# The Improved Generalized tanh-coth Method Applied to Sixth-Order Solitary Wave Equation

Authors: M. Torvattanabun; J. Simmapim; D. Saennuad; T. Somaumchan
Journal: Journal of Mathematics
 (2017)

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## Abstract

The improved generalized tanh-coth method is used in nonlinear sixth-order solitary wave equation. This method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. The new exact solutions consisted of trigonometric functions solutions, hyperbolic functions solutions, exponential functions solutions, and rational functions solutions. The numerical results were obtained with the aid of Maple.


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## Body

## 1. Introduction

Nonlinear evolution equations (NLEEs) play an important role in various branches of scientific disciplines, such as fluid mechanics, optical fibers, plasma physics, chemical physics, biology, solid state physics, oceans engineering, and many other scientific applications. The solitary wave was introduced by Russell more than a century ago [1]. In the past years, many powerful methods for finding exact solutions of NLEEs have been proposed, such as the generalized (G′/G)-expansion method [2], the tanh-coth method [3], the modified sine-cosine method [4], the generalized unified method [5], the improved F-expansion method [6], the generalized Kudryashov method [7], the generalized Riccati equation mapping method [8], the modified Kudryashov method [9], the exp⁡(-ϕ(ξ)) method [10], the lie symmetry analysis method [11], the first integral method [12], and the consistent Riccati expansion [13].Another powerful method has been presented by Malfliet [14], who had customized the tanh technique and called the tanh method. In 2002, Fan and Hona [15] extended the tanh method which is called the extended tanh method, by using U(ξ)=∑k=0MakYk as traveling wave solutions. In 2007, Wazwaz [3] extended and improved this method which is called the tanh-coth method. In this method U(ξ)=∑k=0MakYk+∑k=1MbkY-k is used as traveling wave solutions. In 2008 Gómez and Salas [16] improved and generalized this method which is called the improved generalized tanh-coth method, by using U(ξ)=∑i=0Maiϕ(ξ)i+∑i=M+12Maiϕ(ξ)M-i, where ϕ is the solution of the generalized Riccati equation. Afterwards, several researchers applied this method to obtain new exact solutions for nonlinear PDEs [17–20].In 2017, Christou [21] studies solitons occurring in electrical nonlinear transmission lines; there are called electrical solitons. The problem is applied to Ohm’s law of solid state physics by using Taylor-series expansions.In this paper, we focus on using the improved generalized tanh-coth method for finding exact solutions of the sixth-order solitary wave equation:(1)Vtt=aV+bV2-cV3+gVxx+hVxxxxxx,which was proposed by Christou [21] and (2)a=h2C0Lb=12F0c=13F02g=12δ+1h412C0Lh=60δ+1h6360C0L.In Section 2, we briefly describe the improved generalized tanh-coth method; in Section 3, the improved generalized tanh-coth method is applied to the sixth-order solitary wave equations. The last section is short summary and discussion.


## 2. The Improved Generalized tanh-coth Method

Consider the nonlinear partial differential equation in the variablesx and t(3)P1u,ux,ut,ux,t,ux,x,ut,t,…=0.The traveling wave transformation is given by (4)ux,t=Uξ,ξ=x-λt+ξ0,where λ is the wave speed. We can reduce (3) to the ordinary differential equation (5)P2U,U′,U′′,…=0.According to the improved generalizedtanh-coth method, we seek the exact solution of (3) that can be expressed in the following form: (6)Uξ=∑i=0Maiϕξi+∑i=M+12MaiϕξM-i,where M is a positive integer that will be determined by balancing the highest order derivative term with the highest order nonlinear term. The coefficients ai are constants (aM≠0 and a-M≠0) that are determined later while the new variable ϕ(ξ) is the solution to the generalized Riccati equation (7)ϕ′ξ=α+βϕξ+γϕξ2,where α, β, and γ are constants. The solutions of generalized Riccati equation are given by [18].Case 1 (exponential function solutions).
Whenα=0(8)ϕξ=β-γ+βe-βξ.Case 2 (trigonometric and hyperbolic function solution).
Whenβ=0, (9)ϕξ=αγγtan⁡αγξ,α>0,γ>0,αγγtanh⁡αγξ,α>0,γ<0,-αγγtanh⁡--αγξ,α<0,γ>0,αγγtan⁡-αγξ,α<0,γ<0.Case 3 (exponential function solutions).
Whenγ=0, (10)ϕξ=-α+βeβξβ.Case 4 (rational function solution).
Whenα=β=0, (11)ϕξ=-1γξ.Case 5 (rational function solution).
Whenβ2≠0 and β2=4αγ, (12)ϕξ=-2αβξ+2β2ξ.Case 6 (trigonometric function solution).
Whenβ2<4αγ and γ≠0, (13)ϕξ=4αγ-β2tan⁡1/24αγ-β2ξ-β2γ.Case 7 (hyperbolic function solution).
Whenβ2>4αγ and γ≠0, (14)ϕξ=β2-4αγtanh⁡1/2β2-4αγξ-β2γ.
We substitute (6) into (5) and collect all terms with the same order of ϕj(ξ); we get a polynomial in ϕ(ξ). Equating each coefficient of the polynomial to zero, we will give a system of algebraic equations involving the parameters ai, α, γ, and β. Solving the system, we can construct a variety of exact solutions of (5).


