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Open Physics

formerly Central European Journal of Physics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations

Dianchen Lu / Aly R. Seadawy
  • Corresponding author
  • Mathematics Department, Faculty of Science, Taibah University Al-Madinah Al-Munawarah Saudi Arabia
  • Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef Egypt
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  • De Gruyter OnlineGoogle Scholar
/ Mujahid Iqbal
Published Online: 2018-12-31 | DOI: https://doi.org/10.1515/phys-2018-0111

Abstract

In this research work, for the first time we introduced and described the new method, which is modified extended auxiliary equation mapping method. We investigated the new exact traveling and families of solitary wave solutions of two well-known nonlinear evaluation equations, which are generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified forms of Camassa-Holm equations. We used a new technique and we successfully obtained the new families of solitary wave solutions. As a result, these new solutions are obtained in the form of elliptic functions, trigonometric functions, kink and antikink solitons, bright and dark solitons, periodic solitary wave and traveling wave solutions. These new solutions show the power and fruitfulness of this new method. We can solve other nonlinear partial differential equations with the use of this method.

Keywords: Modified extended auxiliary equation mapping method; GZK-BBM equation; modified form of CH equation; exact and Traveling wave solutions; solitary wave solutions

PACS: 02.30.Jr; 04.30.Nk; 05.45.Yv

1 Introduction

A few years ago many authors found the different types of exact traveling and solitary wave solutions of both nonlinear models GZK-BBM and GCH-equations. These both equations are well-known nonlinear evaluation equations and play important role in many scientific fields. The GZK-BBM equation used in the studies of acoustic waves, acoustic-gravity waves, surface waves with long wavelength, hydromagnetic waves, these all mentioned waves have source harmonic crystals, compressible fluids, cold plasma and liquids, respectively. The GCH equation play important role in shallow water waves. In 2005 Wazwaz [1] studied the GZK-BBM equation for the first time and found some complex solutions, kink type solutions, periodic wave solutions and solitons solutions with the help of the sine-cosine method. Wazwaz [2] found the two types of compactons and solitary patterns wave solutions of ZK-BBM equation by applying the extended tanh method. Abdou [3] found the set of exact solutions of ZK-BBM equation with the help of extended F-Expansion method. Mahmoudi et al. [4] investigated the periodic solitary wave solutions of ZK-BBM equation by applying the exp-function method. Wang and Tang [5] studied the existence property of smoothness of traveling wave solutions of ZK-BBM equation by apply the bifurcation theory of planner. Song and Yang [6] with the help of bifurcation technique found the travelingwave, solitary wave and kink type solutions of ZK-BBM equation.

Camassa and Holm [7] derived a Camassa-Holm equation (CH-equation) by using the Hamiltonian methods, which is a completely integrable dispersive water waves equation by holding two terms, which are neglected in the limit of shallow water waves, having small amplitude. After that many authors started to investigate the different types of travelling solitary wave solutions of CH equation by using various methods. Cooper and Shepard [8] found the solitary wave solutions of GCH-equation by using the variational function. Liu et al. [9] improved? CH equation and found traveling wave solutions. Zhang and Bi [10] studied the bifurcation technique of CH-equation. Liu and Tang [11] investigated the bifurcation phenomena and found the periodic solutions of GCH-equation with the help of integrated scheme. Deng et al. [12] found the compacton, kink and anti-kink, periodic solitary wave and solitons solutions of GCH-Degaspersi-Procesi-equation. Kalla and Klein [13] found the multidimensional theta functions independent derivation solutions of GCH equation with the help of technique that is related to Fay’s identity.

Recently, Liu and Song [14] found the smooth periodic and blow-up periodic solutions of GZK-BBM equation by applying the bifurcation method. Khadijo Adem and Masood Khalique [15] investigated the traveling waves solutions and conservation laws of GZK-BBM equation with the help of (G / G)− expansion method. Harun-Or-Roshid et al. [16] found the families of solitary waves solutions of GZK-BBM and RLW equations by using the modified simple equation method. Seadawy et al. [17] found the families of exact travelling and solitary wave solutions of GZK-BBM equation with the help of exp (−φ(ξ))-expansion method. Many other authors have investigated the travelling solitary wave solutions of GZK-BBM equation and GCH-equation see Ref. [18, 19, 20, 26, 27, 28, 29, 30,].

