the functional
variable method(FVM). The advantage of this
method is reliability and efficiency. Using this
method we obtained exact solitary and periodic
wave solutions. The solving procedure is very
simple and the traveling wave solutions of this
equation are demonstrated by hyperbolic and
trigonometric functions. The graphical represen-
tations of some obtained solutions are demon-
strated to better understand their physical fea-
tures.
In recent years, due to the intensive study
of the problems of optimal management of the
agroecosystem for instance, the problem of long-
term forecasting and regulation of groundwater
levels and soil moisture interest in loaded equa-
tions has increased significantly. Among the
works devoted to loaded equations, one should
especially note the works of A. Kneser[22], L.
Lichtenstein[23], A. M. Nakhushev[24], and oth-
ers. A complete explanation of solutions of the
nonlinear loaded PDEs and their uses can be
found in papers[25, 26, 27, 28, 29, 30].
The article is organized as follows. In Sec-
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.78
Bazar Babajanov, Fakhriddin Abdikarimov
E-ISSN: 2224-2880
666
Volume 21, 2022
tion 2, we present some basic information about
the description of the functional variable method.
Section 3 is devoted to solutions of the non-linear
loaded two-dimensional BO equation. In Section
4, we present the graphical representation of the
non-linear loaded two-dimensional BO equation.
Finally, conclusions are presented in Section 5.
2 Explanation of the functional
variable method
We consider NPDE of the form
P(u, ux, uy, ut, uxx, utt, uyy, uxy, ...) = 0,(3)
where Pis a polynomial in u=u(x, y, t) and its
partial derivatives.
Step 1. It is used the following transforma-
tion for a travelling wave solution of eq.3:
u(x, y, t) = u(ξ),(4)
with
∂u
∂x =pdu
dξ ,∂u
∂y =qdu
dξ ,∂u
∂t =−kdu
dξ , ..., (5)
where
ξ=px +qy −kt, p =const, q =const (6)
and kis the speed of the traveling wave.
Substituting eq.4 and eq.5 into NPDE eq.3 we
get the following ODE of the form
F(u, u′, u′′, u′′′ , ...) = 0, u′=du
dξ (7)
here Fis a polynomial in u(ξ), u′(ξ), u′′(ξ),
u′′′(ξ), ....
Step 2. Let
u′=F(u).(8)
It follows that
Zdu
F(u)=ξ+ξ0,(9)
we suppose ξ0= 0 for convenience. Now we can
calculate higher order derivatives of u:
u′′ =1
2
d(F2(u))
du
u′′′ =1
2
d2(F2(u))
du2pF2(u)
u′′′′ =1
2hd3(F2(u))
du3F2(u) + d2(F2(u))
du2
d(F2(u))
du i
(10)
Step 3. Putting eq.10 into eq.7, we obtain
H(u, dF (u)
du ,d2F(u)
du2,d3F(u)
du3, ...)=0.(11)
The key idea of this particular form eq.11 is of
special interest because it admits analytical so-
lutions for a large class of nonlinear wave type
equations. After integration, eq.11 provides the
expression of Fand this, together with eq.8, give
solutions to the original problem.
3 Solutions of the non-linear
loaded two-dimensional
Benjamin-Ono equation
We will find the exact solution of the non-linear
loaded two-dimensional BO equation by the func-
tional variable method. For doing this, in eq.2,
let use the following transformation.
u(x, y, t) = u(ξ), ξ =px +qy −kt. (12)
where p=const, q =const and kis the speed of
the traveling wave.
It is easy to show that after transformation
(12), the nonlinear partial differential eq.2 can be
transformed into an ordinary differential equation
of the form
u′′ =k2+φ(t)u(0,0, t)p2
βp4+γq4u−αp2
βp4+γq4u2.(13)
According to (10), eq.13 can be written as follows
1
2
d(F2(u))
du =k2+φ(t)u(0,0,t)p2
βp4+γq4u−
−αp2
βp4+γq4u2.(14)
Integrating eq.14 and after simple simplification,
we get
F(u) = uqµ(t)−ηu. (15)
where µ(t) = k2+φ(t)u(0,0,t)p2
βp4+γq4,η=2αp2
3(βp4+γq4).
