Some Soliton Solutions of Non Linear Partial Differential Equations by Tan-Cot Method

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 23-28
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Some Soliton Solutions of Non Linear Partial Differential
Equations by Tan-Cot Method
Raj Kumar1
, Mukesh Kumar1
and Anshu Kumar2
1
(Department of Mathematics, MNNIT Allahabad, Uttar Pradesh-211004, India)
2
(Department of Mathematics, Sharda University, Plot No. 32-34, Knowledge Park III, Greater Noida,
Uttar Pradesh-201306, India)
Abstract: Tan-cot method is applied to get exact soliton solutions of non-linear partial differential equations
notably generalized Benjamin-Bona-Mahony, Zakharov-Kuznetsov Benjamin-Bona-Mahony, Kadomtsov-
Petviashvilli Benjamin-Bona-Mahony and Korteweg-de Vries equations, which are important evolution
equations with wide variety of physical applications. Elastic behavior and soliton fusion/fission is shown
graphically and discussed physically as far as possible.
Keywords: Benjamin-Bona-Mahony equation, Tan-Cot method, soliton solutions.
I. Introduction
Authors considered the following non-linear partial differential equations:
𝑢𝑡 + 𝑎𝑢 𝑥 + 𝑏𝑢2
𝑢 𝑥 + 𝛿𝑢 𝑥𝑥𝑡 = 0 Generalised Benjamin-Bona-Mahony (1.1)
𝑢𝑡 + 𝑢 𝑥 + 𝑎𝑢(𝑢2
) 𝑥 − 𝑏(𝑢 𝑥𝑡 + 𝑢 𝑦𝑦 ) 𝑥 = 0 Zakharov-Kuznetsov Benjamin-Bona-Mahony (1.2)
𝑢 𝑥𝑡 + 𝑢 𝑥𝑥 + 2𝑎𝑢2
𝑢 𝑥𝑥 + 𝑘𝑢 𝑦𝑦 + 4𝑎𝑢𝑢 𝑥
2
+ 𝑢 𝑥𝑥𝑥𝑡 = 0 Kadomtsov-Petviashvilli Benjamin-Bona-Mahony 1.3
𝑢𝑡 + 6𝑢𝑢 𝑥 + 𝑢 𝑥𝑥𝑥 = 0, Korteweg-de Vries (1.4)
where 𝑎, 𝑏, 𝛿 and 𝑘 are non-zero arbitrary constants. The equations (1.1) − (1.4) are abbreviated as GKBBM,
ZKBBM, KPBBM and KdV respectively. These equations attracted to a diverse group of researchers in view of
their wide applications in many fields of sciences such as fluid mechanics, plasma and solid state physics,
optical fibers and other related issues of engineering.
The BBM equation was introduced first by Benjamin et al [1]. Furthermore, Zhang et al [2]
investigated it as a regularized version of the KdV equation for shallow water waves. Wang et al [3] and Wang
[4] studied non-linear, dispersive and dissipative effects of BBM equation. The study of Benjamin et al [1]
follows that both the equations KdV and BBM are valid at the same level of approximation, but BBM equation
does have some advantages over the KdV equation from the computational mathematics viewpoint. In certain
theoretical investigations, the BBM equation is superior as a model for long waves, and the word “regularized”
refers to the fact that, from the standpoint of existence and stability, the equation offers considerable technical
advantages over the KdV equation.
Two dimensional generalizations of BBM equation i.e. (1.2) and (1.3) are given by Zakharov-
Kuznetsov and Kadomtsov-Petviashvilli respectively, which can be studied pursuing the literature [5-8] and
references therein. Further, the main mathematical difference between KdV and BBM models can be most
readily appreciated by comparing the dispersion relation for the respective linearised equations [9, 10]. As far as
applications of BBM equation is concerned, it is used in the study of drift waves in plasma or the Rossby waves
in rotating fluids. Other applications of KdV and BBM equations in semiconductor, optical devices can be found
in the work of Zhang et al [2].
