# New Periodic and Localized Traveling Wave Solutions to a Kawahara-Type Equation: Applications to Plasma Physics

Authors: Haifa A. Alyousef; Alvaro H. Salas; M. R. Alharthi; S. A. El-Tantawy
Journal: Complexity
 (2022)

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## Abstract

In this study, some new hypotheses and techniques are presented to obtain some new analytical solutions (localized and periodic solutions) to the generalized Kawahara equation (gKE). As a particular case, some traveling wave solutions to both Kawahara equation (KE) and modified Kawahara equation (mKE) are derived in detail. Periodic and soliton solutions to this family are obtained. The periodic solutions are expressed in terms of Weierstrass elliptic functions (WSEFs) and Jacobian elliptic functions (JEFs). For KE, some direct and indirect approaches are carried out to derive the periodic and localized solutions. For mKE, two different hypotheses in the form of WSEFs are used to derive the periodic and localized solutions. Also, the cnoidal wave solutions in the form of JEFs are obtained. As a realistic physical application, the solutions obtained can be dedicated to studying many nonlinear waves that propagate in plasma.


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## Body

## 1. Introduction

Both ordinary and partial differential equations succeed in modelling and describing many complex nonlinear systems that are widely used in various fields of science such as optical fiber, fluid mechanics, nonlinear optics, biology, ecology, astronomy, oceans, economics, and plasma physics [1–10]. Due to the importance of these applications, the great success has achieved by differential equations in clarifying and interpreting the ambiguity of many complex systems, which prompted many authors to look for different analytical and numerical methods in solving such models [5–11]. In recent years, many new analytical and numerical methods have been discovered, and some improvements have been made to many of the existing methods in order to either obtain real solutions related to realistic problems or to obtain more accurate solutions to many integrable and nonintegrable differential equations [12–16]. In particular, there are a large number of partial differential equations (PDEs) that have been used for modelling a lot of nonlinear phenomena such as solitary waves, shock waves, cnoidal waves, peakons, and compactons that arise in different plasma models [5–11]. One of the most important of these equations and the most famous due to its great success not only in the field of fluid mechanics and plasma physics but also in various fields of science is Korteweg–de Vries (KdV) equation [5]:(1)∂tux,t+α1ux,t∂xux,t+β∂x3ux,t=0,where α1,β are the real coefficients which are related to the physical model under study. This equation and its one-dimensional family including a modified KdV (mKdV) equation [5, 10], a Gardner equation, KdV–Burger’s equation [5], damped KdV/mKdV equation [11], and so on have been widely used until this day in interpreting the mechanism and properties of many nonlinear phenomena that can propagate in plasma physics. This family is characterized by the third-order dispersion, but there is another family characterized by the fifth-order dispersion which is called the family of Kawahara equation (KE) [17].(2)RKE≡∂tux,t+α1ux,t∂xux,t+β∂x3ux,t−γ∂x5ux,t=0.This is a nonlinear dispersive equation which generalizes the well-known KdV equation. Kawahara equation (2), sometimes referred to as the fifth-order KdV/or super KdV equation [18], is a model that describes solitary waves, cnoidal waves, and periodic waves propagating in nonlinear and high-dispersive media. This equation and many related equations with fifth-order dispersion have been extensively studied in literature [19]. It has important applications in the theory of magnetoacoustic waves in plasma and in the theory of shallow water waves with surface tension [17, 18, 20–30]. However, equation (2) fails to explain the nonlinear waves at some critical values of the plasma compositions due to the disappearance of the nonlinear term, i.e., α1=0. Accordingly, modified Kawahara equation (mKE) with higher-order nonlinearity was derived to describe some nonlinear phenomena at the critical plasma compositions:(3)RmKE≡∂tux,t+α2u2x,t∂xux,t+β∂x3ux,t−γ∂x5ux,t=0.Both KE equation (2) and mKE equation (3) are integrable Hamiltonian systems which are due to the many applications related to this family; many methodologies have been applied for analyzing it [17, 18, 20–24, 27–30]. There remain many secrets about the solutions of this family that appear and become clear day after day as a result of using new analytical and numerical methods for solving this family. This is one of our motives for obtaining a new generation of solutions to this family, which can contribute in understanding the mysterious of many phenomena in plasma physics and other fields related to this family. Thus, our aim is to provide new traveling wave (localized and periodic) solutions to the following generalized KE [29] using several new hypotheses and techniques:(4)∂tux,t+αpuPx,t∂xux,t+β∂x3ux,t−γ∂x5ux,t=0,where p is a real number. Note that KE equation (2) can be obtained for p=1, while for p=2, mKE equation (3) is recovered.


## 2. General Analytical Solutions to the Generalized KE

To find a general analytic solution to the evolution equation (4), we suppose(5)u=v1/p,v≡vξξ=x+λt,where λ represents the frame velocity.Inserting ansatz equations (5) into (4) gives the nonlinear ODE:(6)αp4v5v′+p4v4βv3+λv′−γv5+p−1p3v35γv4v′+10γv3v′′−3βv′v′′+p−1p22p−1v2v′−15γv′′2+βv′2−10γv3v′−γp−12p−13p−14p−1v′5+10γp−1p2p−13p−1vv′3v′′=0,where v′≡∂ξv, v′′≡∂ξ2v, v3≡∂ξ3v, v4≡∂ξ4v, and v5≡∂ξ5v.In the following subsections, two important particular cases (p=1 and p=2), i.e., KE equation (2) and mKE equation (3)are analyzed.


### 2.1. Solutions of the Planar Kawahara Equation

In the following sections, we try to find some new solutions including the periodic wave solutions, cnoidal wave solutions, and solitary wave solutions to the planar KE equation (2) p=1.


