Nonlinear DynDOI 10.1007/s11071-013-1189-9

ORIGINAL PAPER

Singular solitons, shock waves, and other solutionsto potential KdV equation

Gang-Wei Wang · Tian-Zhou Xu ·Ghodrat Ebadi · Stephen Johnson ·Andre J. Strong · Anjan Biswas

Received: 4 October 2013 / Accepted: 13 December 2013© Springer Science+Business Media Dordrecht 2013

Abstract This paper addresses the potentialKorteweg–de Vries equation. The singular 1-solitonsolution is obtained by the aid of ansatz method. Sub-sequently, the G ′/G-expansion method and the exp-function approach also gives a few more interestingsolutions. Finally, the Lie symmetry analysis leads toanother plethora of solution to the equation. Theseresults are going to be extremely useful and applica-ble in applied mathematics and theoretical physics.

Keywords Solitons · Lie symmetry · Integrability

G.-W. Wang · T.-Z. XuSchool of Mathematics, Beijing Institute of Technology,Beijing 100081, People’s Republic of China

G. EbadiFaculty of Mathematical Sciences, University of Tabriz,51666-14766 Tabriz, Iran

S. Johnson · A. J. Strong · A. BiswasDepartment of Mathematical Sciences, Delaware StateUniversity, Dover, DE 19901-2277, USA

S. JohnsonLake Forest High School, 5407 Killens Pond Road,Felton, DE 19943, USA

A. Biswas (B)Department of Mathematics, Faculty of Science, KingAbdulaziz University, Jeddah, Saudi Arabiae-mail:biswas.anjan@gmail.com

1 Introduction

The mathematical models of shallow water wavesare a growing area of research in mechanical engi-neering. There have been several models that areproposed to study this dynamics. A few of thesemodels are the well-known Korteweg–de Vries equa-tion (KdV), Boussinesq equation, Benjamin–Bona–Mahoney equation that is also known as the Peregrineequation or regularized long wave (RLW) equation,Kawahara equation, and many others. All these mod-els govern the dynamics of water waves along oceanshores, canals, and other low-lying areas of water.These models are all well studied and very well under-stood. However, in the recent past there are several othermodels that are being reported. A few of such modelsare the Rosenau–Kawahara equation, Rosenau–RLWequation and these models dispersive water wavesalong ocean shores. Another model that is believed toreplicate tsunami waves is the potential KdV (pKdV)equation. This paper will study this equation in orderto extract solitary wave solutions by the application ofseveral integration tools. These models fall in the cat-egory of nonlinear evolution equations (NLEEs) thatarise in mathematical physics.

The mathematical analysis of NLEEs is growing atan alarming rate. There are several aspects of NLEEswhich are being addressed by several scientists acrossthe globe [1–25]. One of the primary focus is theintegrability issue. The integration of these severalNLEEs leads to a plethora of solutions. These solutions

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are extremely important in various areas of appliedmathematics and theoretical physics especially wherethese equations appear. It is due to this pressing rea-son, there are different forms of integration architec-ture, which have been developed in the last couple ofdecades.

A few of these integration tools are variational iter-ation method, Adomian decomposition method, firstintegral method, differential transform method, Liesymmetry analysis, mapping method, and several oth-ers. However, it is imperative to exercise caution whileimplementing these techniques of integration. Whilethese methods reveal results and several forms of solu-tions to these equations, there are a few hindrances thatone experiences. For example, none of these integra-tion tools lead to the computation of soliton radiationor phonons which is an important component of soli-ton solutions. In addition, most of these techniques donot lead to multiple soliton solutions that is also impor-tant in various areas of engineering sciences where thestudy soliton dynamics is extremely important, such asoptical fiber technology.

This paper addresses one such NLEE that is knownas the pKdV equation. This equation is studied inapplied mathematics including the dynamics of shal-low water waves. The integrability aspect of this equa-tion will be the primary focus of this paper. The ansatzmethod will be first employed to extract the singu-lar 1-soliton solution to this equation. Subsequently,three integration tools will be applied to integrate thepKdV equation that will display a range of solutions.The three additional methods of integration that will beapplied are the G ′/G-expansion method, exp-functionapproach, and the Lie symmetry analysis. These resultswill be discussed in the following sections.

