# Exact Traveling Wave Solutions of the Gardner Equation by the ImprovedtanΘϑ-Expansion Method and the Wave Ansatz Method

Authors: Hatıra Günerhan
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Nonlinear partial differential equations (NLPDEs) are an inevitable mathematical tool to explore a large variety of engineering and physical phenomena. Due to this importance, many mathematical approaches have been established to seek their traveling wave solutions. In this study, the researchers examine the Gardner equation via two well-known analytical approaches, namely, the improvedtanΘϑ-expansion method and the wave ansatz method. We derive the exact bright, dark, singular, and W-shaped soliton solutions of the Gardner equation. One can see that the methods are relatively easy and efficient to use. To better understand the characteristics of the theoretical results, several numerical simulations are carried out.


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## Body

## 1. Introduction

The Gardner equation is given as [1–3](1)ut+2αuux+3βu2ux+γuxxx=0,where α, β, and γ>0 are constant values. If the coefficient β>0, equation (1) admits two families of solitons and oscillating wave packets (called breathers), whereas if β<0, only one category of solitons exists [4].Equation (1) is also called the combined KdV-mKdV equation. In recent years, partial differential equations have become one of the most widely used fields of mathematics in various branches of science and engineering [5–12]. In this paper, the Gardner equation is examined by using the improved tanΘϑ-expansion method and the wave ansatz method. Recently, the improved tanΘϑ-expansion method (ITEM) [13–16] and the wave ansatz method [17–21] have been exploited to integrate a variety of nonlinear partial differential evolution equations (NLPDEs). In the past, several years ago, various methods have been proposed to obtain the solitary solution of this equation. In [22, 23], some kind of solutions of equation (1) were obtained. The Gardner equation (1) for α>0,β<0, and γ=0 has been studied in [24]. In [25], the authors have studied the attitudes of some solitary solitons for this equation. Many powerful analytic solution methods for solving nonlinear equation (1) have appeared in the open literature, such as the Hirota bilinear method [26], mapping method [25], similarity transformation method [27], generalized exponential rational function method, Jacobi elliptical solution finder method [28], fractional homotopy perturbation transform method [29], Coffey’s series expansion method [30], a unified method including solitary wave solutions, triangular periodic solutions, and Jacobi periodic wave solutions, as well as rational solutions [23], Wadati’s inverse scattering transform and Hirota methods [31, 32], consistent Riccati expansion (CRE) [33], planar dynamical systems approach method [34], Kudryashov method [35], Lie symmetry group method [36], ill-posedness results [37], classification of single traveling wave solutions [38], spectral collocation method [5], the Gardner equation with time-dependent coefficients and forcing term, have been investigated in [39, 40]. For more methods, we refer the readers to [22–40] and the references therein.This paper consists of several sections. In Section2, a brief description of the improved tanΘϑ-expansion method is reviewed. With the aid of this method, we will retrieve several sets of solutions for the Gardner equation in Section 3. In Section 4, the method of wave ansatz method is considered and the corresponding solutions in terms of bright, dark, singular, and W-shaped soliton solutions. Furthermore, some 3D profiles of acquired solutions are also depicted in this section. However, to the best of the authors’ knowledge, these two approaches have not been applied for equation (1) in previous studies. Finally, Section 5 concludes the paper.


