Dissipative Kawahara ion-acoustic solitary and cnoidal waves in a degenerate magnetorotating plasma

Abstract

The two-dimensional fluid quantum hydrodynamic (QHD) is adopted as the basis for a discussion of the effects of Landau quantization magnetic field, the Coriolis force, and collisional frequency on the (non)linear properties of the dissipative ion-acoustic waves (IAWs). By employing the reductive perturbation technique (RPT), the damped Korteweg–de Vries (KdV) equation which contains the lowest perturbation order actions is derived. It was found that with an increase in amplitude, the soliton width and the velocity diverge from the prediction of the damped KdV equation as observed in some laboratory experiments, i.e. the damped KdV approximation becomes invalid to describe the system. Therefore, it is necessary to investigate the effect of higher-order which leads to the damped Kawahara equation. This equation is a completely non-integrable differential equation. Thus, a new approximate solution which is called a semi-analytical solution is derived in detail. The obtained results can help in understanding the features of quantum IAWs in dense and slowly rotating astrophysical plasmas and maybe understand the quantum Hall effect of novel materials like graphene and topological insulators.

Keywords:

• Damped Kawahara equation
• dissipative solitary and cnoidal waves
• quantum plasma
• magnetoplasmas
• Landau quanitization
• Coriolis force

1. Introduction

Decades ago, numerous investigations have been carried out to study a degenerate plasma rotating in a magnetic field due to their applications in astrophysical plasmas of high-density [1,2] as well as in modern technology [3–6]. In such degenerate plasma, in addition to the effect of the degenerate electron which obeys Fermi–Dirac statistics [7,8] and the non-localized particles that cause the dispersive effects of particles, one also encounters two effects (the Pauli paramagnetism and the Landau diamagnetism) due to applying a strong magnetic field on degenerate plasma. Furthermore, the plasma rotating flow in the presence of a magnetic field causes a Coriolis force that may have a significant influence in the cosmos and in rotating plasma devices [9–16]. Where the Coriolis force has the inclination to output an effective magnetic field like the effect when the ionized medium rotates. Thus, many scientists have been pointed out to study the dynamics of a rotating magnetized plasma time and again. The excitations of heavy nucleus-acoustic (HNA) have been investigated by analysing the Zakharov–Kuznetsov (ZK) Burger's (ZKB) equation in a degenerate magneto-rotating degenerate plasma [17]. The effects of quantum, the rotation angle, and the strength of the magnetic field on the linear and nonlinear propagation of IAWs in an electron-ion (e−i) dense plasma have been illustrated by Sahu and Sinha [18]. Saini et al. [19] studied the influence of rotational effect on the HNA periodic structures in relativistic magneto-rotating quantum plasma. Shukla and Eliasson [20] studied the magnetic field effect on the waves of plasma and develop a quantum magnetohydrodynamic equation.

Cnoidal waves (CWs) are nonlinear periodic wave structures that have solutions discovered by Korteweg and de Vries in 1985 from the KdV equation with periodic boundary conditions expounds characteristics of the cnoidal wave (CW) (flatter troughs and narrow crests). This periodic wave has solutions of the KdV equation that can include the terms of the Jacobian elliptic cosine (cn) function [21]. Recently, periodic (cnoidal) waves have received a lot of attention due to it is extensive applications in diverse areas of physics such as in photorefractive bismuth titanate crystal [22] and in shallow water [23,24] and it is also believed that these waves are generated in plasmas defocusing regime [25]. Mahmood and Haas [26] reported the study on the ion-acoustic CW (IACWs) in a degenerate plasma. They found that the structures of both compressive and rarefactive IACWs are modified significantly by the quantum parameters. Plasma nonextensivity effects on IACWs in e−p−i  plasma investigated by Farhadkiyaei and Dorranian [27]. The characteristics of the propagation of IACWs in a magnetized quantum plasma consisting of both spin-up and spin-down state species discussed by Kaur et al. [28]. It is noticed that the quantum parameter has strongly impressed with the dynamics of IACWs. Cnoidal waves were investigated in a magnetized quantum dusty plasma with negatively and positively charged dust grains, inertialess electrons, and ions by Sethi and Saini [29].