## 3. The Improved Generalized tanh-coth Method Applied to Sixth-Order Solitary Wave Equation

We use the wave transformationsV(x,t)=V(ξ), ξ=x-λt+ξ0, to reduce (1) to the following ODE: (15)λ2-aV′′ξ-bV2ξ′′+cV3ξ′′+gV′′′′ξ-hV′′′′′′ξ=0.Balancing the highest order term V′′′′′′ with the highest order nonlinear term V3′′′ in (13), we have(16)M+6=3M+2;then M=2. Consequently, we set (17)Vξ=a0+a1ϕξ+a2ϕξ2+a3ϕξ-1+a4ϕξ-2.Using (6) and (14) in (13) and equating all the coefficients of power of ϕ(ξ) to be zero, we obtain a system of algebraic equations in the unknowns a0, a1, a2, a3, a4, λ, α, β, and γ. (18)ϕξ-8:-5040α6za4+42α2ca43=0,ϕξ-7:-720α6za3-19440α5βza4+90α2ca3a42+8αβca43=0,ϕξ-6:165ca3a42αβ+72ca43αγ+60ca0a42α2+60ca32α2a4-29400za4α4β2-13440za4α5γ-2520za3α5β+36ca43β2-20ba42α2-120ga4α4=0,ϕξ-5:75ca3a42β2+66ca43βγ+72ca0a3a4α2+108ca0a42αβ+150ca3a42αγ+108ca32a4αβ-38640za4α4βγ-24ba3a4α2-36ba42αβ+36ca1a42α2-21840za4α3β3-3360za3α4β2-1680za3α5γ-336ga4α3β+12ca33α2-24ga3α4=0,ϕξ-4:96ca0a42αγ+36ca1a3a4α2+135ca3a42βγ-4200za3α4βγ-40152za4α3β2γ-42ba3a4αβ+96ca32a4αγ+63ca1a42αβ-12096za4α4γ2-8106za4α2β4-2100za3α3β3-240ga4α3γ-330ga4α2β2-60ga3α3β+21ca33αβ+48ca32a4-12ba0a4α2-32ba42αγ+18ca02a4α2+18ca0a32α2+48ca0a42β2+18ca2a42α2-6aa4α2+6λ2a4α2+30ca43γ2-6ba32α2-16ba42β2+126ca0a3a4αβ=0,ϕξ-3:54ca0a3a4β2+84ca0a42βγ+12ca2a3a4α2+30ca2a42αβ-20ba0a4αβ-440ga4α2βγ-3584za3α3β2γ-17920za4α2β3γ-22960za4α3βγ2-36ba3a4αγ+30ca02a4αβ+54ca1a42αγ+84ca32a4βγ+12ca0a1a4α2+30ca0a32αβ-1330za4β5α-1232za3α4γ2-602za3α2β4-130ga4αβ3-40ga3α3γ-50ga3α2β2+18ca33αγ+60ca3a42γ2+10a4λ2αβ-4ba0a3α2-10aa4αβ-4ba1a4α2-10ba32αβ-18ba3a4β2-28ba42βγ+6ca02a3α2+2ca1a42β2+6ca1a32α2-2aa3α2+2a3λ2α2+9ca33β2+108ca0a3a4αγ+60ca1a3a4αβ=0,ϕξ-2:24ca0a32αγ+9ca1a32αβ+24ca1a3a4β2+24ca2a42αγ-6ba0a3αβ-16ba0a4αγ-232ga4αγβ2-60ga3α2βγ-116za3α2βγ-1848za3α3βγ2-13320za4α2β2γ2-3096za4αβ2γ-30ba3a4βγ-6ba1a4αβ+9ca02a3αβ+24ca02a4αγ+45ca1a42βγ-3968za4α3γ3-63za3β5α-136ga4α2γ2-15ga3αβ3+15ca33βγ+36ca32a4γ2+3a3λ2αβ+8a4λ2αγ-8ba0a4β2-3aa3αβ-8aa4αγ-8ba32αγ+12ca02a4β2+12ca0a32β2+36ca0a42γ2+12ca2a42β2-4aa4β2+4a4λ2β2-4a32bβ2-12ba42γ2-64za4β6-16ga4β4+90ca0a3a4βγ+18ca0a1a4αβ+48ca1a3a4αγ+18