2 Singular soliton solution

The dimensionless structure of the pKdV equation thatis going to be studied in this paper is given by [3]

qt + a (qx )2 + bqxxx = 0, (1)

where a and b are constant quantities, while x and tare the independent variables that represent the spatialand temporal variables, respectively, while q(x, t) isthe dependent variable that represents the wave profile.This section will focus on the singular 1-soliton solu-tion to this equation that will be obtained by the ansatzmethod. The starting ansatz therefore is given by

q(x, t) = A cothp[B(x − vt)], (2)

where A and B are free parameters while v is the veloc-ity of the soliton. The unknown exponent is p, wherep > 0, whose value will fall out during the course ofderivation of the soliton solution. Substituting (2) into(1) and simplifying, we get

v(

cothp+2 τ − cothp τ)

+ ap AB(

coth2p+4 τ − 2 coth2p+2 τ + coth2p τ)

− bB2[(p + 2)(p + 4) cothp+6 τ

− (p + 2) {2(p + 1) + (p + 4)} cothp+4 τ

+{

p2 + 2(p + 1)(p + 2)}

cothp+2 τ

− p2 cothp τ]

= 0, (3)

where the notation

τ = B(x − vt) (4)

was introduced.By the balancing principle, equating the exponents

2p and p + 2 leads to

2p = p + 2, (5)

so that

p = 2. (6)

It needs to be noted here that the same value of p isrecovered when the exponent pairs 2p + 2 and p + 4or 2p +4 and p +6 are equated. Now from (3), settingthe coefficients of the linearly independent functionscothp+ j τ , for j = 0, 2, 4 and 6, to zero, we get

v = 4bB2, (7)

v = 28bB2 − 2a AB, (8)

A = 12b

aB. (9)

Now, equating the two values of the soliton velocityfrom (7) and (9) also gives the relation between thesoliton free parameters as seen in (9). The only con-straint condition that is evident from (9) is given by

a �= 0. (10)

Hence, finally, the singular 1-soliton solution to (1) isgiven by

q(x, t) = A coth2[B(x − vt)], (11)

where the relation between the free parameters of thesoliton is given by (9) while the velocity of the soli-ton is given in (7) or (9). The following two sectionsdisplay other solutions that are obtained by two otherintegration techniques.

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Korteweg–de Vries equation

3 G′/G-expansion method

In this section, we will demonstrate the G ′/G-expansionmethod on pKdV equation. The pKdV equation thatwill be studied is given by [3]

qt + a (qx )2 + bqxxx = 0, (12)

which by the wave transformation

q(x, t) = q(τ ), τ = B(x − vt), (13)

is converted to

− vq ′(τ ) + aB(q ′)2

(τ ) + bB2q ′′′(τ ) = 0, (14)

where a and b are arbitrary constants. By balancingprincing principle, we get 2(m + 1) = m + 3, hencem = 1. We then suppose that Eq. (14) has the followingformal solutions:

q = A1

(G ′

G

)+ A0, (15)

where A0 and A1 are unknown constants to be deter-mined by solving a set of algebraic equation by Mapleand this leads to [4]

A1 = 6bB

a, v = bB2(λ2 − 4μ), (16)

where μ and A0 are arbitrary constants.When � = λ2 − 4μ > 0, we obtain hyperbolic

function solutions:

q(x, t) = q(τ )

= A0 − 6bλB

2a

+6b Bε

a

(c1 sinh(ετ ) + c2 cosh(ετ )

c1 cosh(ετ ) + c2 sinh(ετ )

)

(17)

and ε = 12

√λ2 − 4μ.

When � = λ2 − 4μ < 0, we obtain trigonometricfunction solutions:

q(x, t) = q(τ )

= A0 − 6bλB

2a

+6b Bε

a

(−c1 sin(ετ ) + c2 cos(ετ )

c1 cos(ετ ) + c2 sin(ετ )

),

(18)

where ε = 12

√4μ − λ2.