#### 2.1.1. Periodic and Solitary Wave Solutions

For planar KE equation (2), ODE equation (6) reduces to(7)−γv5+βv3+αvv′+λv′=0.Integrating equation (7) once over ξ gives us(8)c0−γv4+βv′′+12αv2+λv=0,where c0 is the integration constant.Multiplying equation (8) by v′ and then integrating it again, we get(9)c0v+c1+12γv′′2+12βv′2−γv3v′+16αv3+12λv2=0,where c1 the new constant of integration. The solution of equation (2) via several approaches is discussed as follows.We seek a solution in the ansatz form:(10)v=∑k=0Ndjφj,where φ≡φξ is a solution to the following Helmholtz equation [31].(11)φ′′+Aφ+Bφ2=0,φ′2=−Aφ2−23Bφ3+2c2.Balancing the highest linear and nonlinear terms in equation (11) gives N=2, so that(12)v=d0+d1φ+d2φ2.Inserting ansatz equation (12) into equation (9), we obtain(13)∑j=06Fjφj=0,where the values of Fjj=0,1,⋯6 are defined in Appendix A, and by solving the following system,(14)Fj=0,we get(15)c0=−157122αγ24598321A4γ4−54418A2β2γ2+1457β4−28561γ2λ2,c1=−128960854α2γ319394118562A6γ6−183532986A4β2γ4+2331348747A4γ5λ+134862A2β4γ2−27589926A2β2γ3λ+94922β6+738699β4γλ−4826809γ3λ3,c2=13Aγ+β263640B2γ31690A2γ2−403Aβγ+31β2,d0=−1507αγ1183A2γ2−910Aβγ+31β2+507γλ,d1=140B13Aγ+β39α,d2=140B2γ3α.Equation (11) has many formulas for its general solutions such as(16)φ=−A2B−6B℘ξ+e0;A2/12,g3,where ℘≡℘ξ+e0;A2/12,g3 indicates the Weierstrass elliptic function (WSEF) and the values of e0 and g3 are undetermined parameters which can be obtained from the initial conditions.Also, the general solution to equation (11) can be expressed by(17)φ=e1−6e1A+Be1A+2Be1+12℘ξ+e0;A2/12,1/216A3−6B2e12A−4B3e13,where the values of e0 and e1 are determined from the initial conditions.Thus, a periodic solution to KE equation (2) according to the values of parameters given in system equation (15) and the value of φ given in equation (16) is obtained as(18)u=910βAγ−7098A2γ2−910Aβγ−31β2−507γλ507αγ−280β13α℘x+λt+ξ0;A2/12,g3+1680γα℘2x+λt+ξ0;A2/12,g3.The constantsA, B, λ, and ξ0 are the arbitraries.Using relation equation (9) with of the parameters given in system equation (15) and the value of φ given in equation (17), the solution to KE equation (2) can be expressed by(19)u=−31β2−13γ39λ−70Aβ+91A2γ+1820Bγe0β+13Aγ+13Bγe0507αγ−280Be0A+Be0β+13Aγ+26Bγe013αA+2Be0+12℘x+λt+ξ0;A2/12,1/216A3−2B2e023A+2Be0+1680B2γe02A+Be02αA+2Be0+12℘2x+λt+ξ0;A2/12,1/216A3−2B2e023A+2Be0,where the constants A, B, λ, and e0 are the arbitraries.From the periodic solution equation (18), the soliton solutions can be obtained using the following hypotheses:(20)cg2=A2/12=43,g3=β3549A2γ2−31β24745520γ3=827,which lead to A,β=4,52γ, and by rearrange solution (18), the following soliton solution is obtained:(21)=−1168γ+3λ3α−1120γα℘x+λt;43,−827,+1680γα℘x+λt;43,−8272.Moreover, the solitary wave solution can be obtained by(22)g2=A2/12=1/12,g3=β3549A2γ2−31β24745520γ3=−1/216,which leads to A,β=1,13γ; then, the periodic solution (18) reduces to the following soliton solution:(23)=−73γ+3λ3α−280γ℘x+λt+ξ0;1/12,−1/216α+1680γ℘x+λt+ξ0;1/12,−1/2162α.Also, the periodic solution to KE (2) can be derived directly in terms of WSEF ℘≡℘x+λt+ξ0;g2,g3 by inserting the following ansatz in (9).(24)u=a0+a1℘+a2℘2,which leads to(25)∑j=06Sjφj=0,where the values of Sjj=0,1,⋯ are defined in Appendix B , and by solving the following system,(26)Sj=0,we have(27)a1=−280βk213α,a2=1680γk4α,c1=−114394744αγ2k11661αa0γk507αa0γk+1014γλ+146β2k−1285245γ2λ2+449159β4k2+1702506β2γkλ,c2=121894405624α2γ3k1521αa0γ507αa0γk6929αa0γk+9126γλ+2189β2k+5277β2k−507γλ507γλ+73β2k−7128781549β2γ2λ2+9635389β6k2+75627162β4γkλ,g2=−507αa0γk+507γλ+31β2k85176γ2k5,g3=−β169αa0γk+169γλ+31β2k3163680γ3k7.Collecting both equations (24) and (27), we finally get(28)u=a0−280βk213α℘kx+tλ+ξ0;−31kβ2+507γλ+507kαγa085176k5γ2,−β31kβ2+169γλ+169kαγa03163680k7γ3+1680γk4α℘kx+tλ+ξ0;−31kβ2+507γλ+507kαγa085176k5γ2,−β31kβ2+169γλ+169kαγa03163680k7γ32.Solution equation (28) satisfies KE equation (2).


#### 2.1.2. Cnoidal and Solitary Wave Solutions

We look for a solution to KE equation (2) in the ansatz form(29)ux,t=vξ=∑j=0Ndjcnjξ,where ξ=ωx+λt and N is an integer and positive number. From the balance between the highest-order linear N+4 and nonlinear 2N terms of equation (8), we have N=4. Substituting ansatz equation (29) into equation (6) gives a very complicated system. By solving this system using Mathematica package, we found that the coefficients of the odd power in ansatz equation (29) vanish. Thus, the solution of KE equation (2) could be written in the following ansatz:(30)u=A+Bcn2ξ,m+Ccn4ξ,m.Inserting ansatz equation (30) into KE equation (2), we get(31)RKE=∑j=03Wjcn2j=0,the values of Wjj=0,1,⋯3 are given in Appendix C, and by solving the system,(32)Wj=0,we have(33)A=1507αγ−31β2+3640βγ1−2mω+169γ112γ14m−1m−1ω2−3λ,B=−280mω13α52γ2m−1ω−β,C=1680γm2ω2α,where ω is a solution to the following cubic equation:(34)31β3−56784βγ2m2−m+1ω2+703040γ3m−2m+12m−1ω3=0.Finally, the cnoidal wave solutions to KE equation (2) are obtained as(35)=−31β2+3640βγ1−2mω+169γ112γ14m−1m−1ω2−3λ507αγ,−280mω52γ2m−1ω−β13αcn2ξ,m+1680γm2ω2αcn4ξ,m,where λ is the arbitrary constant, ξ=ωx+λt, and ω is a root to equation (34).The cnoidal wave solution equation (35) can be directly reduced to the soliton solution for m⟶1 as(36)u=−36β2+169γλ169αγ+105β2169αγsech412β13γx+λt.Moreover, solution equation (36) coincides with the obtained one by means of the tanh method:(37)u=105β2169αγsech412β13γx−36β2169γt.The obtained solutions can be employed for investigating the propagation of nonlinear structures in different plasma models. For instance, we can apply these solutions to study cnoidal and solitary waves in the ultracold neutral plasma (UCNP) which is composed of strongly coupled positive ions and non-Maxwellian electron distributions [32–35]. Based on this model and for Maxwellian electrons, the values of the coefficients α,β are given by (to prevent stuffing and repetition, all the details can be found [32])(38)α=λphandβ=12λph,and the phase velocity λph of the ion-acoustic waves (IAWs) reads(39a)λ=1+σ∗,where σ∗≡σ∗Te,Ti represents the effective temperature ratio which is a function of electron and ion temperatures Te,Ti [32–35]. For Te,Ti=25K,1K, we get σ∗=0.401169, and for Te,Ti=900K,1K, we obtain σ∗=0.0120837 [32–35]. With respect to the coefficient of the fifth-order dispersion γ, in general, it has a small value 0<γ≪1. The impact of effective temperature ratio σ∗ on the profile of the cnoidal wave solution equation (35) and the solitary wave solution equation (36) for γ,λ=0.1,0.1 is shown in Figures 1 and 2, respectively. It is observed that increasing the electron temperature, i.e., decreasing σ∗, leads to the enhancement (reduction) of the amplitude (width) of both localized and periodic waves.Figure 1
The profile of the periodic wave solution equation (35) to KE equation (2) plotted in x,t plane.
(a)(b)Figure 2
The profile of the solitary wave solution equation (36) to KE equation (2) is plotted in x,t plane.
(a)(b)