ca2a3a4αβ=0,ϕξ-1:18ca0a32βγ+36ca0a3a4γ2+6ca1a32αγ+6ca2a3a4β2+18ca2a42βγ-4ba0a3αγ-12ba0a4βγ-120ga4αβγ2-22ga3αβ2γ-720za3α2β2γ2-114za3αβ4γ-3696za4α2βγ3-2352za4αβ3γ2-4ba1a4αγ+6ca02a3αγ+18ca02a4βγ+6ca0a1a4β2-126za4β5γ-22za3α3γ3-30ga4β3γ-16ga3α2γ2+2a3λ2αγ+6a4λ2βγ-2ba0a3β2-2aa3αγ-6aa4βγ-2ba1a4β2-6ba32βγ-12ba3a4γ2+3ca02a3β2+18ca1a42γ2+3ca1a32β2-aa3β2+a3λ2β2+6ca33γ2-za3β6-ga3β4+12ca0a1a4αγ+36ca1a3a4βγ+12ca2a3a4αγ=0,ϕξ0:a3λ2βγ+a1λ2βα+6ca22a4α2+6ca2a42γ2+6ca0a12α2+6ca0a32γ2+6ca02a4γ2+6ca02a2α2-62za2β4α2-272za2γ2α4-za1β5α-62za4β4γ2-22za4α2γ4-za3β5γ-16ga4αγ3-14ga4β2γ2-ga3β3γ-ga1β3α-14ga2β2α2-16ga2γα3-4ba0a4γ2-4ba0a2α2-aa3βγ-aa1βα-584za2β2α3γ-52za1β3γα2-136za1βγ2α3-136za3α2βγ3-52za3αβ3γ2-584za4αβ2γ3-8ga1βγα2-8ga3αβγ2-2ba0a3βγ-2ba0a1βα-2ba1a4βγ-2ba2a3αβ+3ca02a3βγ+3ca02a1βα+12ca1a3a4γ2+3ca12a3αβ+12ca1a2a3α2+3ca1a32βγ-2ba12α2-2ba32γ2-2aa4γ2-2aa2α2+2a4λ2γ2+2a2λ2α2+6ca0a1a4βγ+6ca0a2a3αβ+6ca1a2a4αβ+6ca2a3a4βγ=0,ϕξ:18ca22a4αβ-4ba2a3αγ-4ba0a1γα+6ca02a1γα+18ca02a2αβ+6ca0a2a3β2+18ca0a12αβ+36ca0a1a2α2+6ca12a3αγ+6ca1a2a4β2-12ba0a2αβ-22ga1β2γα-120ga2βα2γ-114za1β4γα-720za1β2γ2α2-2352za2β3γα2-3696za2βγ2α3-6ba12αβ-12ba1a2α2-2ba2a3β2+3ca02a1β2+18ca22a3α2+3ca12a3β2+2a1λ2γα+6a2λ2αβ-126za2β5α-272za1γ3α3-16ga1γ2α2-30ga2β3α-2ba0a1β2-2aa1γα-6aa2αβ+6ca13α2-za1β6-ga1β4-aa1β2+a1λ2β2+12ca0a2a3αγ+36ca1a2a3αβ+12ca1a2a4αγ=0,ϕξ2:24ca1a2a3β2+9ca12a3βγ+24ca22a4αγ-16ba0a2αγ-60ga1βαγ2-232ga2β2αγ-1848za1βγ3α2-1176za1β3αγ2-3096za2β4γα-13320za2β2γ2α2-30ba1a2αβ-6ba2a3βγ-6ba0a1βγ+9ca02a1βγ+24ca02a2αγ+45ca22a3αβ+24ca0a12αγ-3968za2γ3α3-63za1β5γ-15ga1β3γ-136ga2γ2α2+3a1λ2βγ+8a2λ2αγ-8ba0a2β2-3aa1βγ-8aa2αγ-8ba12αγ+12ca02a2β2+36ca0a22α2+12ca0a12β2+15ca13αβ+36ca12a2α2+12ca22a4β2-4aa2β2+4a2λ2β2-12ba22α2-4ba12β2-64za2β6-16ga2β4+90ca0a1a2αβ+18ca0a2a3βγ+48ca1a2a3αγ+18ca1a2a4βγ=0,ϕξ3:6ca12a3γ2+60ca1a22α2+10a2λ2βγ-1330za2β5γ-602za1β4γ2-1232za1γ4α2-50ga1β2γ2-40ga1αγ3-130ga2β3γ-4ba0a1γ-10aa2βγ-10ba12βγ-18ba1a2β2-28ba22αβ-4ba2a3γ2+6ca02a1γ2+27ca22a3β2+18ca13αγ+108ca0a1a2αγ+60ca1a2a3βγ-20ba0a2βγ-440ga2βαγ2-3584za1β2αγ3-22960za2βγ3α2+12ca1a2a4γ2+84ca12a2αβ+30ca22a4βγ-17920za2β3αγ2+54ca22a3αγ-36ba1a2αγ+30ca02a2βγ+30ca0a12βγ+54ca0a1a2β2+84ca0a