Getting inspiration from many applications of BBM, ZKBBM, KPBBM and KdV equations in real life
problems, authors attained the solitory solutions of these equations. The term soliton coined by Zabusky and
Kruskal [11] after their findings that waves like particles retained their shapes and velocities after interactions.
The first published observation [12] of a solitary wave i.e. a single and localized wave was made by a naval
architect John Scott Russel in 1834. Russel explored his experiences in his report to the British Association [13].
In order to understand the non-linear phenomena of equations (1.1) − (1.4) in a better way it is important to
seek their more exact solutions. A variety of useful methods notably Algebraic method [14], Exp-function
method [15], Adomian modified method [16], Inverse scattering method [17], Tanh function method [18],
Variational method [19], Similarity transformation methods using Lie group theory [20], Homotopy
perturbation method [21], Jacobi elliptic function expansion method [22], Hirota method [23], Backlund
transformations method [24, 25], F-expansion method [26], Differential transformations method [27], Darboux
transformations [28], Balance method [29], Sine-cosine method [30] and Tan-Cot method [31] were applied to
investigate the solutions of non-linear partial differential equations.
Authors have been motivated from these researches and applied the Tan-Cot method to obtain soliton
solutions of the equations (1.1) − (1.4). The tan-cot method is a direct and effective algebraic method for

Some Soliton Solutions Of Non Linear Partial Differential Equations By Tan-Cot Method
handling many non-linear evolution equations. It is a good tool to solve non-linear partial differential equations
with genuine non-linear dispersion where solitary patterns solutions are generated [31].
The goal of the present work is to get exact soliton solutions of GBBM, ZKBBM, KPBBM and KdV equations
by using tan-cot method. Elastic behavior and soliton fusion/fission of these solutions discussed physically as
far as possible.
II. Methodology
Consider non-linear partial differential equation in the form:
𝐻(𝑢, 𝑢 𝑥 , 𝑢 𝑦 , 𝑢𝑡 , 𝑢 𝑛
𝑢 𝑥, 𝑢 𝑥𝑦 , 𝑢 𝑥𝑡 , … , 𝑢 𝑥𝑥𝑡 , … ) = 0, (2.1)
where u(x, y, t) is the soliton solution of (2.1), ux and ut etc. are partial derivatives of u with respect to x and t
respectively. The term un
ux shows non-linearity in (2.1). Since equation (2.1) admits soliton solution. Hence,
we can use the transformation:
𝑢 𝑥, 𝑦, 𝑡 = 𝑓 𝑋 , (2.2)
where X = x + y − ct + d, c is the speed of the travelling wave and d is a constant. It follows:
∂u
∂x
=
∂u
∂y
=
∂f
∂X
,
∂2u
∂x ∂t
= −c
d2f
dX2 = −c f etc. (2.3)
Making use of (2.3) into (2.1) yields a non-linear ordinary differential equation (ODE):
𝐺 𝑓, 𝑓, 𝑓 𝑛
𝑓, 𝑓, … = 0 (2.4)
The equation (2.4) is then integrated as long as all terms involve derivatives of 𝑓, where we ignore the constant