## 2. The ImprovedtanΘϑ-Expansion Method

In this section, the main algorithm of the improvedtanΘϑ-expansion method (ITEM) is explained as follows:Step 1: using a new definition of wave variableϑ=μx−θt, a general partial differential equation (PDE) such as(2)Nu,ux,ut,uxx,⋯=0is transformed into an ordinary differential equation (ODE)(3)Nu,u′,−μu′,u″,μ2u″,⋯=0.Step 2: suppose that(4)uϑ=SΘ=∑k=0mAkp+tanΘϑ2k+∑k=1mBkp+tanΘϑ2−kcould be constructed as a solution of equation (2), where Ak0≤k≤m and Bk1≤k≤m with Am≠0,Bm≠0 are unknown parameters, so that Θ=Θϑsatisfies(5)Θ′ϑ=asinϑ+bcosϑ+c.Taking (5) into account, some solutions are as follows:Category 1: whilea2+b2−c2<0 and b−c≠0, then(6)Θϑ=−2tan−1−ab−c+c2−a2−b2b−ctanc2−a2−b22ϑ+C.Category 2: whilea2+b2−c2>0 and b−c≠0, then(7)Θϑ=−2tan−1−ab−c−b2+a2−c2b−ctanhb2+a2−c22ϑ+C.Category 3: whilea2+b2−c2>0,b≠0, andc=0, then(8)Θϑ=2tan−1ab+b2+a2btanhb2+a22ϑ+C.Category 4: whilea2+b2−c2<0,c≠0, and b=0, then(9)Θϑ=2tan−1−ac+c2−a2ctanc2−a22ϑ+C.Category 5: whilea2+b2−c2>0,b−c≠0 and a=0, then(10)Θϑ=2tan−1b+cb−ctanhb2−c22ϑ+C.Category 6: whilea=0 and c=0, it resulted that(11)Θϑ=tan−1e2bϑ+C−1e2bϑ+C+1,2ebϑ+Ce2bϑ+C+1.Category 7: whileb=0 and c=0, it resulted that(12)Θϑ=tan−12eaϑ+Ce2aϑ+C+1,e2aϑ+C−1e2aϑ+C+1.Category 8: whilea2+b2=c2, it resulted that(13)Θϑ=−2tan−1b+caϑ+C+2a2ϑ+C.Category 9: whilea=b=c=κ, it resulted that(14)Θϑ=2tan−1eκϑ+C−1.Category 10: whilea=c=κ and b=−κ, it resulted that(15)Θϑ=−2tan−1eκϑ+C−1+ekδϑ+C.Category 11: whilec=a, it resulted that(16)Θϑ=−2tan−1a+bebϑ+C−1a−bebϑ+C−1.Category 12: whilea=c, it resulted that(17)Θϑ=2tan−1b+cebϑ+C+1b−cebϑ+C−1.Category 13: whilec=−a, it resulted that(18)Θϑ=2tan−1ebϑ+C+b−aebϑ+C−b−a.Category 14: whileb=−c, it resulted that(19)Θϑ=−2tan−1aeaϑ+Cceaϑ+C−1.Category 15: whileb=0 and a=c, it resulted that(20)Θϑ=−2tan−1cϑ+C+2cϑ+C.Category 16: whilea=0 and b=c, it resulted that(21)Θϑ=2tan−1cϑ+C.Category 17: whilea=0 and b=−c, it resulted that(22)Θϑ=−2tan−11cϑ+C.Category 18: whilea=0 and b=0, it resulted that(23)Θϑ=cϑ+C.Category 19: whileb=c, it resulted that(24)Θϑ=2tan−1eaϑ+C−ca,whereAk,Bkk=1,2,⋯,m,a,b and c are the unknown parameters that need to be calculated. To determine the natural number m, one can use the homogeneous balance rule.Step 3: inserting the formal scheme of (4) into equation (3), and then setting each coefficient of tanΘϑ/2k,cotΘϑ/2k,k=0,1,2,⋯ to zero, we will arrive to a set of nonlinear equations for A0,Ak,Bkk=1,2,⋯,m,a,b,c,μ,θ and p.Step 4: solving the algebraic equations in Step 3, it resulted that substitutingA0,A1,B1,⋯,Am,Bm,μ,θ,p in (4).