### 2.2. Solutions of Planar Modified Kawahara Equation

The generalized KE equation (4) can reduce the planar mKE equation (3) for p=2. Making the traveling wave transformation u=vξ, where ξ=x+λt+ξ0, we get(40)λv′+αv2v′+βv3−γv5=0,and integrating equation (40) twice over ξ, we obtain(41)ℝ≡c0v+c1−γv3v′+1/12αv4+12λv2+12βv′2+12γv′′2=0.Some new localized and periodic solutions to equation (41) are discussed using different approaches in the following sections.


#### 2.2.1. First Ansatz in Terms of WSEFs

The following ansatz is introduced to find a periodic wave solution to equation (41) in terms of WSEFs:(42)v=A+B℘,where ℘≡℘ξ+ξ0;g2,g3, g2 and g3 denote the elliptic invariants, while the other parameters A, B, g2, and g3 are the constants and will be determined later.Inserting the ansatz equations (42) into (40), we obtain(43)A2α+18γg2+λ+2AαB+12β℘+αB2−360γ℘2=0.Equating the coefficients of℘0, ℘, and ℘2 to zero and solving the obtained system, we have(44)A=±β10αγ,B=∓610γα,andg2=−β2−10γλ180γ2.Note thatg3 is an arbitrary constant. Using the initial condition v0=v0, we can get(45)A+B℘ξ0;g2,g3=v0,which leads to(46)ξ0=℘−1v0−AB;g2,g3.By substituting the values ofA,B,g2 given in equation (44) into the ansatz equation (42), we finally obtain the solutions of cnoidal wave as(47)u1,2x,t=±β10αγ∓610γα℘x+λt+ξ0;−β2−10γλ180γ2,g3,and these solutions satisfy the evolution equation (3).For the following choice,(48)g2=43andg3=−827,the solitary wave solutions are recovered:(49)u1,2x,t=±β10αγ∓610γα℘x+λt+ξ0;43,−827.


#### 2.2.2. Second Ansatz in Terms of WSEFs

The following rational hypothesis/ansatz is assumed to find some analytical solution to equation (41):(50)vξ=A+B1+C℘,where A, B, and C are the undetermined constants and ℘≡℘ξ+ξ0;g2,g3.Inserting the ansatz equations (50) into (41), we have(51)∑j=06Zj℘j=0,where the coefficients Zjj=0,1,…,6 are defined in Appendix D, and by solving the following system,(52)Z0=0,Z1=0,…,Z6=0,we obtain the nontrivial solution:(53)A=60γ+βC10αγC,λ=−β210γ−65C210αγBC+180γ,c0=γ−3/210/α86400γ3+β3C3−2160βγ2C−120BCβC−60γ150C3,c1=1400αγ2C4−800γ3αB2C2+3610αγBC+3240γ+56β2γ3/2C210αBC+180γ+2880βγ5/2C10αBC+120γ+β4−C4,g2=10α/γBC+18015C2,g3=BC15C310α/γ+120.Thus, the traveling wave solutions to mKE equation (3) are expressed by(54)u=60γ+βC10αγC+B1+C℘x−β2/10γ+610αγBC+180γ/5C2t+ξ0;10α/γBC+180/15C2,10α/γ+120/15C3BC.The values of the constantsB and C are arbitrary.The solitary wave solutions can be obtained from the periodic solution equation (54) according to the following choices:(55)g2=1/12,g3=−1/216,g2=43,g3=−827.For the choices equation (55), the soliton solutions are obtained:(56)u=β10αγ+125γ2α4−91+6℘x−β2+15γ2/10γt+ξ0;1/12,−1/216,(57)u=β10αγ+10γα1−182+3℘x−β2+15γ2/10γt+ξ0;4/3,−8/27.


#### 2.2.3. Third Ansatz in Terms of JEFs

Using the following ansatz in equation (41),(58)vξ=A+Bcn2,we have(59)∑j=02nYjcnj=0,where the coefficients Yjj=0,2,4,6 are defined in Appendix E, and by solving the following system,(60)Yj=0,we get(61)A=−−β−20γω+40γmω10αγ,B=610γmωα,λ=−β2+240γ2ω2+240γ2m2ω2−240γ2mω210γ,c0=11510αγ3/2−240βγ2ω2+3200γ3ω3+3200γ3m3ω3−240βγ2m2ω2−4800γ3m2ω3+240βγ2mω2−4800γ3mω3+β3,c1=−1400αγ2β4−1120β2γ2ω2−12800βγ3ω3+32000γ4ω4+32000γ4m4ω4−12800βγ3m3ω3−64000γ4m3ω4−1120β2γ2m2ω2+19200βγ3m2ω3+96000γ4m2ω4+1120β2γ2mω2+19200βγ3mω3−64000γ4mω4.cn≡ cnξ,m and ξ=x+λt+ξ0.Using equation (61), the following cnoidal wave solution is obtained:(62)=β+20γω−40γmω10αγ+610γmωαcn2ωx−tβ2+240m2γ2ω2−240mγ2ω2+240γ2ω210γ|m.For lettingm⟶1, solution equation (62) can recover the soliton solution as(63)u=β−20γω10αγ+610γωαsech2ωx−tβ2+240γ2ω210γ.Furthermore, solution equation (54) can be reduced to the following cnoidal wave solution using the relation between WSEFs and JEFs [36] (more details are inserted in Appendix F):(64)=60γ+βC+βCm−120γm10αγCm+1+1810γmαCm+1cn23Cm+1x−β210γ−216m2−m+1γC2m+12t+ξ0,m.Also, the soliton solution can be obtained from solution equation (64) for letting m⟶1:(65)u=2βC−60γ210αγC+9C10γαsech232Cx−β2C2−540γ210γC2t+ξ0.The propagation of higher-order ion-acoustic structures in a collisionless and unmagnetized plasma consisting of inertialess nonextensive electrons and positrons and inertial warm ions and nonextensive electrons as well as positrons [10]is investigated. El-Tantawy[10] derived both two-coupled KdV equations and two-coupled modified KdV (mKdV) equations for studying the KdV and mKdV solitons collisions. For some external perturbation or at some certain conditions, the derivatives fifth-order should be taken into consideration which leads to both KE equation (2) and mKE (3). Now, to analyze the obtained solutions, we can use the same values of the coefficients of mKdV equation (26) in [10]. Based on this plasma model α2=α,β=2.14663,0.698771 at q=0.6 and α2=α,β=0.269511,0.957799 at q=1.2, where q indicates the nonextensive parameter, the profile of the periodic solution equation (47) is illustrated as shown in Figure 3 for the parameter values γ,λ,g3,ξ0,v0=0.1,−0.1,5,0,0. Moreover, the profiles of the solitary wave solutions equation (48) are shown in Figure 4 using the same parameters used in Figure 3 replacing g2,g3=4/3,−8/27. It is clear that the two solutions have opposite polarity, i.e., positive and negative potential. Furthermore, it is noticed that both amplitude and width increase with the increase of the nonextensive parameter q.Figure 3
The profile of the cnoidal wave solution equation (47) to mKE equation (3) is plotted in x,t plane.
(a)(b)(c)(d)Figure 4
The profile of the solitary wave solution equation (49) to mKE equation (3) is plotted in x,t plane.
(a)(b)(c)(d)