The nonlinear system of partial differential equations is very useful to study the physical nature in many different scientific fields, such as engineering, physics, geophysics, optics, chemistry, biology, material science, computer science, mechanics, electricity, ultrasound, thermodynamics and so on. The solitary and travelling wave solutions of NPDEs have many applications to understanding the process and physical phenomena in many areas of applied science. In the last five decades a lot of new methods have been developed by many groups of mathematicianas and engineers to investigate the (NPDEs). For example some important methods such as, exp-function method; modified Extended tanh-expansion method; modified simple equation method; homotopy perturbation method; novel (G /G)− expansion method; extended modified direct algebraic method; generalized kudryashov method; modified extended Kudryashov method; exp (−φ(ξ))-expansion method; extended Jacobian method; extended trial equation method and so on [31, 32, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,].

The main aim of this research is to investigate the exact traveling and solitary wave solutions of GZK-BBM and simplified modified form of CH-equations. These new solutions are obtained with the help of new method, which is modified extended auxiliary equation mapping method. The arrangement of this article is organized as fellows. Description of the ,modified extended auxiliary equation mapping method is given in Section 2. Section 3 deals with the investigation of the solitary wave solutions of GZK-BBM-equation and simplified modified CH-equation by using the described method. Finally, the conclusion are presented in Section 4.

2 Modified extended auxiliary equation mapping method

Consider the general form of (2+1)-dimensional NPDEs as

F(U,Ut,Ux,Uy,Uxt,...)=0,(1)

here F denotes the polynomial function of U(x, y, t) and its all derivatives which contained highest order nonlinear terms and highest order partial derivatives. Herewe explain the important steps of the new method as:

Step1. We apply the traveling wave transformations as

Ux,y,t=Uξ,ξ=lx+my+ωt,(2)

where l and m are the wave numbers and ω is the frequency of the wave. We obtained the ODE of Eq.1 as

P(U,U,U,U,...)=0,(3)

here P is the polynomial function in U(ξ) and its derivatives.

Step2. We consider the general solution of Eq.(2), in the following form

U(ξ)=i=0naiΨ(x)i+i=1nbiΨ(x)i+i=2nc2Ψ(x)i2Ψ(x)+i=1ndiΨ(x)Ψ(x)i(4)

where a0, a1, ...an , b1, b2, ...bn , c2, c3, ...cn , d1, d2, ...dn are constants parameter to be find later, the values of Ψ(ξ) and its derivative Ψ (ξ) satisfy to the given auxiliary equation

Ψ(ξ)=β1Ψ2(ξ)+β2Ψ3(ξ)+β3Ψ4(ξ);Ψ(ξ)=β1Ψ(ξ)+32β2Ψ2(ξ)+2β3Ψ3(ξ);Ψ(ξ)=β1+3β2Ψ(ξ)+6β3Ψ2(ξ)Ψ(ξ);Ψ(ξ)=12Ψ(ξ)(2β12+15β1β2Ψ(ξ)+5(3β22+8β1β3)Ψ2(ξ)+60β2β3Ψ3(ξ)+48β32Ψ4(ξ)).(5)

Where βis are real constants, which determine later such that βn ≠0.

Step3. Balance of the highest order nonlinear terms and highest order partial derivatives in Eq. (3) is found to be the positive integer n of Eq. (3)

Step4. Substituting Equation 5 into Equation 4 and combining each coefficients of Ψi(ξ)(i = 1, 2, 3, ...n), then making a every coefficient equal to zero and obtaining a families of algebraic equations, solving this system of equations with the help of Mathematica, the constants a0, a1, ...an , b1, b2, ...bn , c2, c3, ...cn , d1, d2, ...dn can be determined.

