This article discusses about solutions of Benjamin-Bona-Mahony (BBM) equation and Modified Regularized Long Wave (MRLW) equation. BBM equation is a model describing the propagation of long wave with small amplitude on one directional space. This equation was developed to resolve the shortcoming of classic Korteweg-de-Vries (KdV) equation which fails to model the wave when the wavenumbers value is high. Meanwhile, MRLW equation represents the dispersed wave phenomenon such as shallow water and phonon packet on nonlinear crystal. The solutions of these equations are known as a solitary wave (soliton). This solution can be determined by various methods. Here, we apply the sine-cosine function method and analyze in detail the resulting solitary waves.

1.
M. D.
Groves
,
Study of Water Waves
,
J. Nonlinear Math. Phys.
,
2004
, Vol.
11
,
435
460
.
2.
Marwan
,
On The Maximal Temporal Amplitude of Down Stream Running Nonlinear Water Waves
,
Tamkang J. Math.
,
2010
, Vol.
41
(
1
):
51
69
.
3.
M.
Ramli
,
S.
Munzir
,
T.
Khairuman
, and
V.
Halfiani
,
Amplitude Increasing Formula of Bichromatic Wave Propagation Based on Fifth Order Side Band Solution of Korteweg de Vries Equation
,
Far East J. Math Sci.
,
2014
, Vol.
90
(
1
):
97
117
.
4.
M.
Ramli
,
Nonlinear Evolution of Wave Group With Three Frequencies
,
Far East J. Math Sci.
,
2015
, Vol.
97
(
8
):
925
937
.
5.
M.
Ramli
,
Amplitude Amplification Factor of Bi-chromatic Waves Propagation in Hydrodynamic Laboratory
,
IAENG Int. J. App. Math.
,
2016
, Vol.
46
(
1
):
29
34
.
6.
V.
Halfiani
and
Marwan
,
Deformation of Bichromatic Wave Groups Based On Third Order Side Band Solution of Benjamin-Bona-Mahony Equation
,
Proc. IOP Conference Series
(accepted).
7.
M.J.
Ablowitz
,
Nonliniear Dispersive Waves: Asymptotic Analisis and solitons
,
2011
,
Cambridge University Press
,
New York
.
8.
N.N.
Akhmediev
and
A.
Ankiewicz
,
Solitons: Nonlinear Pulses and Beams
,
1997
,
Chapman and Hall
,
London
.
9.
S.K.
El-Labany
,
W.M.
Moslem
,
N.A.
El-Bedwehy
,
R.
Sabry
, and
H.N. Abd
El-Razek
,
Rogue Wave in Titans Atmosphere
,
Astrophys. Space Sci.
,
2012
, Vol.
338
:
38
.
10.
M.
Bacha
,
S.
Boukhalfa
, and
M.
Tribeche
,
Ion-Acoustic Rogue Waves in a Plasma with a q-Nonextensive Electron Velocity Distribution
,
Astrophys. Space Sci.
,
2012
, Vol.
341
:
591
595
.
11.
A.U.
Rahman
, and
S.
Ali
,
Solitary and Rogue Waves in Fermi-Dirac Plasmas: Relativistic Degeneracy Effects
,
Astrophys. Space Sci.
,
2014
, Vol.
351
:
165
172
.
12.
S.A.
El-Wakil
,
E. M.
Abulwafa
,
A.
Elhanbaly
, and
E.K.
El-Shewy
,
Rogue Waves for Kadomstev-Petviashvili Equation in Electron-Positron-Ion Plasma
,
Astrophys. Space Sci.
,
2014
, Vol.
353
:
501
506
.
13.
J.P.
Boyd
, and
Y.C.
Guan
,
Weakly Nonlinear Wavepackets in The Korteweg-de Vries Equation: The KdV/NLS Connection
,
Mathematics and Computers in Simulation
,
2001
, Vol.
55
(
4-6
):
317
328
.
14.
M.C.
Bruckner
,
W.P.
Düll
, and
G.
Schneider
,
Validity of The KdV Equation for The Modulation of Periodic Traveling Waves in The NLS Equation
,
J. Math. Anal. Appl.
,
2014
, Vol.
414
:
166
175
.
15.
G.
Schneider
,
Justification of the NLS Approximation for the KdV Equation Using the Miura Transformation
,
Advances in Mathematical Physics
,
2011
, Article ID 854719:
4
pages
16.
V.
Halfiani
,
Salmawaty
, and
M.
Ramli
,
An Envelope Equation of Benjamin Bona Mahony Wave Group
,
Far East J. Math Sci.
,
2017
, Vol.
102
(
6
)
17.
Y.M. Abo
Essa
,
I.A.
Ibrahim
, and
E.D.
Rahmo
,
The Numerical Solution of the MRLW Equation using the Multigrid Method
.
Sci. Res. Appl. Math.
,
2014
, Vol.
5
:
3328
3334
.
18.
J.
Chen
,
S.
Lai
, and
Y.
Qing
,
Exact Solutions to a Generalized Benjamin-Bona-Mahony Equation
.
Int. J. Comput. Int. Sys.
,
2007
, DOI:.
19.
A.R.
Seadawy
,
A.
Sayed
,
Travelling Wave Solutions of the Benjamin-Bona-Mahony Water Wave Equation
,
Abstr. Appl. Anal.
,
2014
, Article ID 926838 :
7
pages.
20.
A. K.
Khalifa
,
K. R.
Raslan
,
H.M.
Alzubaidi
.
A Collocation Method with Cubic B-Splines for Solving the MRLW Equation
,
J. Comput. Appl. Math
,
2008
, Vol.
212
:
406
418
.
21.
A. K.
Khalifa
,
K. R.
Raslan
,
H.M.
Alzubaidi
.
Numerical Study using ADM for the Modified Regularized Long Wave Equation
,
Appl. Math. Modelling
,
2008
, Vol.
32
:
2962
2972
.
22.
R.
Arora
,
A.
Kumar
,
Soliton Solution for the BBM and MRLW Equations by Cosine-function Method
.
Appl. Math.
,
2011
, Vol.
1
(
2
) :
59
61
.
23.
Z.U.A.
Zafar
,
Soliton Solution of IMKDV, KDV, GKDV and MRLW Equations by Sine-hyperbolic Function Method
.
Int. J. Adv. Eng. Glob. Technol.
,
2013
, Vo.
1
(
3
) ISSN No. .
24.
M.A.
Wazwaz
.
Partial Dierential Equations and Solitary Waves Theory
.
2009
,
Springer
,
Berlin Heidelberg
.
25.
T.B.
Benjamin
,
J.L.
Bona
, and
J.J.
Mahony
,
Model Equation For Long Wave In Nonlinear Dispersive Systems
,
Philos. Trans. Roy. Soc. London
,
1972
, Vol.
A272
:
47
78
.
26.
D.H.
Peregrin
,
Calculations of the Development of an Undular Bore
,
Journal of Fluid Mechanics
,
1996
, Vol.
25
(
2
):
321
330
.
This content is only available via PDF.