## 2.1. Solutions of the Planar Kawahara Equation

In the following sections, we try to find some new solutions including the periodic wave solutions, cnoidal wave solutions, and solitary wave solutions to the planar KE equation (2) p=1.


### 2.1.1. Periodic and Solitary Wave Solutions

For planar KE equation (2), ODE equation (6) reduces to(7)−γv5+βv3+αvv′+λv′=0.Integrating equation (7) once over ξ gives us(8)c0−γv4+βv′′+12αv2+λv=0,where c0 is the integration constant.Multiplying equation (8) by v′ and then integrating it again, we get(9)c0v+c1+12γv′′2+12βv′2−γv3v′+16αv3+12λv2=0,where c1 the new constant of integration. The solution of equation (2) via several approaches is discussed as follows.We seek a solution in the ansatz form:(10)v=∑k=0Ndjφj,where φ≡φξ is a solution to the following Helmholtz equation [31].(11)φ′′+Aφ+Bφ2=0,φ′2=−Aφ2−23Bφ3+2c2.Balancing the highest linear and nonlinear terms in equation (11) gives N=2, so that(12)v=d0+d1φ+d2φ2.Inserting ansatz equation (12) into equation (9), we obtain(13)∑j=06Fjφj=0,where the values of Fjj=0,1,⋯6 are defined in Appendix A, and by solving the following system,(14)Fj=0,we get(15)c0=−157122αγ24598321A4γ4−54418A2β2γ2+1457β4−28561γ2λ2,c1=−128960854α2γ319394118562A6γ6−183532986A4β2γ4+2331348747A4γ5λ+134862A2β4γ2−27589926A2β2γ3λ+94922β6+738699β4γλ−4826809γ3λ3,c2=13Aγ+β263640B2γ31690A2γ2−403Aβγ+31β2,d0=−1507αγ1183A2γ2−910Aβγ+31β2+507γλ,d1=140B13Aγ+β39α,d2=140B2γ3α.Equation (11) has many formulas for its general solutions such as(16)φ=−A2B−6B℘ξ+e0;A2/12,g3,where ℘≡℘ξ+e0;A2/12,g3 indicates the Weierstrass elliptic function (WSEF) and the values of e0 and g3 are undetermined parameters which can be obtained from the initial conditions.Also, the general solution to equation (11) can be expressed by(17)φ=e1−6e1A+Be1A+2Be1+12℘ξ+e0;A2/12,1/216A3−6B2e12A−4B3e13,where the values of e0 and e1 are determined from the initial conditions.Thus, a periodic solution to KE equation (2) according to the values of parameters given in system equation (15) and the value of φ given in equation (16) is obtained as(18)u=910βAγ−7098A2γ2−910Aβγ−31β2−507γλ507αγ−280β13α℘x+λt+ξ0;A2/12,g3+1680γα℘2x+λt+ξ0;A2/12,g3.The constantsA, B, λ, and ξ0 are the arbitraries.Using relation equation (9) with of the parameters given in system equation (15) and the value of φ given in equation (17), the solution to KE equation (2) can be expressed by(19)u=−31β2−13γ39λ−70Aβ+91A2γ+1820Bγe0β+13Aγ+13Bγe0507αγ−280Be0A+Be0β+13Aγ+26Bγe013αA+2Be0+12℘x+λt+ξ0;A2/12,1/216A3−2B2e023A+2Be0+1680B2γe02A+Be02αA+2Be0+12℘2x+λt+ξ0;A2/12,1/216A3−2B2e023A+2Be0,where the constants A, B, λ, and e0 are the arbitraries.From the periodic solution equation (18), the soliton solutions can be obtained using the following hypotheses:(20)cg2=A2/12=43,g3=β3549A2γ2−31β24745520γ3=827,which lead to A,β=4,52γ, and by rearrange solution (18), the following soliton solution is obtained:(21)=−1168γ+3λ3α−1120γα℘x+λt;43,−827,+1680γα℘x+λt;43,−8272.Moreover, the solitary wave solution can be obtained by(22)g2=A2/12=1/12,g3=β3549A2γ2−31β24745520γ3=−1/216,which leads to A,β=1,13γ; then, the periodic solution (18) reduces to the following soliton solution:(23)=−73γ+3λ3α−280γ℘x+λt+ξ0;1/12,−1/216α+1680γ℘x+λt+ξ0;1/12,−1/2162α.Also, the periodic solution to KE (2) can be derived directly in terms of WSEF ℘≡℘x+λt+ξ0;g2,g3 by inserting the following ansatz in (9).(24)u=a0+a1℘+a2℘2,which leads to(25)∑j=06Sjφj=0,where the values of Sjj=0,1,⋯ are defined in Appendix B , and by solving the following system,(26)Sj=0,we have(27)a1=−280βk213α,a2=1680γk4α,c1=−114394744αγ2k11661αa0γk507αa0γk+1014γλ+146β2k−1285245γ2λ2+449159β4k2+1702506β2γkλ,c2=121894405624α2γ3k1521αa0γ507αa0γk6929αa0γk+9126γλ+2189β2k+5277β2k−507γλ507γλ+73β2k−7128781549β2γ2λ2+9635389β6k2+75627162β4γkλ,g2=−507αa0γk+507γλ+31β2k85176γ2k5,g3=−β169αa0γk+169γλ+31β2k3163680γ3k7.Collecting both equations (24) and (27), we finally get(28)u=a0−280βk213α℘kx+tλ+ξ0;−31kβ2+507γλ+507kαγa085176k5γ2,−β31kβ2+169γλ+169kαγa03163680k7γ3+1680γk4α℘kx+tλ+ξ0;−31kβ2+507γλ+507kαγa085176k5γ2,−β31kβ2+169γλ+169kαγa03163680k7γ32.Solution equation (28) satisfies KE equation (2).