22αβ+12ca0a2a3γ2-2aa1γ2+2a1λ2γ2+9ca13β2=0,ϕξ4:135ca1a22αβ+36ca1a2a3γ2+96ca12a2αγ-4200za1βγ4α-40152za2β2αγ3+63ca22a3βγ-42ba1a2βγ+96ca0a22αγ-8106za2β4γ2-12096za2γ4α2-2100za1β3γ3-60ga1βγ3-330ga2β2γ2-240ga2γ3α-12ba0a2γ2-32ba22αγ+18ca02a2γ2+18ca0a12γ2+48ca0a22β2+21ca13βγ+48ca12a2β2+18ca22a4γ2-6aa2γ2-6aa2γ2+6a2λ2γ2+30ca23α2-6ba12γ2-16ba22β2+126ca0a1a2βγ=0,ϕξ5:-336ga2βγ3-24ba1a2γ2-36ba22βγ+36ca22a3γ2+75ca1a22β2+66ca23αβ-21840za2β3γ3-3360za1β2γ4-1680za1γ5α+72ca0a1a2γ2+108ca0a22βγ+150ca1a22αγ+108ca12a2βγ-38640za2βγ4α+12ca13γ2-24ga1γ4=0,ϕξ6:60ca0a22γ2+60ca12a2γ2+72ca23αγ-29400za2β2γ2-13440za2γ5α-2520za1βγ5+165ca1a22βγ+36ca23β2-20ba22γ2-120ga2γ4=0,ϕξ7:-19440βγ5za2-720a1zγ6+8ca23βγ+90ca1a22γ2=0,ϕξ8:-5040a2zγ6+42ca23γ2=0.Solving the system of algebraic equations with the aid of Maple, using (18), we obtain the following results.First Set
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720
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.Case 1.
Whenβ=0, α>0, and γ>0, ϕ(ξ)=αγ/γtan⁡(αγξ), the periodic solutions of (1) are (20)V1x,t=1304030hcγαh+30hcg+10bhhc+230hcγα2ccot2⁡αγx+λt+ξ0,V2x,t=1304030hcγαh+30hcg+10bhhc+230hcγα2ccot2⁡αγx-λt+ξ0,where(21)λ=130-30ch720α2γ2ch2-30ach-10b2h+3cg2ch.Case 2.
Whenβ=0, α<0, and γ>0, ϕ(ξ)=-αγ/γtanh⁡(--αγξ), the periodic solutions of (1) are (22)V3x,t=1304030hcγαh+30hcg+10bhhc-230hcαγccoth2⁡--αγx+λt+ξ0,V4x,t=1304030hcγαh+30hcg+10bhhc-230hcαγccoth2⁡--αγx-λt+ξ0,where (23)λ=130-30ch720α2γ2ch2-30ach-10b2h+3cg2ch.Case 3.
Whenγ=0, ϕ(ξ)=(-α+βeβξ)/β, the combined formal single kink solutions of (1) are (24)V5x,t=130530hcβ2h+30hcg+10bhhc+2β230hcαc-α+βeβx+λt+ξ0+2β230hcα2c-α+βeβx+λt+ξ02,V6x,t=130530hcβ2h+30hcg+10bhhc+2β230hcαc-α+βeβx-λt+ξ0+2β230hcα2c-α+βeβx-λt+ξ02,where (25)λ=130-30ch45β4ch2-30ach-10b2h+3cg2ch.Case 4.
Whenβ2≠0 and β2=4αγ, ϕ(ξ)=-2α(βξ+2)/β2ξ, the rational solutions of (1) are (26)V7x,t=1301530hcβ2h+30hcg+10bhch-β330hcx-λt+ξ0cβx-λt+ξ0+2+β530hcx-λt+ξ022cβx-λt+ξ0+22,V8x,t=1301530hcβ2h+30hcg+10bhch-β330hcx+λt+ξ0cβx+λt+ξ0+2+β530hcx+λt+ξ022cβx+λt+ξ0+22,where (27)λ=130-30ch-30ach-10b2h+3cg2ch.Case 5.