When � = λ2 − 4μ = 0, we obtain trigonometricfunction solutions:

q(x, t) = q(τ )

= A0 − 6bλB

2a+ 6b B

a

(c2

c1 + c2 Bx

). (19)

4 Exp-function method

In this approach, it is assumed that the governing equa-tion can be expressed in the following form [5]:

q(τ ) =∑d

n=−c anenτ

∑sm=−p bmemτ

, (20)

where c, d, p, and s are positive integers which areunknown to be further determined; an and bm areunknown constants. In this case, the pKdV equationthat will be studied is given by [3]

qt + a (qx )2 + bqxxx = 0, (21)

which by the wave transformation is converted to

− vq ′(τ ) + aB(q ′)2

(τ ) + bB2q ′′′(τ ) = 0, (22)

where a and b are arbitrary constants. Using the ansatz(20), for the linear term of highest order q ′2 and q ′′′ bysimple calculation, we have

(q ′)2 = c1e2(d+s)τ + · · ·

c2e4sτ · · · (23)

and

q ′′′ = c3e(d+7s)τ + · · ·c4e8sτ + · · · , (24)

where ci are determined coefficients only for sim-plicity. Balancing highest order of exp-function inq ′2 and q ′′′, we have

2d + 2s − 4s = d + 7s − 8s, (25)

which leads to the result

d = s. (26)

Similarly, to determine the values of c and p, we bal-ance the linear term of lowest order in Eq. (22)

(q ′)2 = d1e−2(c+p)τ + · · ·

d2e−4pτ + · · · (27)

and

q ′′′ = · · · + d3e−(c+7p)τ

· · · + d4e−8pτ, (28)

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where di are determined coefficients only for simplic-ity. Balancing lowest order of exp-function in Eqs. (27)and (28), we have

− (2c + 2p) + 4p = −(c + 7p) + 8p, (29)

which leads to the result

c = p. (30)

Since the final solution does not strongly depend uponthe choice of values of c and d, so for simplicity, we setp = c = 1, d = s = 1, the function Eq. (20) becomes

q(τ ) = a−1e−τ + a0 + a1eτ

b−1e−τ + b0 + b1eτ. (31)

Substituting Eq. (31) into Eq. (22), and equating to zero,the coefficients of all powers of enτ yield a set of alge-braic equations for a1, a0, a−1, b1, b0, v, B, and b−1.

Solving the system of algebraic equations with the aidof Maple, we obtain the following several set of solu-tions.

First set:

a1 = b1 = 0, v = bB2, a−1 = − b−1(−aa0+6bBb0)

ab0,

(32)

where a, b0 �= 0, τ = B(x − bB2t). Here, a, b, a0,

a−1, b0, b−1, and B are free parameters.

Second set:

a−1 = −6(6a1 Bbb−1 + aa20)bBb−1

a2a20

, (33)

v = bB2, b1 = −a2a20

36B2b2b−1, b0 = 0,

where a, a0, b, b−1, B �= 0, τ = B(x − bB2t) anda, b, a0, b−1, a1 are free parameters.

Third set:

b−1 = −az, v = bB2, a−1 = (6bBb1 − aa1)z

b1, (34)

where

z = (a0b1 − a1b0)(6b0b1bB + a0ab1 − aa1b0)

36(B2b2b31)

(35)

and a, b, a0, b0, a1, b1, B are free parameters withthe restriction bBb1 �= 0.

Fourth set:

a0 =0, b0 =0, v=4bB2, a−1 =−b−1(12bBb1−aa1)

ab1,

(36)

where a, b, B, a1, b1, b−1 are free parameters withthe condition ab1 �= 0.

5 Lie symmetry analysis

This section will focus on the Lie symmetry analy-sis to the pKdV equation. In the first subsection, thebasic mathematical phenomena of Lie symmetry analy-sis will be described. Next, this analysis will be appliedto the pKdV equation and several solutions will beretrieved.