Step5. Substituting the values of obtained constants and Ψ(ξ) into Equation 3, then required solutions of Equation 1 are obtained.

3 Application of the proposed method

Now we applying the modified extended auxiliary equation mapping method to investigate the families of new solitary wave solutions for the (2+1)-dimensional GZK-BBM-equation and simplified modified form of CH-equation.

3.1 Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation

We consider a (2+1)-dim GZK-BBM-equation as

Ut+Ux+α(U3)x+γUxt+Uyyx=0,(6)

where γ and α are non zero constants. Consider the travel-ingwave transformation U(x, y, t) = U(ξ), ξ = lx+my+ωt, by this transformation we obtained ordinary differential equation of Eq.6 as

ωU+lU+3αlU2U+γm3U+γl2ωU=0,(7)

we integrate the Eq.7 once time according to ξ and integration constant equal to zero, then we obtained as

ωU+lU+αlU3+γl2ωU+γm3U=0,(8)

Balance the highest order nonlinear term and highest order partial derivative in Eq.8 obtained the value of n =

1. The general solution of Eq.8 takes form of

Uξ=a0+a1Ψ(ξ)+b1Ψ(ξ)+d1Ψ(ξ)Ψ(ξ)(9)

Substituting Equation 9 into Equation 8 and combining each coefficients of Ψj(ξ)Ψi(ξ)(j = 0,1;i = 1, 2, 3, ...n), then making a every coefficient equal to zero and obtaining a set of algebraic equations. We solve this system of equations with the aid of Mathematica. The parameters a0, a1, b1, d1 can be determined as

Case-I

a0=0,a1=a1,b1=0,d1=±lωαβ1l,β3=±αa12β1ll+ω,m=±(1)2/3β1(γ)l2ω+2l+2ω3β13γ3.(10)

Substituting the Equation 9, then solitary wave solutions of Equation 6 are obtained in simplified forms as:

U1x,y,t=ϵlωcsch12β1lx+my+ωt+ξ02αl2ϵcoth12β1lx+my+ωt+ξ0+2a1β11+ϵcoth12β1lx+my+ωt+ξ0β2(11)U2x,y,t=2ϵlωηcoshβ1lx+my+ωt+ξ0+1αlη+coshβ1lx+my+ωt+ξ0+ϵsinhβ1lx+my+ωt+ξ02a1β1β3/2αlη+coshβ1lx+my+ωt+ξ0η+coshβ1lx+my+ωt+ξ0+ϵsinhβ1lx+my+ωt+ξ0(12)U3x,y,t=ϵlωηp2+1coshβ1lx+my+ωt+ξ0psinhβ1lx+my+ωt+ξ0+1/αlcoshβ1lx+my+ωt+ξ0+ηp2+1coshβ1lx+my+ωt+ξ0+ηp2+1+ϵsinhβ1lx+my+ωt+ξ0+p+a11ϵsinhβ1lx+my+ωt+ξ0+pcoshβ1lx+my+ωt+ξ0+ηp2+1(13)

Case-II

a0=±lωαl,a1=a1,b1=d1=0,m=±2(l+ω)β1γl2ω3β13γ3,β2=±αa1β1llω,β3=αa12β1l4(l+ω).(14)

Substituting Eq.14, only the positive value of a0 into Eq.9, the solutions of Eq.6 are given as:

U4x,y,t=lωαla1β11+ϵcoth12β1lx+my+ωt+ξ0β2(15)U5x,y,t=lωαl12a1β1β31+ϵsinhβ1lx+my+ωt+ξ0η+coshβ1lx+my+ωt+ξ0(16)U6x,y,t=lωαl+a11ϵsinhβ1lx+my+ωt+ξ0+pcoshβ1lx+my+ωt+ξ0+ηp2+1(17)

Case-III

a0=a1=b1=0,d1=±lωαβ1l,m=±(1)2/32(l+ω)β1γl2ω3β13γ3.(18)

Substituting Eq.18 into Eq.9, then the solutions of Eq.6 can be obtained as:

U7x,y,t=ϵlωcsch12β1lx+my+ωt+ξ02αl2ϵcoth12β1lx+my+ωt+ξ0+2(19)U8x,y,t=ϵlωηcoshβ1lx+my+ωt+ξ0+1/αlη+coshβ1lx+my+ωt+ξ0η+coshβ1lx+my+ωt+ξ0+ϵsinhβ1lx+my+ωt+ξ0(20)U9x,y,t=ϵlωηp2+1coshβ1lx+my+ωt+ξ0+psinhβ1lx+my+ωt+ξ01/αlcoshβ1lx+my+ωt+ξ0+ηp2+1coshβ1lx+my+ωt+ξ0+ηp2+1+ϵsinhβ1lx+my+ωt+ξ0+p(21)
Solitary wave solution given in Eq.(11) when a1 = 0.5, β1 = 2, β2 = 4, ϵ = 10, η = −8, ξ0 = 0.5, l = 0.8, m = 0.2, ω = −8, y = 1, a = 1
Figure 1

Solitary wave solution given in Eq.(11) when a1 = 0.5, β1 = 2, β2 = 4, ϵ = 10, η = −8, ξ0 = 0.5, l = 0.8, m = 0.2, ω = −8, y = 1, a = 1

Solitary wave solution given in Eq.(12) when a1 = 0.5, β1 = 4, β2 = 8, β3 = 4, ϵ = 28, η = −18, ξ0 = 0.3, l = 0.2, m = 0.4, ω = −5, y = 1, a = 1
Figure 2

Solitary wave solution given in Eq.(12) when a1 = 0.5, β1 = 4, β2 = 8, β3 = 4, ϵ = 28, η = −18, ξ0 = 0.3, l = 0.2, m = 0.4, ω = −5, y = 1, a = 1

Case-IV

a0=±lω2αl,a1=b1=0,d1=±lω2αβ1l(22)

Substituting Eq.22, only the positive value of a0 into Eq.9, then the solutions of Eq.6, can be given as:

U10x,y,t=lωϵsinhβ1lx+my+ωt+ξ01csch12β1lx+my+ωt+ξ02+24αlϵcoth12β1lx+my+ωt+ξ0+1(23)U11x,y,t=lω1+2ϵ+2η2+2η(ϵ+2)coshβ1lx+my+ωt+ξ0+cosh2β1lx+my+ωt+ξ0+2ϵsinhβ1lx+my+ωt+ξ0η+coshβ1lx+my+ωt+ξ04αlη+coshβ1lx+my+ωt+ξ0η+coshβ1lx+my+ωt+ξ0+ϵsinhβ1lx+my+ωt+ξ0(24)U12x,y,t=lωϵηp2+1coshβ1lx+my+ξ0+tωpsinhβ1lx+my+ξ0+tω+1coshβ1lx+my+ξ0+tω+ηp2+1ϵsinhβ1lx+my+ξ0+tω+p+coshβ1lx+my+ξ0+tω+ηp2+1+12αl(25)

Case-V

a0=a0,a1=0,b1=2a0β1β2,d1=0,m=13ω3β13γ3(26)

Substituting Eq.26 into Eq.9, then the solutions of Eq.6 can be obtained as:

U13x,y,t=a012ϵcoth12β1lx+my+ωt+ξ0+1(27)U14x,y,t=a014β1β3β3β2+β2ϵsinhβ1lx+my+ωt+ξ0η+coshβ1lx+my+ωt+ξ0(28)U15x,y,t=a012β1β2ϵsinhβ1lx+my+ωt+ξ0+pcoshβ1lx+my+ωt+ξ0+ηp2+1+1(29)

Case-VI

a0=±6β1γω2+92α4β12γ2ω49,a1=b1=0,d1=±6β1γω2+92αβ14β12γ2ω49,l=32β1γωω,m=±1334β1γω2β1γω23322/3β12/3γ2/3ω3.(30)

Substituting Eq.30, only positive value of a0 and d1 into Eq.9, the solitary wave solutions of Eq.6 can be obtained in the simplified form as:

U16x,y,t=146β1γω2+92α4β12γ2ω49β1ϵcsch12β1lx+my+ωt+ξ02αβ14β12γ2ω49ϵcoth12β1lx+my+ωt+ξ0+1(31)U17x,y,t=126β1γω2+91α4β12γ2ω49+β1ϵηcoshβ1lx+my+ωt+ξ0+1/η+coshβ1lx+my+ωt+ξ0η+coshβ1lx+my+ωt+ξ0+ϵsinhβ1lx+my+ωt+ξ0αβ14β12γ2ω49(32)U18x,y,t=126β1γω2+91α4β12γ2ω49+ϵηp2+1coshβ1lx+my+ωt+ξ0+1psinhβ1lx+my+ωt+ξ0β1coshβ1lx+my+ωt+ξ0+ηp2+1coshβ1lx+my+ωt+ξ0+ηp2+1+ϵsinhβ1lx+my+ωt+ξ0+pαβ14β12γ2ω49(33)
Solitary wave solution given in Eq.(13) when a1 = 0.5, β1 = 4, ϵ = 8, η = −5, ξ0 = 0.3, l = 0.9, m = 0.2, ω = −5, y = 1, a = 0.4, p = 0.2
Figure 3

Solitary wave solution given in Eq.(13) when a1 = 0.5, β1 = 4, ϵ = 8, η = −5, ξ0 = 0.3, l = 0.9, m = 0.2, ω = −5, y = 1, a = 0.4, p = 0.2

Solitary wave solution given in Eq.(21) when a1 = 0.4, β1 = 2, ϵ = 18, η = −15, ξ0 = 0.3, l = 0.5, m = 0.2, ω = −5, y = 1, a = 0.4, p = 0.2
Figure 4

Solitary wave solution given in Eq.(21) when a1 = 0.4, β1 = 2, ϵ = 18, η = −15, ξ0 = 0.3, l = 0.5, m = 0.2, ω = −5, y = 1, a = 0.4, p = 0.2

Case-VII

a0=±1β12γ2ω42αβ1γω21β1γω2+2,a1=b1=0,d1=±1β12γ2ω42αβ1β1γω21β1γω2+2,l=2β1γω+ω,m=(1)2/3ω.(34)

Substituting Eq.34, only the positive value of a0 and d1 into Eq.9, then the solutions of Eq.6 can be get in the simplified form as:

U19x,y,t=1β12γ2ω42αβ1γω21β1γω2+2β1ϵcsch12β1lx+my+ωt+ξ02αβ1β1γω21β1γω2+2ϵcoth12β1lx+my+ωt+ξ0+122(35)U20x,y,t=121β12γ2ω41αβ1γω21β1γω2+2+β1ϵηcoshβ1lx+my+ωt+ξ0+1/η+coshβ1lx+my+ωt+ξ0η+coshβ1lx+my+ωt+ξ0+ϵsinhβ1lx+my+ωt+ξ0αβ1β1γω21β1γω2+2(36)U21x,y,t=121β12γ2ω41αβ1γω21β1γω2+2+ϵηp2+1coshβ1lx+my+ωt+ξ0+1psinhβ1lx+my+ωt+ξ0β1coshβ1lx+my+ωt+ξ0+ηp2+1ηp2+1+coshβ1lx+my+ωt+ξ0+ϵsinhβ1lx+my+ωt+ξ0+pαβ1β1γω21β1γω2+2(37)

3.2 Simplified modified form of Camassa-Holm equation

We consider a simplified modified form of CH-equation as

Ut+2βUxUxxt+δU2Ux=0,(38)

where β and δ are non zero constants. Consider the wave transformation as; U(x, y, t) = U(ξ), ξ = kx + ωt. By this transformation we obtained ordinary differential equation of Eq.38 as

ωU+2βkUk2ωU+δkU2U=0,(39)

we integrate Eq.39 once time according to the ξ and constant of integration equal to zero, then we obtained as

ωU+2βkUk2ωU+13δkU3=0.(40)

We balance the nonlinear term and highest order derivative in Eq.40 allow obtaining the value of n = 1. The general solution of Eq.40 takes the form of:

Uξ=a0+a1Ψ(ξ)+b1Ψ(ξ)+d1Ψ(ξ)Ψ(ξ)(41)

Substituting Eq.41 into Eq.40 and combining each coefficients of Ψj(ξ)Ψi(ξ)(j = 0,1; i = 1, 2, 3, ...n), then making a every coefficient equal to zero and obtaining a set of algebraic equations. We solve this system of equations with the aid of Mathematica. The parameters a0, a1, b1, d1 can be determined as

Case-1

a0=0,a1=±6ββ3kβ1(δ)k22δ,b1=0,d1=±6βkβ1(δ)k22δ,ω=4βkβ1k2+2.(42)

Substituting the Eq.42, only for the positive value of a1 and d1 in Eq.41, the solutions of Eq.38 are given as:

U1x,y,t=32βk2β1β3ϵcoth12β1kx+ωt+ξ0+12+β1β2ϵcsch12β1kx+ωt+ξ02β2δβ1k2+2ϵcoth12β1kx+ωt+ξ0+1(43)U2x,y,t=32kβ2β1ϵηcoshβ1kx+ωt+ξ0+1+η+coshβ1kx+ωt+ξ0+ϵsinhβ1kx+ωt+ξ02β1β3β3η+coshβ1kx+ωt+ξ0η+coshβ1kx+ωt+ξ0+ϵsinhβ1kx+ωt+ξ0δβ1k2+2(44)U3x,y,t=6βkϵηp2+1coshβ1kx+ωt+ξ0+psinhβ1kx+ωt+ξ01β1+sinhβ1kx+ωt+ξ0+p2coshβ1kx+ωt+ξ0+2ηp2+1+ϵsinhβ1kx+ωt+ξ0+pϵsinhβ1kx+ωt+ξ0+pδβ1k2+2ϵsinhβ1kx+ωt+ξ0+pcoshβ1kx+ξ0+tω+ηp2+1+1(45)
Solitary wave solution given in Eq.(44) when β1 = 2, β2 = 4, β3 = 2, ϵ = 5, η = −3, ξ0 = 0.5, k = 2, ω = 8, α = 0.2, δ = −2
Figure 5

Solitary wave solution given in Eq.(44) when β1 = 2, β2 = 4, β3 = 2, ϵ = 5, η = −3, ξ0 = 0.5, k = 2, ω = 8, α = 0.2, δ = −2

Solitary wave solution given in Eq. (49) when β1 = 4, β3 = 2, ϵ = −5, η = 2, ξ0 = 0.3, k = 2, ω = 8, p = 0.6, α = 0.2, δ = −4
Figure 6

Solitary wave solution given in Eq. (49) when β1 = 4, β3 = 2, ϵ = −5, η = 2, ξ0 = 0.3, k = 2, ω = 8, p = 0.6, α = 0.2, δ = −4

Case-II

a0=±6ββ1kβ1(δ)k22δ,a1=±26ββ3kβ1(δ)k22δ,b1=d1=0,ω=4βkβ1k2+2,β2=±2β1β3.(46)

Substituting Equation 46, only the positive value of a0 and a1 into Equation 41, the solutions of Equation 38 can be given as:

U4x,y,t=6βk2β1β3ϵcoth12β1kx+ωt+ξ0+1β1β2β2δβ1k2+2(47)
Solitary wave solution given in Eq. (51) when β1 = 1, β2 = 2, β3 = 1, ϵ = 5, η = 3, ξ0 = 0.2, k = 2, ω = 8, α = 1, δ = 2
Figure 7

Solitary wave solution given in Eq. (51) when β1 = 1, β2 = 2, β3 = 1, ϵ = 5, η = 3, ξ0 = 0.2, k = 2, ω = 8, α = 1, δ = 2

Solitary wave solution given in Eq. (52) when β1 = 2, β2 = 4, β3 = 2, ϵ = 18, η = −15, ξ0 = 0.8, k = 2, ω = 8, α = 8, δ = 2
Figure 8