Whenβ2<4αγ and γ≠0, ϕ(ξ)=(4αγ-β2tan⁡((1/2)4αγ-β2ξ)-β)/2γ, the periodic solutions of (1) are (28)V9x,t=1304030hcγαh+530hcβ2h+30hcg+10bhhc+2β30hcαc2γ4αγ-β2tan⁡1/24αγ-β2x+λt+ξ0-β+230hcα2c4γ24αγ-β2tan⁡1/24αγ-β2x+λt+ξ0-β2,V10x,t=1304030hcγαh+530hcβ2h+30hcg+10bhhc+2β30hcαc2γ4αγ-β2tan⁡1/24αγ-β2x-λt+ξ0-β+230hcα2c4γ24αγ-β2tan⁡1/24αγ-β2x-λt+ξ0-β2,where (29)λ=130-30ch720α2γ2ch2-360αγβ2ch2+45β4ch2-30ach-10b2h+3cg2ch.Case 6.
Whenβ2>4αγ and γ≠0, ϕ(ξ)=(β2-4αγtanh⁡((1/2)β2-4αγξ)-β)/2γ, the periodic solutions of (1) are(30)V11x,t=1304030hcγαh+530hcβ2h+30hcg+10bhhc+2β30hcαc2γβ2-4αγtanh⁡1/2β2-4αγx+λt+ξ0-β+230hcα2c4γ2β2-4αγtanh⁡1/2β2-4αγx+λt+ξ0-β2,V12x,t=1304030hcγαh+530hcβ2h+30hcg+10bhhc+2β30hcαc2γβ2-4αγtanh⁡1/2β2-4αγx-λt+ξ0-β+230hcα2c4γ2β2-4αγtanh⁡1/2β2-4αγx-λt+ξ0-β2,where(31)λ=130-30ch720α2γ2ch2-360αγβ2ch2+45β4ch2-30ach-10b2h+3cg2ch.Second Set
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720
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.Case 1.
Whenα=0, ϕ(ξ)=β/(-γ+βe-βξ), the combined formal single kink solutions of (1) are (33)V13x,t=130530hcβ2h+30hcg+10bhhc+2β230hcγc-γ+βe-βx+λt+ξ0+2β230hcγ2c-γ+βe-βx+λt+ξ02,V14x,t=130530hcβ2h+30hcg+10bhhc+2β230hcγc-γ+βe-βx-λt+ξ0+2β230hcγ2c-γ+βe-βx-λt+ξ02,where (34)λ=130-30ch45β4ch2-30ach-10b2h+3cg2ch.Case 2.
Whenβ=0, α>0, and γ>0, ϕ(ξ)=αγ/γtan⁡(αγξ), the periodic solutions of (1) are (35)V15x,t=1304030hcγαh+30hcg+10bhhc+230hcαγctan2⁡αγx+λt+ξ0,V16x,t=1304030hcγαh+30hcg+10bhhc+230hcαγctan2⁡αγx-λt+ξ0,where (36)λ=130-30ch720α2γ2ch2-30ach-10b2h+3cg2ch.Case 3.
Whenβ=0, α<0, and γ>0, ϕ(ξ)=-αγ/γtanh⁡(--αγξ), the periodic solutions of (1) are (37)V17x,t=1304030hcγαh+30hcg+10bhhc+230hc-αγ2ctanh2⁡--αγx+λt+ξ0,V18x,t=1304030hcγαh+30hcg+10bhhc+230hc-αγ2ctanh2⁡--αγx-λt+ξ0,where (38)λ=130-30ch720α2γ2ch2-30ach-10b2h+3cg2ch.Case 4.
Whenα=β=0, ϕ(ξ)=-1/γξ, the rational solutions of (1) are (39)V19x,t=13030hcg+10bhhc+230hcγ2cγx+λt+ξ02,V20x,t=13030hcg+10bhhc+230hcγ2cγx-λt+ξ02,V21x,t=13030hcg+10bhhc-230hcγ2cγx+λt+ξ02,V22x,t=13030hcg+10bhhc-230hcγ2cγx-λt+ξ02,V23x,t=-13030hcg+10bhhc+230hcγ2cγx+λt+ξ02,V24x,t=-13030hcg+10bhhc+230hcγ2cγx-λt+ξ02,V25x,t=-13030hcg+10bhhc-230hcγ2cγx+λt+ξ02,V26x,t=-13030hcg+10bhhc-230hcγ2cγx-λt+ξ02,where (40)λ=130-30ch-30ach-10b2h+3cg2ch.Case 5.