### 2.1.2. Cnoidal and Solitary Wave Solutions

We look for a solution to KE equation (2) in the ansatz form(29)ux,t=vξ=∑j=0Ndjcnjξ,where ξ=ωx+λt and N is an integer and positive number. From the balance between the highest-order linear N+4 and nonlinear 2N terms of equation (8), we have N=4. Substituting ansatz equation (29) into equation (6) gives a very complicated system. By solving this system using Mathematica package, we found that the coefficients of the odd power in ansatz equation (29) vanish. Thus, the solution of KE equation (2) could be written in the following ansatz:(30)u=A+Bcn2ξ,m+Ccn4ξ,m.Inserting ansatz equation (30) into KE equation (2), we get(31)RKE=∑j=03Wjcn2j=0,the values of Wjj=0,1,⋯3 are given in Appendix C, and by solving the system,(32)Wj=0,we have(33)A=1507αγ−31β2+3640βγ1−2mω+169γ112γ14m−1m−1ω2−3λ,B=−280mω13α52γ2m−1ω−β,C=1680γm2ω2α,where ω is a solution to the following cubic equation:(34)31β3−56784βγ2m2−m+1ω2+703040γ3m−2m+12m−1ω3=0.Finally, the cnoidal wave solutions to KE equation (2) are obtained as(35)=−31β2+3640βγ1−2mω+169γ112γ14m−1m−1ω2−3λ507αγ,−280mω52γ2m−1ω−β13αcn2ξ,m+1680γm2ω2αcn4ξ,m,where λ is the arbitrary constant, ξ=ωx+λt, and ω is a root to equation (34).The cnoidal wave solution equation (35) can be directly reduced to the soliton solution for m⟶1 as(36)u=−36β2+169γλ169αγ+105β2169αγsech412β13γx+λt.Moreover, solution equation (36) coincides with the obtained one by means of the tanh method:(37)u=105β2169αγsech412β13γx−36β2169γt.The obtained solutions can be employed for investigating the propagation of nonlinear structures in different plasma models. For instance, we can apply these solutions to study cnoidal and solitary waves in the ultracold neutral plasma (UCNP) which is composed of strongly coupled positive ions and non-Maxwellian electron distributions [32–35]. Based on this model and for Maxwellian electrons, the values of the coefficients α,β are given by (to prevent stuffing and repetition, all the details can be found [32])(38)α=λphandβ=12λph,and the phase velocity λph of the ion-acoustic waves (IAWs) reads(39a)λ=1+σ∗,where σ∗≡σ∗Te,Ti represents the effective temperature ratio which is a function of electron and ion temperatures Te,Ti [32–35]. For Te,Ti=25K,1K, we get σ∗=0.401169, and for Te,Ti=900K,1K, we obtain σ∗=0.0120837 [32–35]. With respect to the coefficient of the fifth-order dispersion γ, in general, it has a small value 0<γ≪1. The impact of effective temperature ratio σ∗ on the profile of the cnoidal wave solution equation (35) and the solitary wave solution equation (36) for γ,λ=0.1,0.1 is shown in Figures 1 and 2, respectively. It is observed that increasing the electron temperature, i.e., decreasing σ∗, leads to the enhancement (reduction) of the amplitude (width) of both localized and periodic waves.Figure 1
The profile of the periodic wave solution equation (35) to KE equation (2) plotted in x,t plane.
(a)(b)Figure 2
The profile of the solitary wave solution equation (36) to KE equation (2) is plotted in x,t plane.
(a)(b)


## 2.1.1. Periodic and Solitary Wave Solutions

For planar KE equation (2), ODE equation (6) reduces to(7)−γv5+βv3+αvv′+λv′=0.Integrating equation (7) once over ξ gives us(8)c0−γv4+βv′′+12αv2+λv=0,where c0 is the integration constant.Multiplying equation (8) by v′ and then integrating it again, we get(9)c0v+c1+12γv′′2+12βv′2−γv3v′+16αv3+12λv2=0,where c1 the new constant of integration. The solution of equation (2) via several approaches is discussed as follows.We seek a solution in the ansatz form:(10)v=∑k=0Ndjφj,where φ≡φξ is a solution to the following Helmholtz equation [31].(11)φ′′+Aφ+Bφ2=0,φ′2=−Aφ2−23Bφ3+2c2.Balancing the highest linear and nonlinear terms in equation (11) gives N=2, so that(12)v=d0+d1φ+d2φ2.Inserting ansatz equation (12) into equation (9), we obtain(13)∑j=06Fjφj=0,where the values of Fjj=0,1,⋯6 are defined in Appendix A, and by solving the following system,(14)Fj=0,we get(15)c0=−157122αγ24598321A4γ4−54418A2β2γ2+1457β4−28561γ2λ2,c1=−128960854α2γ319394118562A6γ6−183532986A4β2γ4+2331348747A4γ5λ+134862A2β4γ2−27589926A2β2γ3λ+94922β6+738699β4γλ−4826809γ3λ3,c2=13Aγ+β263640B2γ31690A2γ2−403Aβγ+31β2,d0=−1507αγ1183A2γ2−910Aβγ+31β2+507γλ,d1=140B13Aγ+β39α,d2=140B2γ3α.Equation (11) has many formulas for its general solutions such as(16)φ=−A2B−6B℘ξ+e0;A2/12,g3,where ℘≡℘ξ+e0;A2/12,g3 indicates the Weierstrass elliptic function (WSEF) and the values of e0 and g3 are undetermined parameters which can be obtained from the initial conditions.Also, the general solution to equation (11) can be expressed by(17)φ=e1−6e1A+Be1A+2Be1+12℘ξ+e0;A2/12,1/216A3−6B2e12A−4B3e13,where the values of e0 and e1 are determined from the initial conditions.Thus, a periodic solution to KE equation (2) according to the values of parameters given in system equation (15) and the value of φ given in equation (16) is obtained as(18)u=910βAγ−7098A2γ2−910Aβγ−31β2−507γλ507αγ−280β13α℘x+λt+ξ0;A2/12,g3+1680γα℘2x+λt+ξ0;A2/12,g3.The constantsA, B, λ, and ξ0 are the arbitraries.Using relation equation (9) with of the parameters given in system equation (15) and the value of φ given in equation (17), the solution to KE equation (2) can be expressed by(19)u=−31β2−13γ39λ−70Aβ+91A2γ+1820Bγe0β+13Aγ+13Bγe0507αγ−280Be0A+Be0β+13Aγ+26Bγe013αA+2Be0+12℘x+λt+ξ0;A2/12,1/216A3−2B2e023A+2Be0+1680B2γe02A+Be02αA+2Be0+12℘2x+λt+ξ0;A2/12,1/216A3−2B2e023A+2Be0,where the constants A, B, λ, and e0 are the arbitraries.From the periodic solution equation (18), the soliton solutions can be obtained using the following hypotheses:(20)cg2=A2/12=43,g3=β3549A2γ2−31β24745520γ3=827,which lead to A,β=4,52γ, and by rearrange solution (18), the following soliton solution is obtained:(21)=−1168γ+3λ3α−1120γα℘x+λt;43,−827,+1680γα℘x+λt;43,−8272.Moreover, the solitary wave solution can be obtained by(22)g2=A2/12=1/12,g3=β3549A2γ2−31β24745520γ3=−1/216,which leads to A,β=1,13γ; then, the periodic solution (18) reduces to the following soliton solution:(23)=−73γ+3λ3α−280γ℘x+λt+ξ0;1/12,−1/216α+1680γ℘x+λt+ξ0;1/12,−1/2162α.Also, the periodic solution to KE (2) can be derived directly in terms of WSEF ℘≡℘x+λt+ξ0;g2,g3 by inserting the following ansatz in (9).(24)u=a0+a1℘+a2℘2,which leads to(25)∑j=06Sjφj=0,where the values of Sjj=0,1,⋯ are defined in Appendix B , and by solving the following system,(26)Sj=0,we have(27)a1=−280βk213α,a2=1680γk4α,c1=−114394744αγ2k11661αa0γk507αa0γk+1014γλ+146β2k−1285245γ2λ2+449159β4k2+1702506β2γkλ,c2=121894405624α2γ3k1521αa0γ507αa0γk6929αa0γk+9126γλ+2189β2k+5277β2k−507γλ507γλ+73β2k−7128781549β2γ2λ2+9635389β6k2+75627162β4γkλ,g2=−507αa0γk+507γλ+31β2k85176γ2k5,g3=−β169αa0γk+169γλ+31β2k3163680γ3k7.Collecting both equations (24) and (27), we finally get(28)u=a0−280βk213α℘kx+tλ+ξ0;−31kβ2+507γλ+507kαγa085176k5γ2,−β31kβ2+169γλ+169kαγa03163680k7γ3+1680γk4α℘kx+tλ+ξ0;−31kβ2+507γλ+507kαγa085176k5γ2,−β31kβ2+169γλ+169kαγa03163680k7γ32.Solution equation (28) satisfies KE equation (2).