5.1 Lie symmetry analysis to (1)

In this section, we will perform Lie group method forEq. (1). If (1) is invariant under a one-parameter Liegroup of point transformations

t∗ = t + ετ(x, t, q) + O(ε2),

x∗ = x + εξ(x, t, q) + O(ε2),

q∗ = q + εη(x, t, q) + O(ε2), (37)

with infinitesimal generator

V = τ(x, t, q)∂

∂t+ξ(x, t, q)

∂

∂x+η(x, t, q)

∂

∂q. (38)

If the vector field (38) generates a symmetry of (1),then V must satisfy Lie’s symmetry condition:

pr (3)V (�1)|�1=0 = 0, (39)

where �1 = qt + a (qx )2 + bqxxx . Applying the third

prolongation pr (3)V to Eq. (1), we find the followingsystem of symmetry equations then the invariant con-dition reads as

ηt + 2aηx qx + ηxxx = 0, (40)

where

ηt = Dx (η) − qx Dx (ξ) − qt Dx (τ )

= Dx (η − ξqx − τqt ) + ξqxx + τqxt

= ηx +(ηq −ξx )qx − τx qt −ξqq2x −τqqx qt , (41)

ηt = Dt (η) − qx Dt (ξ) − qt Dt (τ )

= Dt (η − ξqx − τqt ) + ξqxt + τqtt

= ηt −ξt qx +(ηq −τt )qx −τt qt −ξqqx qt − τqq2t ,

(42)

ηxx = Dx (ηx ) − qxt Dx (τ ) − qxx Dx (ξ) . . . , (43)

ηxxx = Dx (ηxx ) − qxxt Dx (τ ) − qxxx Dx (ξ) . . . . (44)

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Korteweg–de Vries equation

Here, Di denotes the total derivative operator and isdefined by

Di = ∂

∂xi+ qi

∂

∂q+ qi j

∂

∂q j+ · · · , i = 1, 2, (45)

and (x1, x2) = (t, x).Then, in terms of the Lie symmetry analysis method,

one can obtain

τ = 3c1t + c2, ξ = c1x + c3, η = −c1q + c4, (46)

where c1, c2, c3, and c4 are arbitrary constants.We obtain the corresponding (four-dimensional Lie

algebra) geometric vector fields of (1) is as follows:

V1 = ∂

∂t, V2 = ∂

∂x,

V3 = ∂

∂q, V4 = x

∂

∂x+ 3t

∂

∂t− q

∂

∂q. (47)

It is easy to check that the symmetry generatorsfound in (47) form a closed Lie algebra

[V1, V2] = [V2, V1] = [V1, V3]= [V3, V1] = [V3, V2] = [V2, V3] = 0.

[V1, V4] = −[V4, V1] = 3V1, [V2, V4]= −[V4, V2] = V2,

[V3, V4] = −[V4, V3] = −V3. (48)

To obtain the group transformation which is gener-ated by the infinitesimal generators Vi for i = 1, 2, 3, 4,we need to solve the following initial problems

d(x(ε))

dε= ξ

(x(ε), t(ε), q(ε)

), x(0) = x,

d(t(ε))

dε= τ

(x(ε), t(ε), q(ε)

), t(0) = t,

d(u(ε))

dε= η

(x(ε), t(ε), q(ε)

), q(0) = q, (49)

where ε is a parameter. Thus, one can obtain the Liesymmetry group

g : (x, t, q) → (x, t, q). (50)

Exponentiating the infinitesimal symmetries of Eq. (1),we get the one-parameter groups gi (ε) generated by Vi

for i = 1, 2, 3, 4

g1 : (x, t, q) �→ (x, t + ε, q),

g2 : (x, t, q) �→ (x + ε, t, q),

g3 : (x, t, q) �→ (x, t, q + ε),

g4 : (x, t, q) �→ (e−εx, e−3εt, eεq). (51)

The symmetry groups g1 and g2 demonstrate thetime- and space-invariance of the equation. The well-known scaling symmetry turns up in g4. Consequently,we can obtain the corresponding theorem:

Theorem 1 If q = f(x ,t) is a solution of (1), so are thefunctions

g1(ε) · f (x, t) = f (x − ε, t),

g2(ε) · f (x, t) = f (x, t − ε),

g3(ε) · f (x, t) = f (x, t) − ε,

g4(ε) · f (x, t) = e−ε f (e−εx, e−3εt). (52)

5.2 Symmetry reductions and exact group-invariantsolutions

5.2.1 V1

The group-invariant solution corresponding to V1 isq = f (ξ), where ξ = t is the group-invariant, thesubstitution of this solution into Eq. (1) gives the triv-ial solution q(x, t) = C, C is a constant.