of integration (Since 𝑓 ⟶ 0 as 𝑋 ⟶ ±∞). To get solution of (2.1) by applying Tan-Cot method, the function
f(X) can be expressed either in the following forms:
𝑓 𝑋 = 𝛼𝑡𝑎𝑛 𝛽
𝜇𝑋 , 𝜇𝑋 ≤
𝜋
2
(2.5a)
or 𝑓 𝑋 = 𝛼𝑐𝑜𝑡 𝛽
𝜇𝑋 , 𝜇𝑋 ≤
𝜋
2
, (2.5b)
where α, β and μ (wave number) are the unknown parameters. The derivatives of (2.5a) are:
𝑓 𝑋 =𝛼𝛽𝜇[𝑡𝑎𝑛 𝛽−1
𝜇𝑋 + 𝑡𝑎𝑛 𝛽+1
𝜇𝑋 ], (2.6)
𝑓 𝑋 =𝛼𝛽𝜇2
[(𝛽 − 1)𝑡𝑎𝑛 𝛽−2
𝜇𝑋 + 2𝛽𝑡𝑎𝑛 𝛽
𝜇𝑋 + (𝛽 + 1)𝑡𝑎𝑛 𝛽+2
𝜇𝑋 ], (2.7)
[(𝛽 − 2)(𝛽 − 1)𝑡𝑎𝑛 𝛽−3
𝜇𝑋 + 3𝛽2
− 3𝛽 + 2 𝑡𝑎𝑛 𝛽−1
𝜇𝑋 + 3𝛽2
+ 3𝛽 + 2 𝑡𝑎𝑛 𝛽+1
𝜇𝑋
+(𝛽 + 2)(𝛽 + 1)𝑡𝑎𝑛 𝛽+3
𝜇𝑋 ], (2.8)
[(𝛽 − 3)(𝛽 − 2)(𝛽 − 1)𝑡𝑎𝑛 𝛽−4
𝜇𝑋 + 4 𝛽 − 1 𝛽2
− 2𝛽 + 2 𝑡𝑎𝑛 𝛽−2
𝜇𝑋
+2𝛽 3𝛽2
+ +5 𝑡𝑎𝑛 𝛽
𝜇𝑋 + 4 𝛽 + 1 𝛽2
+ 𝛽 + 2 𝑡𝑎𝑛 𝛽+2
𝜇𝑋
+ 𝛽 + 1 𝛽 + 2 𝛽 + 3 𝑡𝑎𝑛 𝛽+4
𝜇𝑋 ], (2.9)
and so on.
Authors found all solutions considering the form α tanβ
μX . Therefore, derivatives are calculated intentionally
only for this form.
We can obtain an equation in different powers of tangent functions substituting the values of derivatives from
(2.6) − (2.9) into (2.4). Then we collect the coefficients of each pair of tangent functions with same exponent
from (2.4), where each term has to vanish. Consequently, one can obtain a system of algebraic equations in
unknown parameters c, α, β and μ. Solving this system, we can get the soliton solutions of partial differential
equation (2.1) by substituting the values of these parameters in (2.5a).
III. Results And Discussions
1. Benjamin-Bona-Mahony (BBM) equation
Authors studied the BBM equation in the form (1.1). Taking X = x − ct + d and applying the above
procedure on (1.1), it leads to:
a − c f + bf2
f − cδf = 0, a ≠ c (3.1.1)
Integrating 3.1.1 with respect to X and ignoring constant of integration, one can obtain:
a − c f +
b
3
f3
− cδf = 0, (3.1.2)
Making use of (2.5a) and (2.7) into (3.1.2) gives:
a – c αtanβ
μX +
b
3
α3
tan3β
μX − cδαβ β − 1 μ2
tanβ−2
μX − 2cδαβ2
μ2
tanβ
μX
−cδαβ β + 1 μ2
tanβ+2
μX = 0 (3.1.3)
Equating the exponents of second and third term of tangent functions in the equation (3.1.3), then collecting the
coefficients of the terms involved tangent functions of the same exponent, where each term has to vanish. We
obtain the following system of algebraic equations:
3𝛽 = 𝛽 − 2;
𝑏
3
𝛼2
− 2𝑐𝛿𝜇2
= 0; 𝜇2
=
𝑎−𝑐
2𝑐𝛿
(3.1.4)

On solving the system (3.1.4), we obtain:
𝛽 = −1; α= −
3(𝑎−𝑐)
𝑏
; μ=
(𝑎−𝑐)
2𝑐𝛿
(3.1.5)
It follows immediately:
𝑎−𝑐
𝑏
> 0 and
𝑎−𝑐
𝑐𝛿
> 0 (3.1.6)
Thus, soliton solution of BBM equation (1.1) is furnished by:
𝑢 𝑥, 𝑡 = −
3(𝑎−𝑐)
𝑏
cot
𝑎−𝑐
2𝑐𝛿
𝑥 − 𝑐𝑡 + 𝑑 , where 0 < 𝑥 − 𝑐𝑡 + 𝑑 <
𝜋 𝑐𝛿
2(𝑎−𝑐)
(3.1.7)
For a = δ = d = 2, b = 4 and c = 1, equation 3.1.7 becomes:
𝑢 𝑥, 𝑡 = −
3
2
𝑐𝑜𝑡
𝑥−𝑡+2
2
(3.1.8)
Evolutional profile of 𝑢(𝑥, 𝑡) through the expression (3.1.8) is shown in the figure 1 for the domain −10 ≤
𝑥, 𝑡 ≤ 10. It is obvious from figure 1 that elastic behavior of multi solitons can be viewed.
2. Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation
Authors found the soliton solutions of the ZKBBM governed by (1.2). Equation (1.2) can be rewritten
in simplified form as:
ut + ux + 2au2
ux − buxxt − buxyy = 0 (3.2.1)
The wave variable X = x + y − ct + d carries the equation (3.2.1) into an ODE represented by:
1 − c f + 2af2
f – b(1 − c)f = 0, c ≠ 1 (3.2.2)
Integrating (3.2.2) and ignoring the arbitrary constant of integration, one can get:

1 − c f +
2a
3
f3
− b(1 − c) f = 0 (3.2.3)
Substituting the derivatives from equations (2.5) and (2.7) into (3.2.3), it provides us:
α 1 – c tanβ
μX +
2a
3
α3
tan3β
μX − b 1 − c αβ β − 1 μ2
tanβ−2
μX − 2b 1 − c αβ2
μ2
tanβ
μX
−b(1 − c)αβ(β + 1)μ2
tanβ+2
μX = 0 (3.2.4)
Equating the exponents of second and third tangent functions in the equation (3.2.4), and then we collect the
coefficients of each pair of tangent functions of the same exponent, where each term has to vanish. Finally, we
obtain the following system:
3β = β − 2; 1 − 2bβ2
μ2
= 0;
2a
3
α2
− b(1 − c)β(β − 1)μ2
= 0 (3.2.5)
Thus we obtain:
β = −1; α= −
3(1−c)
2a
; μ=
1
2b
(3.2.6)
It follows:
𝑏 > 0 and
1−c
a
> 0 (3.2.7)
In view of (3.2.6), solution of ZKBBM equation (1.2) is given by:
u x, y, t = −
3(1−c)
2a
cot
1
2b
x + y − ct + d , with 0 < 𝑥 + 𝑦 − 𝑐𝑡 + 𝑑 <
𝜋 𝑏
2
(3.2.8)
Setting arbitrary constants a = −2, b = c = 2 and d = 0 in (3.2.8), it becomes:
u x, y, t = −
3
2
cot
x+y−2t
2
(3.2.9)
Solutions graphs via expression (3.2.9) are shown in the figure 2 for −2 ≤ x ≤ 2, 0 ≤ y ≤ 4 at t = 0.01 and