Whenβ2≠0 and β2=4αγ, ϕ(ξ)=-2α(βξ+2)/β2ξ, the rational solutions of (1) are (41)V27x,t=1301530hcβ2h+10bh+30chgch-β30chβx-λt+ξ0+2cx-λt+ξ0+4β30chβx-λt+ξ0+22cx-λt+ξ02,V28x,t=1301530hcβ2h+10bh+30chgch-β30chβx+λt+ξ0+2cx+λt+ξ0+4β30chβx+λt+ξ0+22cx+λt+ξ02,where (42)λ=130-30ch-30ach-10b2h+3cg2ch.Case 6.
Whenβ2<4αγ and γ≠0, ϕ(ξ)=(4αγ-β2tan⁡((1/2)4αγ-β2ξ)-β)/2γ, the periodic solutions of (1) are (43)V29x,t=1304030hcγαh+530hcβ2h+30hcg+10bhhc+β30hcc4αγ-β2tan⁡124αγ-β2x+λt+ξ0-β+30hc2c4αγ-β2tan⁡124αγ-β2x+λt+ξ0-β2,V30x,t=1304030hcγαh+530hcβ2h+30hcg+10bhhc+β30hcc4αγ-β2tan⁡124αγ-β2x-λt+ξ0-β+30hc2c4αγ-β2tan⁡124αγ-β2x-λt+ξ0-β2,where(44)λ=130-30ch720α2γ2ch2-360αγβ2ch2+45β4ch2-30ach-10b2h+3cg2ch.Case 7.
Whenβ2>4αγ and γ≠0, ϕ(ξ)=(β2-4αγtanh⁡((1/2)β2-4αγξ)-β)/2γ, the periodic solutions of (1) are (45)V31x,t=1304030hcγαh+530hcβ2h+30hcg+10bhhc+β30hccβ2-4αγtanh⁡12β2-4αγx+λt+ξ0-β+30hc2cβ2-4αγtanh⁡12β2-4αγx+λt+ξ0-β2,V32x,t=1304030hcγαh+530hcβ2h+30hcg+10bhhc+β30hccβ2-4αγtanh⁡12β2-4αγx-λt+ξ0-β+30hc2cβ2-4αγtanh⁡12β2-4αγx-λt+ξ0-β2,where(46)λ=130-30ch720α2γ2ch2-360αγβ2ch2+45β4ch2-30ach-10b2h+3cg2ch.


## 4. Summary and Discussion

In conclusion, the improved generalized tanh-coth method is proposed to obtain more general exact solutions of the sixth-order solitary wave equations. It can be observed that the method used is a powerful and more general tool for finding the exact solutions. By using the method, we have obtained new exact solutions in terms of the hyperbolic functions, the trigonometric functions, the exponential functions, and the rational functions. The results reveal that the improved generalized tanh-coth method has significant effects on the wave behavior and can also be applied to other NLEEs in mathematical physics.The solutions of generalized Riccati equation (7): there are more than seven cases. We can extend the improved generalized tanh-coth method by considering expanding solutions of generalized Riccati equation in this method.



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*Source: 8751097-2017-07-17.xml*