From eq.8 and eq.15 we deduce that
du
upµ(t)−ηu =dξ. (16)
After integrating eq.16, we have
u(x, y, t) = −6(k2+φ(t)u(0,0,t)p2)
αp2×
×
eqk2+φ(t)u(0,0,t)p2
βp4+γq4(px+qy−kt)
1−eqk2+φ(t)u(0,0,t)p2
βp4+γq4(px+qy−kt)
2
.(17)
The function u(0,0, t) can be easily obtained
based on expression eq.17.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.78
Bazar Babajanov, Fakhriddin Abdikarimov
E-ISSN: 2224-2880
667
Volume 21, 2022
We get two types of solutions of the non-linear
loaded two-dimensional BO equation eq.2 as fol-
lows:
1) When k2+φ(t)u(0,0,t)p2
βp4+γq4>0, we get the soli-
tary solution
u(x, y, t) = −6(k2+φ(t)u(0,0,t)p2)
αp2×
×cth2qµ(t)px +qy −kt
2−1.(18)
where µ(t) = k2+φ(t)u(0,0,t)p2
βp4+γq4.
2) When k2+φ(t)u(0,0,t)p2
βp4+γq4<0, we get the peri-
odic solution
u(x, y, t) = 6(k2+φ(t)u(0,0,t)p2)
αp2×
×ctg2qµ(t)px +qy −kt
2+ 1.(19)
where µ(t) = k2+φ(t)u(0,0,t)p2
βp4+γq4.
The graphs of solutions of the non-linear
loaded two-dimensional BO equation by using
distinct values of random parameter will be
demonstrated.
If k=−1, α=−6, β= 1, γ= 1, φ(t) = t,
p= 1, q= 1 then we have
u(x, y, t) = (1 + tu(0,0, t)) ×
×
cth2
s1 + tu(0,0, t)
2
x+y+t
2
−1
.
(20)
If k=−1, α= 6, β=−1, γ=−1, φ(t) =
t2,p= 1, q= 1, then we have
u(x, y, t) = 1 + t2u(0,0, t)×
×
ctg2
s(1 + t2u(0,0, t))
2
x+y+t
2
+ 1
.
(21)
4 Graphical representation of
the non-linear loaded
two-dimensional
Benjamin-Ono equation
After visualizing the graphs of the soliton solu-
tions (Figure 1) and the periodic wave solutions
(Figure 2), the use of distinct values of random
parameters is demonstrated to better understand
their physical features. It is known that the pa-
rameters included in the solutions have a deep
connection with the amplitudes and velocities. A
soliton or solitary wave in the concept of mathe-
matical physics defined as a self-reinforcing wave
package that retains its shape. It propagates at a
constant amplitude and velocity. Solitons are so-
lutions of a common class of nonlinearly partially
differential equations with weak linearity describ-
ing physical systems. The existence of periodic
travelling waves usually depends on the parame-
ter values in a mathematical equation. If there is
a periodic travelling wave solution, then there is
typically a family of such solutions, with different
wave speeds. In this regard, we can explore some
of the non-linear phenomena that take place in
physics, applied mathematics and technology.
Figure 1: Solitary wave solution of the non-linear
loaded two-dimensional BO equation for k=−1,
α=−6, β= 1, γ= 1, φ(t) = t,p= 1, q= 1.
Figure 2: Periodic wave solution of the non-linear
loaded two-dimensional BO equation for k=−1,
α= 6, β=−1, γ=−1, φ(t) = t2,p= 1,
q= 1.
5 Conclusion
The functional variable method has been success-
fully used to obtain the soliton solutions and the
periodic solutions of the non-linear loaded two-
dimensional BO equation. We have shown that
this method can provide a useful way to efficiently
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.78
Bazar Babajanov, Fakhriddin Abdikarimov
E-ISSN: 2224-2880
668
Volume 21, 2022
find the exact structures of solutions to a variety
of non-linear wave equations. After visualizing
the graphs of the soliton solutions and the peri-
odic wave solutions, the use of distinct values of
random parameters is demonstrated to better un-
derstand their physical features. It is known that
the parameters included in the solutions have a
deep connection with the amplitudes and veloci-
ties. In this regard, we can explore some of the
non-linear phenomena that take place in physics,
applied mathematics and technology. We con-
clude that when revealing the internal mecha-
nisms of physical phenomena, it will be necessary
to find an exact solution to the problem.
References:
[1] Benjamin T.B, Internal waves of permanent
form in fluids of great depth, Journal of Fluid
Mechanics, Vol.29, No.3, 1967, pp. 559-592.
https://doi.org/10.1017/S002211206700103X.
[2] Ono H, Algebraic solitary waves in strati-
fied fluids, Journal of the Physical Society
of Japan, Vol.39, No.4, 1975, pp. 1082-1091.
https://doi.org/10.1143/JPSJ.39.1082.