5.2.2 V2 (Stationary solution)

For the generator V2, we have

q = f (ξ), (53)

where ξ = x is the group-invariant. Substituting Eq.(53) into Eq. (1), we reduce it to the following ODE

a( f ′)2 + b f ′′′ = 0, (54)

where f ′ = d fdξ

.

5.2.3 V2 + λV1 (Traveling wave solutions)

For the linear combination V2 + λV1, we have

q = f (ξ), (55)

where ξ = x − λt is the group-invariant. Substitutionof (55) into the (1), we reduce it to the following ODE

λ f ′ + a( f ′)2 + b f ′′′ = 0, (56)

where f ′ = d fdξ

.

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It should be noted that if λ �= 0, then (56) is thetraveling wave solution to the equation, where λ �= 0denotes the wave speed. In particular, if λ = 0, then weget (53). Now, we consider the traveling wave solutionsof the (56).

By balancing the highest order partial derivativeterm and the nonlinear term in (56), we get the valueof m,

(u(ξ))2m+2 = (u(ξ))m+3 , (57)

so one can get m = 1. In this case, we get

f (ξ) = a0 + a1ϕ, (58)

where a0, a1 are constants to be determined, and ϕ(ξ)

satisfies

ϕ′ = A + Bϕ + Cϕ2. (59)

Substituting the ansatz (58) along with Eq. (59) into Eq.(55), collecting coefficients of monomials of ϕ with theaid of Maple, then setting each coefficients to zero, wecan deduce

A = A, B = B, C = C, a = a, b = λ

4 AC − B2 ,

λ = λ, a1 = − 6Cλ(4 AC − B2

)a

, a0 = a0. (60)

In view of Eq. (59) which has a lot of fundamentalsolutions (27 solutions) [16] , one can find a numberof exact traveling wave solutions for Eq. (1), some ofwhich are listed as follows:

Family 1 : When B2 − 4AC > 0 and BC �= 0 (orAC �= 0),

q(x, t) = a0 − a11

2C

×[

B +√

B2 − 4AC tanh

(√B2 − 4AC

2ξ

)]. (61)

q(x, t) = a0 − a11

2C

×[

B+√

B2 − 4AC coth

(√B2 − 4AC

2ξ

)]. (62)

q(x, t) = a0 − a11

2C

[B+

√B2 − 4AC

×(

tanh(√

B2−4ACξ)

± isech(√

B2−4ACξ))]

.

(63)

q(x, t) = a0 − a11

2C

[B+

√B2 − 4AC

×(

coth(√

B2−4ACξ)

± icsch(√

B2−4ACξ))]

.

(64)

q(x, t) = a0 − a11

4C

[2B +

√B2 − 4AC

×(

tanh

(√B2−4AC

4ξ

)+coth

(√B2−4AC

4ξ

))].

(65)

q(x, t) = a0+a11

2C

[−B+

√(E2+F2)(B2−4AC)−E

√B2−4AC cosh(

√B2−4ACξ)

Esinh(√

B2−4ACξ)+F

].

(66)

q(x, t)=a0+a11

2C

[−B−

√(F2−E2)(B2−4AC)+E

√B2−4AC sinh(

√B2−4ACξ)

Ecosh(√

B2−4ACξ)+F

],

(67)

where E and F are two non-zero real constants andsatisfies F2 − E2 > 0.

q(x, t) = a0

+a1

[2A cosh(

√B2−4AC

2 ξ)√

B2−4AC sinh(√

B2−4AC2 ξ)−B cosh(

√B2−4AC

2 ξ)

].

(68)q(x, t) = a0

+a1

[−2A sinh(

√B2−4AC

2 ξ)

−√B2−4AC cosh(

√B2−4AC

2 ξ)+B sinh(√

B2−4AC2 ξ)

].