## 2.1.2. Cnoidal and Solitary Wave Solutions

We look for a solution to KE equation (2) in the ansatz form(29)ux,t=vξ=∑j=0Ndjcnjξ,where ξ=ωx+λt and N is an integer and positive number. From the balance between the highest-order linear N+4 and nonlinear 2N terms of equation (8), we have N=4. Substituting ansatz equation (29) into equation (6) gives a very complicated system. By solving this system using Mathematica package, we found that the coefficients of the odd power in ansatz equation (29) vanish. Thus, the solution of KE equation (2) could be written in the following ansatz:(30)u=A+Bcn2ξ,m+Ccn4ξ,m.Inserting ansatz equation (30) into KE equation (2), we get(31)RKE=∑j=03Wjcn2j=0,the values of Wjj=0,1,⋯3 are given in Appendix C, and by solving the system,(32)Wj=0,we have(33)A=1507αγ−31β2+3640βγ1−2mω+169γ112γ14m−1m−1ω2−3λ,B=−280mω13α52γ2m−1ω−β,C=1680γm2ω2α,where ω is a solution to the following cubic equation:(34)31β3−56784βγ2m2−m+1ω2+703040γ3m−2m+12m−1ω3=0.Finally, the cnoidal wave solutions to KE equation (2) are obtained as(35)=−31β2+3640βγ1−2mω+169γ112γ14m−1m−1ω2−3λ507αγ,−280mω52γ2m−1ω−β13αcn2ξ,m+1680γm2ω2αcn4ξ,m,where λ is the arbitrary constant, ξ=ωx+λt, and ω is a root to equation (34).The cnoidal wave solution equation (35) can be directly reduced to the soliton solution for m⟶1 as(36)u=−36β2+169γλ169αγ+105β2169αγsech412β13γx+λt.Moreover, solution equation (36) coincides with the obtained one by means of the tanh method:(37)u=105β2169αγsech412β13γx−36β2169γt.The obtained solutions can be employed for investigating the propagation of nonlinear structures in different plasma models. For instance, we can apply these solutions to study cnoidal and solitary waves in the ultracold neutral plasma (UCNP) which is composed of strongly coupled positive ions and non-Maxwellian electron distributions [32–35]. Based on this model and for Maxwellian electrons, the values of the coefficients α,β are given by (to prevent stuffing and repetition, all the details can be found [32])(38)α=λphandβ=12λph,and the phase velocity λph of the ion-acoustic waves (IAWs) reads(39a)λ=1+σ∗,where σ∗≡σ∗Te,Ti represents the effective temperature ratio which is a function of electron and ion temperatures Te,Ti [32–35]. For Te,Ti=25K,1K, we get σ∗=0.401169, and for Te,Ti=900K,1K, we obtain σ∗=0.0120837 [32–35]. With respect to the coefficient of the fifth-order dispersion γ, in general, it has a small value 0<γ≪1. The impact of effective temperature ratio σ∗ on the profile of the cnoidal wave solution equation (35) and the solitary wave solution equation (36) for γ,λ=0.1,0.1 is shown in Figures 1 and 2, respectively. It is observed that increasing the electron temperature, i.e., decreasing σ∗, leads to the enhancement (reduction) of the amplitude (width) of both localized and periodic waves.Figure 1
The profile of the periodic wave solution equation (35) to KE equation (2) plotted in x,t plane.
(a)(b)Figure 2
The profile of the solitary wave solution equation (36) to KE equation (2) is plotted in x,t plane.
(a)(b)


## 2.2. Solutions of Planar Modified Kawahara Equation

The generalized KE equation (4) can reduce the planar mKE equation (3) for p=2. Making the traveling wave transformation u=vξ, where ξ=x+λt+ξ0, we get(40)λv′+αv2v′+βv3−γv5=0,and integrating equation (40) twice over ξ, we obtain(41)ℝ≡c0v+c1−γv3v′+1/12αv4+12λv2+12βv′2+12γv′′2=0.Some new localized and periodic solutions to equation (41) are discussed using different approaches in the following sections.


### 2.2.1. First Ansatz in Terms of WSEFs

The following ansatz is introduced to find a periodic wave solution to equation (41) in terms of WSEFs:(42)v=A+B℘,where ℘≡℘ξ+ξ0;g2,g3, g2 and g3 denote the elliptic invariants, while the other parameters A, B, g2, and g3 are the constants and will be determined later.Inserting the ansatz equations (42) into (40), we obtain(43)A2α+18γg2+λ+2AαB+12β℘+αB2−360γ℘2=0.Equating the coefficients of℘0, ℘, and ℘2 to zero and solving the obtained system, we have(44)A=±β10αγ,B=∓610γα,andg2=−β2−10γλ180γ2.Note thatg3 is an arbitrary constant. Using the initial condition v0=v0, we can get(45)A+B℘ξ0;g2,g3=v0,which leads to(46)ξ0=℘−1v0−AB;g2,g3.By substituting the values ofA,B,g2 given in equation (44) into the ansatz equation (42), we finally obtain the solutions of cnoidal wave as(47)u1,2x,t=±β10αγ∓610γα℘x+λt+ξ0;−β2−10γλ180γ2,g3,and these solutions satisfy the evolution equation (3).For the following choice,(48)g2=43andg3=−827,the solitary wave solutions are recovered:(49)u1,2x,t=±β10αγ∓610γα℘x+λt+ξ0;43,−827.