6. In this spatial range, elastic single soliton can be observed in the figures 2 a and 2 b .
3. Kadomtsov-Petviashvilli-Benjamin-Bona-Mahony (KPBBM) equation
Authors considered the KPBBM governed by (1.3). Applying the Tan-Cot method on (1.3) we can have
following ODE:
4af f2
+ 1 − c + k f + 2af2
f − cf = 0 (3.3.1)
Integrating (3.3.1) twice w.r.t. X (ignoring the arbitrary constants of integration), we obtain
2𝑎
3
𝑓3
+ 1 − 𝑐 + 𝑘 𝑓 − 𝑐𝑓 = 0, (3.3.2)
Substituting the values of 𝑓 , 𝑓 from of equations (2.5a) and (2.7) into (3.3.2), one can get
1 – c + k αtanβ
μX +
2a
3
α3
tan3β
μX − cαβ(β − 1)μ2
tanβ−2
μX − 2cαβ2
μ2
tanβ
μX −cαβ(β +
1)μ2
tanβ+2
μX = 0 (3.3.3)
Equating the exponents of second and third tangent functions in the equation (3.3.3), and then we collect the
coefficients of each pair of tangent functions of the same exponent, where each term has to vanish. Finally, we
obtain the following system:
3𝛽 = 𝛽 − 2; 1 − 𝑐 + 𝑘 − 2𝑐𝛽2
𝜇2
= 0;
2a
3
𝛼2
− 𝑐𝛽(𝛽 − 1)𝜇2
= 0 (3.3.4)
On solving the system (3.3.4): we get the following values of unknowns 𝛼, 𝛽 and 𝜇 ;
𝛽 = −1; α= −
3(1−𝑐+𝑘)
2𝑎
; μ=
1−𝑐+𝑘
2𝑐
(3.3.5)
It follows immediately:
1−𝑐+𝑘
𝑎
> 0 and
1−𝑐+𝑘
𝑐
> 0 (3.3.6)
In view of (3.3.5), solution of KPBBM equation (1.3) can be written in explicit manner as:
𝑢 𝑥, 𝑦, 𝑡 = −
3(1−𝑐+𝑘)
2𝑎
cot
1−𝑐+𝑘
2𝑐
𝑥 + 𝑦 − 𝑐𝑡 + 𝑑 , (3.3.7)
where 0 < 𝑥 + 𝑦 − 𝑐𝑡 + 𝑑 <
𝜋 𝑐
2(1−𝑐+𝑘)
(3.3.8)
For a = 1, c = 2, d = 0, k = 4, the equation (3.3.7) becomes:
𝑢 𝑥, 𝑦, 𝑡 = −
3
2
cot
3
2
𝑥 + 𝑦 − 2𝑡 (3.3.9)
Evolutional profile for (3.3.9) is shown in the figure 3 for −5 ≤ 𝑥, 𝑦 ≤ 5 at t = 0.2, 5. Fusion and fission of
elastic solitons can be observed in the figures 3 a and 3 b .
4. Korteweg-de Vries(KdV) equation
The KdV equation under our consideration is governed by (1.4). Using the same procedure as we have
used in previous subsections to obtain an ODE:

−𝑐𝑓 + 6𝑓𝑓 + 𝑓 = 0 (3.4.1)
Substituting the devatives of 𝑓 from (2.6) and (2.8) into (3.4.1), we have:
−cαβμ tanβ−1
μX + tanβ+1
μX + 6α2
βμtanβ
μX tanβ−1
μX + tanβ+1
μX
+𝛼βμ3
[ β − 1 β − 2 tanβ−3
μX + 3β2
− 3β + 2 tanβ−1
μX
+ 3β2
+ 3β + 2 tanβ+1
μX + β + 1 β + 2 tanβ+3
μX ] =0 (3.4.2)
The equation (3.4.2) is satisfied if the following system of algebraic equations holds:
β = 2; c= 8μ2
, α= −2μ2
(3.4.3)
By solving the system (3.4.3), one can find the following soliton solution of KdV equation (1.4):
u x, t =
c
4
[1 − sec2
{
c
2 2
x − ct + d }] (3.4.4)
For c = 8, d = 0 equation (3.4.5) becomes:
u x, t = 2 [1 − sec2
(x − 4t)] (3.4.5)
Evolutional profile for (3.4.5) is shown in the figure 4. Elastic multi solitons can be observed at regular
intervals.
IV. Conclusions
In this work, the exact soliton solutions of generalised Benjamin-Bona-Mahony, Zakharov-Kuznetsov-
BBM, Kadomtsov-Petviashvilli BBM, Korteweg-de Vries equation equations have been obtained successfully
by using tan-cot method. The soliton behaviour of solutions with respect to time and space is shown and
discussed physically wherever was possible. Authors are intend to get soliton solutions by applying this method
to a system of non-linear partial differential equations like (2+1)-dimensional Boiti-Leon-Pempinelli (BLP)
system.
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Some Soliton Solutions of Non Linear Partial Differential Equations by Tan-Cot Method

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Some Soliton Solutions of Non Linear Partial Differential Equations by Tan-Cot Method