## 3. Applications of the Gardner Equation via ITEM

In this section, we will examine ITEM for equation (1). To find the traveling solutions for equation (1), we define the wave transformation as u=Uϑ, where ϑ=μx−θt, μ≠0, and θ≠0 to be determined later. Taking u=uϑ into account allows us to rewrite equation (1) as the following ordinary differential equation:(25)−θu′+2αμuu′+3βμu2u′+γμ3u‴=0.Integrating (25) once with respect to ϑ and neglecting the resulted integration constants, we obtain (26)−θu+αμu2+βμu3+γμ3u″=0.Now, we apply the ITEM to obtain traveling wave solutions of the Gardner equation (1). According to this method, the solution of equation (26) can be written in the form of equation (4).Balancing theu″ and u3 in (26), by using homogeneous, one has(27)3m=m+2⟹m=1.Takingp=0 in (27), the solution structure is formulated as(28)uϑ=A0+A1tanΘϑ2+B1cotΘϑ2.Substituting equation (28) into equation (26) and following the necessary steps of ITEM, we have the following sets of coefficients for the nontrivial solutions of (1) as follows:Set 1:(29)μ=−2α3−σβa2+b2−c2,θ=2α3227β−σβa2+b2−c2,(30)A0=−α3β1+ab2+a2−c2,A1=α3βb−cb2+a2−c2,B1=0,(31)uϑ=A0+A1tanΘϑ2,wherea,b,andc are optional constants, and(32)ϑ=−2α3−βγb2+a2−c2x−22α327β−βγb2+a2−c2t,provided thatβ<0.Setting these values in categories 2, 6, 10, and 14 of Section2, respectively, we acquire the following solutions:(33)u1ϑ=α3βtanha2+b2−c22ϑ−1,wherea2+b2−c2>0 and ϑ is given by (32).(34)u2ϑ=−α3β1+tan12tan−1e2bϑ−1e2bϑ+1,2ebϑe2bϑ+1,whereϑ=−2α/3b−βγx−22α3/27bβ−βγt.(35)u3ϑ=−2α3βceaϑceaϑ−1,whereϑ=−2α/3b−βγx−22α3/27bβ−βγt.Set 2:(36)μ=−2α3−σβa2+b2−c2,θ=2α3227β−σβa2+b2−c2,(37)A0=−α3β1+ab2+a2−c2,A1=0,B1=−α3βb+cb2+a2−c2,(38)uϑ=A0+B1cotΘϑ2,wherea,b,andc are optional constants, and(39)ϑ=−2α3−βγb2+a2−c2x−22α327β−βγb2+a2−c2t,provided thatβ<0.Setting these values in categories 3, 5, and 6 of Section2, respectively, we obtain(40)u4ϑ=−α3β1+tanha2+b2/2ϑa2+b2+aa+a2+b2tanha2+b2/2ϑ,whereϑ=−2α/3−βγa2+b2x−22α3/27β−βγa2+b2t.(41)u5ϑ=−α3βtanhb2−c2/2ϑ+1tanhb2−c2/2ϑ,whereb2−c2>0 and ϑ=−2α/3−βγb2−c2x−22α3/27β−βγb2−c2t.(42)u6ϑ=−α3βtan12tan−1e2bϑ−1/e2bϑ+1,2ebϑ/e2bϑ+1−1tan12tan−1e2bϑ−1/e2bϑ+1,2ebϑ/e2bϑ+1,whereϑ=−2α/3b−βγx−22α3/27bβ−βγt.Set 3:(43)a=c=0,μ=2α6b−γβ,θ=−2α327bβ−γβ,(44)A0=−α3β,A1=−α6β,B1=−α6β,(45)uϑ=A0+A1tanΘϑ2+B1cotΘϑ2,whereb is an optional and β<0 must be held.Setting these values in categories 1, 6, and 13 of Section2, respectively, we obtain(46)u7=−α6βtanhb/2ϑ+12tanhb2ϑ,u8=−α6βtan12tan−1e2bϑ−1/e2bϑ+1,2ebϑ/e2bϑ+1+12tan12tan−1e2bϑ−1/e2bϑ+1,2ebϑ/e2bϑ+1,u9=−2α3βe2bϑe2bϑ−b2,where(47)ϑ=2α6b−γβ+2α327bβ−γβt.It is worth to note that one can find some more new exact solitary solutions from solutions (31), (38), and (45).


## 4. Applications of the Wave Ansatz Method

In what follows, and based on the wave ansatz method, several types of soliton wave solutions for the Gardner equation (1) are presented which is based on the wave ansatz method (see the previous study [24]).