[3] Leblond H., Mihalache D, Models of few
optical cycle solitons beyond the slowly
varying envelope approximation, Physics
Reports, Vol.523, No.2, 2013, pp. 61-126.
https://doi.org/10.1016/j.physrep.2012.10.006.
[4] Najafi M, Multiple soliton solutions of the
second order Benjamin–Ono equation, Tur-
kic World Mathematical Society Journal of
Applied and Engineering Mathematics, Vol.2,
No.1, 2012, pp. 94-100.
[5] Matsuno Y, Exact multi-soliton solution
of the Benjamin-Ono equation, Journal of
Physics A: Mathematical and General, Vol.12,
No.4, 1979, p. 619.
[6] Ablowitz M.J., Demirci A., Yi-Ping M, Dis-
persive shock waves in the KadomtsevPetvi-
ashvili and two dimensional Benjamin-
Ono equations, Physica. D: Nonlinear
Phenomena, Vol.333, 2016, pp. 84-98.
https://doi.org/10.1016/j.physd.2016.01.013.
[7] Tutiya Y., Shiraishi J, On some spe-
cial solutions to periodic Benjamin-
Ono equation with discrete Laplacian,
Mathematics and Computers in Simu-
lation, Vol.82, No.7, 2012, pp. 1341-1347.
https://doi.org/10.1016/j.matcom.2010.05.006.
[8] Ambrose D.M., Wilkening J.J, Compu-
tation of Time-Periodic Solutions of the
Benjamin-Ono Equation, Journal of Non-
linear Science, Vol.20, 2010, pp. 277-308.
https://doi.org/10.1007/s00332-009-9058-x.
[9] Roudenko S., Wang Zh., Yang K, Dynamics
of solutions in the generalized Benjamin-Ono
equation: A numerical study, Journal of Com-
putational Physics, Vol.445, 2021, pp. 1-25,
https://doi.org/10.1016/j.jcp.2021.110570.
[10] Liu Y.K., Li B, Dynamics of rogue
waves on multisoliton background in the
Benjamin-Ono equation, Pramana-Journal
of Physics, Vol.88, No.57, 2017, pp. 1-6.
https://doi.org/10.1007/s12043-016-1361-0.
[11] Mostafa M.A.Khater, Shabbir Muham-
mad, A.Al-Ghamdi, M.Higazy, Abun-
dant wave structures of the fractional
Benjamin-Ono equation through two
computational techniques, Journal of
Ocean Engineering and Science, 2022,
https://doi.org/10.1016/j.joes.2022.01.009.
[12] Singh S., Sakkaravarthi K., Murugesan K.,
Sakthivel R, Benjamin-Ono equation: Rogue
waves, generalized breathers, soliton bend-
ing, fission, and fusion, The European Physi-
cal Journal Plus, Vol.35, No.823, 2020, pp. 1-
17. https://doi.org/10.1140/epjp/s13360-020-
00808-8.
[13] Khalid Karam Ali, Rahmatullah Ibrahim
Nuruddeen, Ahmet Yildirim, On the new
extensions to the Benjamin-Ono equation,
Computational Methods for Differential
Equation, Vol.8, No.3, 2020, pp. 424-
445.https://doi.org/10.22034/cmde.2020.32382.1505.
[14] Peter D.Miller, Alfredo N.Wetzel,
The scattering transform for the Ben-
jamin–Ono equation in the small-
dispersion limit, Physica D: Nonlinear
Phenomena, Vol.33, 2016, pp. 185-199,
https://doi.org/10.1016/j.physd.2015.07.012.
[15] Angulo J., Scialom M., Banquet C,
The regularized Benjamin-Ono and BBM
equations: Well-posedness and nonlinear
stability, Journal of Differential Equa-
tions, Vol.50, No.1, 2011, pp. 4011-4036.
https://doi.org/10.1016/j.jde.2010.12.016.
[16] Abdelrahman M.A.E., Zahran E.H.M.,
Khater M.M.A, The exp(φ(ξ))-expansion
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.78
Bazar Babajanov, Fakhriddin Abdikarimov
E-ISSN: 2224-2880
669
Volume 21, 2022
method and its application for solving nonlin-
ear evolution equations, International Jour-
nal of Modern Nonlinear Theory and Ap-
plication, Vol.4, No.1, 2015, pp. 37-47.
https://doi.org/10.4236/ijmnta.2015.41004.