### 2.2.2. Second Ansatz in Terms of WSEFs

The following rational hypothesis/ansatz is assumed to find some analytical solution to equation (41):(50)vξ=A+B1+C℘,where A, B, and C are the undetermined constants and ℘≡℘ξ+ξ0;g2,g3.Inserting the ansatz equations (50) into (41), we have(51)∑j=06Zj℘j=0,where the coefficients Zjj=0,1,…,6 are defined in Appendix D, and by solving the following system,(52)Z0=0,Z1=0,…,Z6=0,we obtain the nontrivial solution:(53)A=60γ+βC10αγC,λ=−β210γ−65C210αγBC+180γ,c0=γ−3/210/α86400γ3+β3C3−2160βγ2C−120BCβC−60γ150C3,c1=1400αγ2C4−800γ3αB2C2+3610αγBC+3240γ+56β2γ3/2C210αBC+180γ+2880βγ5/2C10αBC+120γ+β4−C4,g2=10α/γBC+18015C2,g3=BC15C310α/γ+120.Thus, the traveling wave solutions to mKE equation (3) are expressed by(54)u=60γ+βC10αγC+B1+C℘x−β2/10γ+610αγBC+180γ/5C2t+ξ0;10α/γBC+180/15C2,10α/γ+120/15C3BC.The values of the constantsB and C are arbitrary.The solitary wave solutions can be obtained from the periodic solution equation (54) according to the following choices:(55)g2=1/12,g3=−1/216,g2=43,g3=−827.For the choices equation (55), the soliton solutions are obtained:(56)u=β10αγ+125γ2α4−91+6℘x−β2+15γ2/10γt+ξ0;1/12,−1/216,(57)u=β10αγ+10γα1−182+3℘x−β2+15γ2/10γt+ξ0;4/3,−8/27.


### 2.2.3. Third Ansatz in Terms of JEFs

Using the following ansatz in equation (41),(58)vξ=A+Bcn2,we have(59)∑j=02nYjcnj=0,where the coefficients Yjj=0,2,4,6 are defined in Appendix E, and by solving the following system,(60)Yj=0,we get(61)A=−−β−20γω+40γmω10αγ,B=610γmωα,λ=−β2+240γ2ω2+240γ2m2ω2−240γ2mω210γ,c0=11510αγ3/2−240βγ2ω2+3200γ3ω3+3200γ3m3ω3−240βγ2m2ω2−4800γ3m2ω3+240βγ2mω2−4800γ3mω3+β3,c1=−1400αγ2β4−1120β2γ2ω2−12800βγ3ω3+32000γ4ω4+32000γ4m4ω4−12800βγ3m3ω3−64000γ4m3ω4−1120β2γ2m2ω2+19200βγ3m2ω3+96000γ4m2ω4+1120β2γ2mω2+19200βγ3mω3−64000γ4mω4.cn≡ cnξ,m and ξ=x+λt+ξ0.Using equation (61), the following cnoidal wave solution is obtained:(62)=β+20γω−40γmω10αγ+610γmωαcn2ωx−tβ2+240m2γ2ω2−240mγ2ω2+240γ2ω210γ|m.For lettingm⟶1, solution equation (62) can recover the soliton solution as(63)u=β−20γω10αγ+610γωαsech2ωx−tβ2+240γ2ω210γ.Furthermore, solution equation (54) can be reduced to the following cnoidal wave solution using the relation between WSEFs and JEFs [36] (more details are inserted in Appendix F):(64)=60γ+βC+βCm−120γm10αγCm+1+1810γmαCm+1cn23Cm+1x−β210γ−216m2−m+1γC2m+12t+ξ0,m.Also, the soliton solution can be obtained from solution equation (64) for letting m⟶1:(65)u=2βC−60γ210αγC+9C10γαsech232Cx−β2C2−540γ210γC2t+ξ0.The propagation of higher-order ion-acoustic structures in a collisionless and unmagnetized plasma consisting of inertialess nonextensive electrons and positrons and inertial warm ions and nonextensive electrons as well as positrons [10]is investigated. El-Tantawy[10] derived both two-coupled KdV equations and two-coupled modified KdV (mKdV) equations for studying the KdV and mKdV solitons collisions. For some external perturbation or at some certain conditions, the derivatives fifth-order should be taken into consideration which leads to both KE equation (2) and mKE (3). Now, to analyze the obtained solutions, we can use the same values of the coefficients of mKdV equation (26) in [10]. Based on this plasma model α2=α,β=2.14663,0.698771 at q=0.6 and α2=α,β=0.269511,0.957799 at q=1.2, where q indicates the nonextensive parameter, the profile of the periodic solution equation (47) is illustrated as shown in Figure 3 for the parameter values γ,λ,g3,ξ0,v0=0.1,−0.1,5,0,0. Moreover, the profiles of the solitary wave solutions equation (48) are shown in Figure 4 using the same parameters used in Figure 3 replacing g2,g3=4/3,−8/27. It is clear that the two solutions have opposite polarity, i.e., positive and negative potential. Furthermore, it is noticed that both amplitude and width increase with the increase of the nonextensive parameter q.Figure 3
The profile of the cnoidal wave solution equation (47) to mKE equation (3) is plotted in x,t plane.
(a)(b)(c)(d)Figure 4
The profile of the solitary wave solution equation (49) to mKE equation (3) is plotted in x,t plane.
(a)(b)(c)(d)


## 2.2.1. First Ansatz in Terms of WSEFs

The following ansatz is introduced to find a periodic wave solution to equation (41) in terms of WSEFs:(42)v=A+B℘,where ℘≡℘ξ+ξ0;g2,g3, g2 and g3 denote the elliptic invariants, while the other parameters A, B, g2, and g3 are the constants and will be determined later.Inserting the ansatz equations (42) into (40), we obtain(43)A2α+18γg2+λ+2AαB+12β℘+αB2−360γ℘2=0.Equating the coefficients of℘0, ℘, and ℘2 to zero and solving the obtained system, we have(44)A=±β10αγ,B=∓610γα,andg2=−β2−10γλ180γ2.Note thatg3 is an arbitrary constant. Using the initial condition v0=v0, we can get(45)A+B℘ξ0;g2,g3=v0,which leads to(46)ξ0=℘−1v0−AB;g2,g3.By substituting the values ofA,B,g2 given in equation (44) into the ansatz equation (42), we finally obtain the solutions of cnoidal wave as(47)u1,2x,t=±β10αγ∓610γα℘x+λt+ξ0;−β2−10γλ180γ2,g3,and these solutions satisfy the evolution equation (3).For the following choice,(48)g2=43andg3=−827,the solitary wave solutions are recovered:(49)u1,2x,t=±β10αγ∓610γα℘x+λt+ξ0;43,−827.