### 4.1. Bright Soliton

To retrieve bright optical solutions of the Gardner equation, we use the following scheme [41]:(48)ux,t=AD+coshτn,where(49)τ=Bx−νt,where A, B, and ν are disposal parameters.Putting these values of (48) into (1) and some calculations, one obtains (50)−3βA3nBsinhτD+coshτ3n+1−2αA2nBsinhτD+coshτ2n+1−AB3γnn+2n+1D2−1sinhτD+coshτn+3+AB3Dγnn+12n+1sinhτD+coshτn+2−ABnB2γn2−νsinhτD+coshτn+1=0.From (50), equating the exponents 3n+1 and n+3 yields(51)n=1.So, solving equation (50) turns to the following equation:(52)−ABB2γ−νD+coshτ2−2AB−3B2Dγ+AαD+coshτ3−3ABA2β+2B2γD2−2B2γD+coshτ4=0.Due to the fact that the functions1/D+coshτj for j=2,3,and4 are linearly independent, equation (52) will introduce a system of equations for the unknown parameters. Solving this system, one gets(53)ν=γB2,(54)A=3γB2Dα,(55)D=2α18βγB2+4α2.Putting (55) into (54) yields(56)A=6γB218βγB2+4α2.Thus, for an arbitrary constantB, the 1-soliton solution of (1) is given by(57)ux,t=6γB22α+18βγB2+4α2coshBx−γB3t,provided (58)9B2βγ+2α2>0.


### 4.2. Dark Soliton

To retrieve dark solutions of the equation, we use the structure [41](59)ux,t=A+Btanhτn,where(60)τ=μx−νt,where A, B, μ, and ν are unknown parameters.Inserting (59) into (1) gives(61)−3βnμBA+Btanhτ3n+1+6βnμABA+Btanhτ3n−3βμnA2−B2BA+Btanhτ3n−1−2αnμBA+Btanhτ2n+1+4αnμABA+Btanhτ2n−2αμnA2−B2BA+Btanhτ2n−1−γnμ3n+2n+1B3A+Btanhτn+3+6γnμ3An+12B3A+Btanhτn+2−μn3γμ2n5A2−B2n+1+2γμ23A2−B2−B2νB3A+Btanhτn+1+2Aμn2γμ2n25A2−3B2+2γμ2A2−B2−B2νB3A+Btanhτn−μnA2−B23γμ2n5A2−B2n−1+2γμ23A2−B2−B2νB3A+Btanhτn−1+6γnμ3An−12A2−B22B3A+Btanhτn−2−γnμ3n−1n−2A2−B23B3A+Btanhτn−3=0.After some algebra, we conclude that(62)μ=−2γβγB,(63)ν=3B2β2−α23β,(64)A=−α3β.The dark soliton solution of equation (1) is obtained as (65)ux,t=A+Btanhμx−νtn,to exist, from (62), the following restriction is obtained(66)β<0.


### 4.3. Singular Soliton

To extract the singular solitons of the Gardner equation (1), the following structure is examined by [41](67)ux,t=AD+sinhτn,with τ is defined by (49).Substituting (67) into (1), we obtain(68)−3βA3nBcoshτD+sinhτ3n+1−2αA2nBcoshτD+sinhτ2n+1−AB3γnn+2n+1D2+1coshτD+sinhτn+3+AB3Dγnn+12n+1coshτD+sinhτn+2−ABnB2γn2−νcoshτD+sinhτn+1=0.Considering the balancing principle indicates (51), vanishing all the coefficients of coshτ/D+sinhτj for j=2,3,and4 to zero in (68), one gets(69)ν=B2γ,(70)A=6B2γ−18B2βγ−4α2,(71)D=2α−18B2βγ−4α2.From (70) and (71), one concludes that if (72)9B2βγ+2α2<0holds, the soliton solution(73)ux,t=AD+sinhBx−B3γtis achieved as a singular solution for the Gardner equation (1). In this solution, A is given by (70), D is shown in (71), and B is an optional constant chosen in such a way that (72) holds.