(69)

q(x, t) = a0 + a1

[2A cosh(

√B2 − 4ACξ)√

B2 − 4AC sinh(√

B2 − 4ACξ) − B cosh(√

B2 − 4ACξ) ± i√

B2 − 4ACξ

]. (70)

123

Korteweg–de Vries equation

q(x, t) = a0 + a1

[2A sinh(

√B2 − 4ACξ)√

B2 − 4AC cosh(√

B2 − 4ACξ) − B sinh(√

B2 − 4ACξ) ± √B2 − 4ACξ

]. (71)

q(x, t) = a0+a1

[4A sinh(

√B2−4AC

4 ξ) cosh(√

B2−4AC4 ξ)

−2B sinh(√

B2−4AC4 ξ) cosh(

√B2−4AC

4 ξ)+2√

B2−4AC cosh2(√

B2−4AC4 ξ)−√

B2−4AC

].

(72)

Family 2 : When B2 − 4AC < 0 and BC �= 0 (orAC �= 0)

q(x, t) = a0 + a11

2C

×[−B+

√4AC − B2 tan

(√4AC − B2

2ξ

)]. (73)

q(x, t) = a0 − a11

2C

×[

B+√

4AC − B2 cot

(√4AC − B2

2ξ

)]. (74)

q(x, t) = a0 + a11

2C

[− B

+√

4AC − B2

(tan

(√4AC − B2ξ

)

±sec(√

4AC − B2ξ) )]

. (75)

q(x, t) = a0 − a11

2C

[B

+√

4AC − B2

(cot

(√4AC − B2ξ

)

± csc(√

4AC − B2ξ))]

. (76)

q(x, t) = a0 − a1

× 1

4C

[−2B+

√4AC − B2

(tan

(√4AC − B2

4ξ

)

− cot

(√4AC − B2

4ξ

))]. (77)

q(x, t) = a0 + a11

2C

[− B + ±√

(F2 − E2)(4AC − B2) − E√

4AC − B2 cos(√

4AC − B2ξ)

Esin(√

4AC − B2ξ) + F

]. (78)

q(x, t) = a0 + a11

2C

[− B + ±√

(F2 − E2)(4AC − B2) + E√

4AC − B2 sinh(√

4AC − B2ξ)

E cos(√

4AC − B2ξ) + F

], (79)

where E and F are two non-zero real constants andsatisfies F2 − E2 > 0.

q(x, t) = a0 + a1

[ −2A cos(√

4AC−B2

2 ξ)√

4AC − B2 sin(

√4AC−B2

2 ξ) + B cos(√

4AC−B2

2 ξ)

].

(80)

q(x, t) = a0 + a1

×[

2A sin(

√4AC−B2

2 ξ)√

4AC − B2 cos(√

4AC−B2

2 ξ) − B sin(

√4AC−B2

2 ξ)

].

(81)

q(x, t) = a0 + a1

[ −2A cos(√

4AC − B2ξ)√4AC − B2 sin(

√4AC − B2ξ) + B cos(

√4AC − B2ξ) ± i

√4AC − B2ξ

]. (82)

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G.-W. Wang et al.

q(x, t) = a0 + a1

[2A sin(

√4AC − B2ξ)√

4AC − B2 cos(√

4AC − B2ξ) − B sin(√

4AC − B2ξ) ± √4AC − B2ξ

].

(83)

q(x, t) = a0 + a1

[4A sin(

√4AC−B2

4 ξ) cos(√

4AC−B2

4 ξ)

−2B sin(√

4AC−B2

4 ξ) cos(√

4AC−B2

4 ξ) + 2√

4AC − B2 cos2(√

4AC−B2

4 ξ) − √4AC − B2

].

(84)

Family 3 : When A = 0 and BC �= 0

q(x, t) = a0

+a1

( −Bd

B(d + cosh(Bξ) − sinh(Bξ))

),

(85)

where d is an arbitrary constant.Family 4 : When A = B = 0 and C �= 0

q(x, t) = a0 + a1

( −1

B(ξ) + k

), (86)

where k is an arbitrary constant.5.2. 3. V4 (Scalar group-invariant solutions)For the generator V3, we have

q = t−13 f (ξ), (87)

where ξ = xt− 13 is the group-invariant. Substitution of

(87) into the (1), we get

− 1

3ξ f ′ − 1

3f ′ + a( f ′)2 + b f ′′′ = 0, (88)

where f ′ = d fdξ

.

We note that the reduced equations (88) are nonau-tonomous ODE. In general, it is difficult to get the exactsolutions. However, the power series method [17,18] isan effective tool for dealing with such nonautonomousODE.