Solitary wave solution given in Eq. (52) when β1 = 2, β2 = 4, β3 = 2, ϵ = 18, η = −15, ξ0 = 0.8, k = 2, ω = 8, α = 8, δ = 2

U5x,y,t=6kββ1η+coshβ1kx+ωt+ξ0η+coshleft[β1kx+ωt+ξ0+ϵsinhβ1kx+ωt+ξ0β1β3β3δβ1k2+2η+coshβ1kx+ωt+ξ0(48)U6x,y,t=6βkβ1+2β3ϵsinhβ1kx+ωt+ξ0+pcoshβ1kx+ωt+ξ0+ηp2+11δβ1k2+2(49)

Case-III

a0=0,a1=±23ββ3kβ1δk2δ,b1=d1=0,ω=2βkβ1k21.(50)

Substituting Eq.50, only the positive value of a1 into Eq.41 the solutions of Eq.38 can be given as:

U7x,y,t=23ββ1β3kϵcoth12β1kx+ωt+ξ0+1β2δβ1k21(51)U8x,y,t=3ββ1β3β3kϵsinhβ1kx+ωt+ξ0η+coshβ1kx+ωt+ξ0+1δβ1k21(52)U9x,y,t=23ββ3kϵsinhβ1kx+ωt+ξ0+pcoshβ1kx+ωt+ξ0+ηp2+11δβ1k21(53)

Case-IV

a0=a1=0,b1=0,d1=±23βkδ2β1δk2,ω=2βk2β1k2+1(54)

Substituting Equation 54, only the positive value of d1 into Equation 41, the solutions of Equation 38 can be given as:

U10x,y,t=3ββ1kϵcsch12β1kx+ωt+ξ02δ2β1k2+1ϵcoth12β1kx+ωt+ξ0+1(55)U11x,y,t=23ββ1kϵηcoshβ1kx+ωt+ξ0+1/η+coshβ1kx+ωt+ξ0δ2β1k2+1η+coshβ1kx+ωt+ξ0+ϵsinhβ1kx+ωt+ξ0(56)U12x,y,t=23ββ1kϵηp2+1coshβ1kx+ωt+ξ0psinhβ1kx+ωt+ξ0+1/coshβ1kx+ωt+ξ0+ηp2+1coshβ1kx+ωt+ξ0+ηp2+1+ϵsinhβ1kx+ωt+ξ0+pδ2β1k2+1(57)

Case-V

a0=±2βδ,a1=±22ββ3β1δ,b1=d1=0,k=1β1,ω=4β3β1,β2=±2β1β3(58)

Substituting Equation 58, only the positive value of a0 and a1 into Equation 41, the solutions of Equation 38 can be given as:

U13x,y,t=2ββ22β1β3ϵcoth12β1kx+ωt+ξ0+1β2δU14x,y,t=2β1β1β3β3ϵsinhβ1kx+ωt+ξ0η+coshβ1kx+ωt+ξ0+1β1δ(59)U15x,y,t=2β2β3ϵsinhβ1kx+ωt+ξ0+pcoshβ1kx+ωt+ξ0+ηp2+11β1+1δ

4 Conclusion

We have successfully applied a new method on two nonlinear evaluation equations. We have obtained a new exact traveling and solitary wave solutions of GZK-BBM-equation and simplified modified form of CH-equation by applying the Modified extended auxiliary equation mapping method. As a results, these new solutions are obtained in the form of elliptic functions, trigonometric functions, kink and antikink solitions, bright and dark solitons, periodic solitary wave and travelling wave solutions and also show two and three dimensional graphs with the help of Mathematica. These new families of solutions show the power, effectiveness, capability, realizabilities and fruitfulness of this new method. We can solve other nonlinear physical phenomena, which are related to nonlinear evaluation equations with the help of this new method.

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About the article

Received: 2018-07-06

Accepted: 2018-09-25

Published Online: 2018-12-31


Citation Information: Open Physics, Volume 16, Issue 1, Pages 896–909, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0111.

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© 2018 Dianchen Lu et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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