### 4.4.W-Shaped Soliton

Now, we explore some exact solutions of the Gardner equation in the form of [41](74)ux,t=A+Dsechτ,where τ is the same as (49).Substituting (74) into (1), we, respectively, obtain(75)−BDsechτtanhτ−6B2γ+3D2βsech2τ+6ADβ+2Dαsechτ+3A2β+B2γ+2Aα−ν=0.Now, equation (75) holds whenever we have(76)ν=3B2βγ−α23β,(77)A=−α3β,(78)D=±B2γβ,which will be valid for(79)β<0.Consequently, the solution (74) with sign “+” in equation (78) is obtained as(80)ux,t=−α3β+B2γβsechBx−3B3βγ−Bα23βt.Moreover, for bright soliton pulse and with sign “−” in equation (78), we obtain(81)ux,t=−α3β−B2γβsechBx−3B3βγ−Bα23βt,for a W-shaped soliton pulse, where B is an optional constant.The correctness of all given solutions has been confirmed with Maple by substituting them back into the original equation.To better understand the characteristics of the soliton solution, we plot equations (57), (65), (73) (80), and (81) of equation (1) by taking different values of parameters α,β,andγ within the interval x,t∈−10,10×−10,10 in Figures 1–7, respectively.Figure 1
The 3d surface of equation (57) for the values α=β=γ=1 and B=1.Figure 2
The 3d surface of equation (57) for the values α=γ=1,β=−1, and B=0.3.Figure 3
The 3d surface of equation (65) for the values α=γ=1,β=−1, and B=1.Figure 4
The 3d surface of equation (65) for the values α=γ=1,β=−1, and B=0.3.Figure 5
The 3d surface of equation (73) for the values α=0.5,β=−1,γ=1, and B=2.Figure 6
The 3d surface of equation (80) for the values α=β=1,γ=2, and B=1.Figure 7
The 3d surface of equation (73) for the values α=0.5,β=−1,γ=1, and B=2.


## 4.1. Bright Soliton

To retrieve bright optical solutions of the Gardner equation, we use the following scheme [41]:(48)ux,t=AD+coshτn,where(49)τ=Bx−νt,where A, B, and ν are disposal parameters.Putting these values of (48) into (1) and some calculations, one obtains (50)−3βA3nBsinhτD+coshτ3n+1−2αA2nBsinhτD+coshτ2n+1−AB3γnn+2n+1D2−1sinhτD+coshτn+3+AB3Dγnn+12n+1sinhτD+coshτn+2−ABnB2γn2−νsinhτD+coshτn+1=0.From (50), equating the exponents 3n+1 and n+3 yields(51)n=1.So, solving equation (50) turns to the following equation:(52)−ABB2γ−νD+coshτ2−2AB−3B2Dγ+AαD+coshτ3−3ABA2β+2B2γD2−2B2γD+coshτ4=0.Due to the fact that the functions1/D+coshτj for j=2,3,and4 are linearly independent, equation (52) will introduce a system of equations for the unknown parameters. Solving this system, one gets(53)ν=γB2,(54)A=3γB2Dα,(55)D=2α18βγB2+4α2.Putting (55) into (54) yields(56)A=6γB218βγB2+4α2.Thus, for an arbitrary constantB, the 1-soliton solution of (1) is given by(57)ux,t=6γB22α+18βγB2+4α2coshBx−γB3t,provided (58)9B2βγ+2α2>0.


## 4.2. Dark Soliton

To retrieve dark solutions of the equation, we use the structure [41](59)ux,t=A+Btanhτn,where(60)τ=μx−νt,where A, B, μ, and ν are unknown parameters.Inserting (59) into (1) gives(61)−3βnμBA+Btanhτ3n+1+6βnμABA+Btanhτ3n−3βμnA2−B2BA+Btanhτ3n−1−2αnμBA+Btanhτ2n+1+4αnμABA+Btanhτ2n−2αμnA2−B2BA+Btanhτ2n−1−γnμ3n+2n+1B3A+Btanhτn+3+6γnμ3An+12B3A+Btanhτn+2−μn3γμ2n5A2−B2n+1+2γμ23A2−B2−B2νB3A+Btanhτn+1+2Aμn2γμ2n25A2−3B2+2γμ2A2−B2−B2νB3A+Btanhτn−μnA2−B23γμ2n5A2−B2n−1+2γμ23A2−B2−B2νB3A+Btanhτn−1+6γnμ3An−12A2−B22B3A+Btanhτn−2−γnμ3n−1n−2A2−B23B3A+Btanhτn−3=0.After some algebra, we conclude that(62)μ=−2γβγB,(63)ν=3B2β2−α23β,(64)A=−α3β.The dark soliton solution of equation (1) is obtained as (65)ux,t=A+Btanhμx−νtn,to exist, from (62), the following restriction is obtained(66)β<0.