Now, we seek a solution of Eq. (88) in a power seriesof the following form:

f (ξ) =∞∑

n=0

cnξn . (89)

Substituting (89) into (88), we get

6bc3 + b∞∑

n=1

(n + 1)(n + 2)(n + 3)cn+3ξn

−1

3c1 − 1

3

∞∑n=1

(n + 1)cn+1ξn

−1

3

∞∑n=1

ncnξn + ac1c1

+a∞∑

n=1

(n∑

k=1

(n − k + 2)ckcn−k+2

)ξn = 0. (90)

Now from (94), comparing coefficients, for n = 0,one can get

c3 =13 c1 − ac2

1

6b. (91)

Generally, for n ≥ 1, we obtain

cn+3 = 1

b(n + 1)(n + 2)(n + 3)

(1

3(n + 1)cn+1

−1

3ncn − a

n∑k=1

(n − k + 2)ckcn−k+2

). (92)

From (91) and (92), one can get all the coefficientscn(n ≥ 5) of the power series (89). For arbitrary chosenconstant numbers c0, c1, and c2, the other terms alsocan be determined successively from (91) and (92) in aunique way. In addition, it is easy to prove that the con-vergence of the power series (89) with the coefficientsgiven by (91) and (92) [17,18]. we omit it here. Thusthis power series solution is an exact analytic solution.

123

Korteweg–de Vries equation

Therefore, the power series solution of Eq. (87) canbe written as follows:

f (ξ) = c0 + c1ξ + c2ξ2 + c3ξ

3 +∞∑

n=1

cn+3ξn+3

= c0 + c1ξ + c2ξ2 +

13 c1 − ac2

1

6bξ3

+ 1

b(n + 1)(n + 2)(n + 3)

(1

3(n + 1)cn+1

−1

3ncn − a

n∑k=1

(n − k + 2)ckcn−k+2

)ξn+3.

(93)

Therefore, we get the solution of Eq. (1)

q(x, t)

=[

c0+c1xt−13 +c2(xt−

13 )2+c3(xt−

13 )3+

∞∑n=1

cn+3ξn+3

= c0 + c1xt−13 + c2(xt−

13 )2 +

13 c1 − ac2

1

6b(xt−

13 )3

+ 1

b(n + 1)(n + 2)(n + 3)

(1

3(n + 1)cn+1

−1

3ncn−a

n∑k=1

(n−k+2)ckcn−k+2

)(xt−

13 )n+3

]t−

13 .

(94)

Of course, in physical applications, it will be writtenin the approximate form

q(x, t)=(c0+c1xt−

13 +c2(xt−

13 )2+c3(xt−

13 )3+ · · ·

)t−

13 .

(95)

Remark 1 The exact solution of the (58) and (60) can bederived in the same way. Here, we do not list them forsimplicity.

Remark 2 Through the above discussion, it is easily seenthat the power series method [17,18] is very effective forhigher-order nonlinear or nonautonomous DEs. To thebest of our knowledge, the solutions obtained in this paperhave not been reported in previous literature. Thus, thesesolutions are new solutions of (1). And above all, thesepower series play an important role in the investigation ofphysical phenomena and other natural phenomenon.

6 Conclusions

This paper is essentially a continuation of an earlierpublished paper on pKdV equation where the mapping

method and Lie symmetry analysis were predominantlythe integration architectures [3]. This paper used a set ofdifferent integration tools, namely the G ′/G-expansionmethod, the exp-function method, and Lie symmetryanalysis in order to retrieve additional set of solutions tothe pKdV equation. The singular soliton solution was alsoobtained by the ansatz method. All these results are veryconcrete and consequently stand on a strong footing tolaunch further and deeper investigation into this equation,possibly in presence of perturbation terms.

The soliton perturbation theory will be developed forthis equation, in future, and consequently the adiabaticparameter dynamics will be displayed. In addition, infuture, the stochastic perturbation term will be includedand thus mean free velocity of the soliton will be deter-mined by formulating the corresponding Langevin equa-tion. In addition, the perturbed pKdV equation will beintegrated using the same tools or perhaps different toolsin order to extract further information. This is just a footin the door.

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