## 2.2.2. Second Ansatz in Terms of WSEFs

The following rational hypothesis/ansatz is assumed to find some analytical solution to equation (41):(50)vξ=A+B1+C℘,where A, B, and C are the undetermined constants and ℘≡℘ξ+ξ0;g2,g3.Inserting the ansatz equations (50) into (41), we have(51)∑j=06Zj℘j=0,where the coefficients Zjj=0,1,…,6 are defined in Appendix D, and by solving the following system,(52)Z0=0,Z1=0,…,Z6=0,we obtain the nontrivial solution:(53)A=60γ+βC10αγC,λ=−β210γ−65C210αγBC+180γ,c0=γ−3/210/α86400γ3+β3C3−2160βγ2C−120BCβC−60γ150C3,c1=1400αγ2C4−800γ3αB2C2+3610αγBC+3240γ+56β2γ3/2C210αBC+180γ+2880βγ5/2C10αBC+120γ+β4−C4,g2=10α/γBC+18015C2,g3=BC15C310α/γ+120.Thus, the traveling wave solutions to mKE equation (3) are expressed by(54)u=60γ+βC10αγC+B1+C℘x−β2/10γ+610αγBC+180γ/5C2t+ξ0;10α/γBC+180/15C2,10α/γ+120/15C3BC.The values of the constantsB and C are arbitrary.The solitary wave solutions can be obtained from the periodic solution equation (54) according to the following choices:(55)g2=1/12,g3=−1/216,g2=43,g3=−827.For the choices equation (55), the soliton solutions are obtained:(56)u=β10αγ+125γ2α4−91+6℘x−β2+15γ2/10γt+ξ0;1/12,−1/216,(57)u=β10αγ+10γα1−182+3℘x−β2+15γ2/10γt+ξ0;4/3,−8/27.


## 2.2.3. Third Ansatz in Terms of JEFs

Using the following ansatz in equation (41),(58)vξ=A+Bcn2,we have(59)∑j=02nYjcnj=0,where the coefficients Yjj=0,2,4,6 are defined in Appendix E, and by solving the following system,(60)Yj=0,we get(61)A=−−β−20γω+40γmω10αγ,B=610γmωα,λ=−β2+240γ2ω2+240γ2m2ω2−240γ2mω210γ,c0=11510αγ3/2−240βγ2ω2+3200γ3ω3+3200γ3m3ω3−240βγ2m2ω2−4800γ3m2ω3+240βγ2mω2−4800γ3mω3+β3,c1=−1400αγ2β4−1120β2γ2ω2−12800βγ3ω3+32000γ4ω4+32000γ4m4ω4−12800βγ3m3ω3−64000γ4m3ω4−1120β2γ2m2ω2+19200βγ3m2ω3+96000γ4m2ω4+1120β2γ2mω2+19200βγ3mω3−64000γ4mω4.cn≡ cnξ,m and ξ=x+λt+ξ0.Using equation (61), the following cnoidal wave solution is obtained:(62)=β+20γω−40γmω10αγ+610γmωαcn2ωx−tβ2+240m2γ2ω2−240mγ2ω2+240γ2ω210γ|m.For lettingm⟶1, solution equation (62) can recover the soliton solution as(63)u=β−20γω10αγ+610γωαsech2ωx−tβ2+240γ2ω210γ.Furthermore, solution equation (54) can be reduced to the following cnoidal wave solution using the relation between WSEFs and JEFs [36] (more details are inserted in Appendix F):(64)=60γ+βC+βCm−120γm10αγCm+1+1810γmαCm+1cn23Cm+1x−β210γ−216m2−m+1γC2m+12t+ξ0,m.Also, the soliton solution can be obtained from solution equation (64) for letting m⟶1:(65)u=2βC−60γ210αγC+9C10γαsech232Cx−β2C2−540γ210γC2t+ξ0.The propagation of higher-order ion-acoustic structures in a collisionless and unmagnetized plasma consisting of inertialess nonextensive electrons and positrons and inertial warm ions and nonextensive electrons as well as positrons [10]is investigated. El-Tantawy[10] derived both two-coupled KdV equations and two-coupled modified KdV (mKdV) equations for studying the KdV and mKdV solitons collisions. For some external perturbation or at some certain conditions, the derivatives fifth-order should be taken into consideration which leads to both KE equation (2) and mKE (3). Now, to analyze the obtained solutions, we can use the same values of the coefficients of mKdV equation (26) in [10]. Based on this plasma model α2=α,β=2.14663,0.698771 at q=0.6 and α2=α,β=0.269511,0.957799 at q=1.2, where q indicates the nonextensive parameter, the profile of the periodic solution equation (47) is illustrated as shown in Figure 3 for the parameter values γ,λ,g3,ξ0,v0=0.1,−0.1,5,0,0. Moreover, the profiles of the solitary wave solutions equation (48) are shown in Figure 4 using the same parameters used in Figure 3 replacing g2,g3=4/3,−8/27. It is clear that the two solutions have opposite polarity, i.e., positive and negative potential. Furthermore, it is noticed that both amplitude and width increase with the increase of the nonextensive parameter q.Figure 3
The profile of the cnoidal wave solution equation (47) to mKE equation (3) is plotted in x,t plane.
(a)(b)(c)(d)Figure 4
The profile of the solitary wave solution equation (49) to mKE equation (3) is plotted in x,t plane.
(a)(b)(c)(d)


## 3. Conclusions

New localized and periodic traveling wave solutions to the generalized KE have been derived in detail using different new approaches and ansatz. As a particular case, several traveling wave solutions to both KE and mKE have been obtained using (in)direct methods. For the indirect method, KE has been solved with the help of Helmholtz equation. After that, we can use any solution to the Helmholtz equation in order to express the solution of the planar KE. We used two different formulas for WSEFs to get some periodic solutions to KE. In the direct method, a new ansatz in the terms of WSEFs has been introduced for getting a cnoidal solution to KE. In all cases and at certain conditions, the periodic solutions have been reduced to the localized solitary wave solutions. In the third (direct) method, the periodic and solitary wave solutions have been derived in the form of JEFs, and it was found that the obtained solutions coincide with that obtained by means of the tanh method. The obtained solutions have been used for interpreting several nonlinear structures that propagate in different plasma models. Furthermore, two new hypotheses in terms of WSEFs have been proposed to find some periodic solutions to mKE. Also, the conditions for reducing the periodic solutions of mKE to the localized solitary waves have been presented. The obtained solutions have been employed for investigating many nonlinear structures in different plasma models.



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*Source: 9942267-2022-06-02.xml*