## 4.3. Singular Soliton

To extract the singular solitons of the Gardner equation (1), the following structure is examined by [41](67)ux,t=AD+sinhτn,with τ is defined by (49).Substituting (67) into (1), we obtain(68)−3βA3nBcoshτD+sinhτ3n+1−2αA2nBcoshτD+sinhτ2n+1−AB3γnn+2n+1D2+1coshτD+sinhτn+3+AB3Dγnn+12n+1coshτD+sinhτn+2−ABnB2γn2−νcoshτD+sinhτn+1=0.Considering the balancing principle indicates (51), vanishing all the coefficients of coshτ/D+sinhτj for j=2,3,and4 to zero in (68), one gets(69)ν=B2γ,(70)A=6B2γ−18B2βγ−4α2,(71)D=2α−18B2βγ−4α2.From (70) and (71), one concludes that if (72)9B2βγ+2α2<0holds, the soliton solution(73)ux,t=AD+sinhBx−B3γtis achieved as a singular solution for the Gardner equation (1). In this solution, A is given by (70), D is shown in (71), and B is an optional constant chosen in such a way that (72) holds.


## 4.4.W-Shaped Soliton

Now, we explore some exact solutions of the Gardner equation in the form of [41](74)ux,t=A+Dsechτ,where τ is the same as (49).Substituting (74) into (1), we, respectively, obtain(75)−BDsechτtanhτ−6B2γ+3D2βsech2τ+6ADβ+2Dαsechτ+3A2β+B2γ+2Aα−ν=0.Now, equation (75) holds whenever we have(76)ν=3B2βγ−α23β,(77)A=−α3β,(78)D=±B2γβ,which will be valid for(79)β<0.Consequently, the solution (74) with sign “+” in equation (78) is obtained as(80)ux,t=−α3β+B2γβsechBx−3B3βγ−Bα23βt.Moreover, for bright soliton pulse and with sign “−” in equation (78), we obtain(81)ux,t=−α3β−B2γβsechBx−3B3βγ−Bα23βt,for a W-shaped soliton pulse, where B is an optional constant.The correctness of all given solutions has been confirmed with Maple by substituting them back into the original equation.To better understand the characteristics of the soliton solution, we plot equations (57), (65), (73) (80), and (81) of equation (1) by taking different values of parameters α,β,andγ within the interval x,t∈−10,10×−10,10 in Figures 1–7, respectively.Figure 1
The 3d surface of equation (57) for the values α=β=γ=1 and B=1.Figure 2
The 3d surface of equation (57) for the values α=γ=1,β=−1, and B=0.3.Figure 3
The 3d surface of equation (65) for the values α=γ=1,β=−1, and B=1.Figure 4
The 3d surface of equation (65) for the values α=γ=1,β=−1, and B=0.3.Figure 5
The 3d surface of equation (73) for the values α=0.5,β=−1,γ=1, and B=2.Figure 6
The 3d surface of equation (80) for the values α=β=1,γ=2, and B=1.Figure 7
The 3d surface of equation (73) for the values α=0.5,β=−1,γ=1, and B=2.


## 5. Concluding Remarks and Observations

In this research, we exerted the improvedtanΘϑ-expansion and wave ansatz method as two useful mathematical tools to construct solitary solutions for the Gardner equation. These two methods for equation (1) have not been reported in the literature so far, to achieve the category of bright, dark, singular, and W-shaped soliton solutions. For a better understanding of the solutions, numerical results have also been included. On the other hand, the results are quite reliable for solving the Gardner equation. The results attest to the efficiency of the proposed method. These two powerful methods can also be applied to other nonlinear partial differential equations with time-dependent coefficients and their systems.



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