[17] Babajanov B., Abdikarimov F, The Appli-
cation of the Functional Variable Method
for Solving the Loaded Non-linear Evalua-
tion Equations, Frontiers in Applied Math-
ematics and Statistics, 2022, 8:912674.
https://doi.org/10.3389/fams.2022.912674.
[18] Babajanov B., Abdikarimov F, Soli-
ton Solutions of the Loaded Modified
Calogero-Degasperis Equation, Inter-
national Journal of Applied Mathemat-
ics, Vol.35, No.3, 2022, pp. 381-392.
http://dx.doi.org/10.12732/ijam.v35i3.2.
[19] Babajanov B., Abdikarimov F, Expansion
Method for the Loaded Modified Zakharov-
Kuznetsov Equation, Advanced Mathematical
Models and Applications, Vol.7, No.2, 2022,
pp. 168-177.
[20] Babajanov B., Abdikarimov F, Solitary and
periodic wave solutions of the loaded modi-
fied Benjamin-Bona-Mahony equation via the
functional variable method, Researches in
Mathematics, Vol.30, No.1, 2022, pp. 10-20.
https://doi.org/10.15421/242202.
[21] Yakhshimuratov, A.B., Babajanov, B.A. In-
tegration of equations of Kaup system kind
with self-consistent source in class of periodic
functions, Ufa Mathematical Journal, Vol.12,
No.1, 2020, pp. 103–113.
[22] Kneser A, Belastete integralgleichun-
gen, Rendiconti del Circolo Matematico di
Palermo, Vol.37, No.1, 1914, pp. 169-197.
https://doi.org/10.1007/BF03014816.
[23] Lichtenstein L, Vorlesungen ¨uber einige
Klassen nichtlinearer Integralgleichungen und
Integro-Differential-Gleichungen nebst An-
wendungen, Berlin, Springer-Verlag, 1931,
https://doi.org/10.1007/978-3-642-47600-6.
[24] Nakhushev A.M, Loaded equations and their
applications, Differential Equations, Vol.9,
No.1, 1983, pp. 86-94.
[25] Nakhushev A.M, Boundary value problems
for loaded integro-differential equations of hy-
perbolic type and some of their applications to
the prediction of ground moisture, Differential
Equations, Vol.15, No.1, 1979, pp. 96-105.
[26] Baltaeva U.I, On some boundary value
problems for a third order loaded integro-
differential equation with real parame-
ters, The Bulletin of Udmurt Univer-
sity. Mathematics. Mechanics. Computer
Science, Vol.3, No.3, 2012, pp. 3-12.
https://doi.org/10.20537/vm120301.
[27] Kozhanov A.I, A nonlinear loaded parabolic
equation and a related inverse problem, Math-
ematical Notes, Vol.76, No.5, 2004, pp. 784-
795. https://doi.org/10.4213/mzm156.
[28] Khasanov A.B., Hoitmetov U.A, On in-
tegration of the loaded mKdV equation in
the class of rapidly decreasing functions,
The bulletin of Irkutsk State Univer-
sity. Series Mathematics, Vol.38, 2021,
pp. 19-35. https://doi.org/10.26516/1997-
7670.2021.38.19.
[29] Urazboev G.U., Baltaeva I.I., Rakhi-
mov I.D, Generalized G’/G - extension
method for loaded Korteweg-de Vries
equation, Siberian Journal of Industrial
Mathematics, Vol.24, No.4, 2021, pp. 72-76.
https://doi.org/10.33048/SIBJIM.2021.24.410.
[30] Yakhshimuratov, A.B., Matyokubov,
M.M, Integration of a loaded Ko-
rtewegde Vries equation in a class of
periodic functions, Russian Mathe-
matics, Vol.60, No.2, 2016, pp. 72-76.
https://doi.org/10.3103/S1066369X16020110.
Contribution of individual
authors to the creation of a
scientific article (ghostwriting
policy)
Bazar Babajanov and Fakhriddin Abdikarimov
conceived of the presented idea. Bazar Baba-
janov developed the theory and performed the
computations. Fakhriddin Abdikarimov veri-
fied the analytical methods. Both authors dis-
cussed the results and contributed to the final
manuscript. Both authors contributed to the ar-
ticle and approved the submitted version.
Follow: www.wseas.org/multimedia/contributor-
role-instruction.pdf
Creative Commons Attribution
License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.78
Bazar Babajanov, Fakhriddin Abdikarimov
E-ISSN: 2224-2880
670
Volume 21, 2022