Dissipative IAWs has been a topic of investigation in plasmas, where, dissipation may grow due to collisions inter the microparticles of plasma, as well as the ion-neutral collisions. Masood et al. [30] have investigated the characteristics of the IAWs in the electron–positron-ion $(e-p-i)$ magnetoplasma by using the quantum hydrodynamic model (QHD). Pakzad [31] has examined the IA shock waves (IASWs) in quantum dissipative plasma. He has concluded the KdV Burger'S (KdVB) equation and discussed the effects of kinematic viscosity and relativistic ions on waves. Ghosh et al. [32] have studied the effects of collisions between ions on the characteristics of IAWs in pair-ion plasma. They have derived the damped KdV and solved it to get the weakly dissipative solitons. Sahu et al. [33] have used the QHD and the dissipative KdV equation paradigm to study the nonlinear dynamics of IAWs in collisional degenerate electron-ion plasma with relativistic electrons. Farooq et al. [34] have reported DIASWs in collisional and magneto-rotating plasma, by using the modified Zakharov–Kuznetsov (mZK) equation. The propagation of the nonlinear IAWs in dense plasma has been studied by Ghosh and Sahu [35] by deriving Kadomtsev–Petviashvili (KP) Burger's (KPB) equation. It is noted from the above studies, the authors have investigated the weakly nonlinear dissipative IAWs with small dispersion which can be modelled by the KdV equation or its family [36,37]: (1) ${\varphi }_t+\mathrm{F\varphi }{\varphi }_x+G{\varphi }_{\mathrm{xxx}}=0,$(1) under certain states of affairs in the laboratory experiments [38], we observed the KdV equation cannot give an accurate description of many nonlinear waves at very small or even close to zero of dispersion coefficient F, so it must be taken the next higher-order perturbation to preserve the harmony between the dispersion and nonlinearity [39,40]. In such cases, we can obtain new types of evolution equations with the fifth-order dispersion term which is called the Kawahara-type equation, or the fifth-order KdV-type equation such as Kawahara equation (KE) [41–56] (2) ${\varphi }_t+\mathrm{F\varphi }{\varphi }_x+G{\varphi }_{\mathrm{xxx}}-K{\varphi }_{\mathrm{xxxxx}}=0.$(2) In most realistic models, we can not ignore the effects on the frictional forces, which leads to some complicated differential equations such as the damped KE $({\varphi }_t+\mathrm{F\varphi }{\varphi }_x+G{\varphi }_{\mathrm{xxx}}-K{\varphi }_{\mathrm{xxxxx}}+\mathrm{R\varphi }=0)$. In general, there is great growth and development in devising a huge number of evolution (differential and integral) equations that could be used for describing many natural phenomena lately, e.g. [49–61].

Sahu [62] studied the nonplanar IAWs in a collisional unmagnetized quantum e−i plasma. He derived the nonplanar Kakutani and Kawahara equation by using the RPT. It noted that the quantum effects can grow the dissipation of the IAWs structures in the presence of collision.

Our main thrust in this paper will be to address the problem at very small or even zero dispersion coefficient G and study the effects of Landau quantization magnetic field, the Coriolis force, and the collisional frequency on the dissipative Kawahara SWs and CWs in quantum e−i plasma. The structure of this manuscript goes as follows. The basic formalism describing the plasma model and the dispersion relation with the orbital quantum motion of electrons and Coriolis force are introduced in Section 2. In Section 3, the damped KdV equation and damped KE are obtained by applying the RPT and the approximate solution of the damped KE solution is presented. The effects of plasma parameters on the waves are discussed in Section 4. Finally, our results are epitomized in Section 5 of the manuscript.

2. Basic formalism and dispersion relation

We shall now consider a homogenous, collisional, and magnetized quantum plasma consisting of positive ions and inertialess electrons embedded in an external $z-$direction magnetic field, i.e. $B=B_0\hat{z}$. The propagation of electrostatic structure is considered in the xz-plane, i.e. $\mathrm{\nabla }=({\mathrm{\partial }}_x,0,{\mathrm{\partial }}_z)$, and is governed by the QHD model as the following: (3) $\begin{array}{rl}{\mathrm{\partial }}_t{n}_i+\mathrm{\nabla }.\left(n_i{\mathbf{v}}_i\right)& =0,\end{array}$(3) (4) $\begin{array}{rl}\frac{m_i}{e}({\mathrm{\partial }}_t{\mathbf{v}}_i+({\mathbf{v}}_i.\mathrm{\nabla }){\mathbf{v}}_i)& =\mathbf{E}+\frac{1}{c}({\mathbf{v}}_i\times \mathbf{B})\\ & +\frac{1}{e}(2({\mathbf{v}}_i\times \mathbf{\Omega })-\nu {\mathbf{v}}_i),\end{array}$(4) (5) $\begin{array}{rl}\mathrm{\nabla }\mathbf{.}\mathbf{E}& \mathbf{=}4\mathrm{\pi e}(n_i-n_e).\end{array}$(5) Here, $n_{i(e)}$ represents the ion (electron) current density, $v_i$ is the ion fluid velocity, $E(\mathbf{=}\mathbf{-}\mathrm{\nabla }\mathrm{\Phi })$ denotes the electric field, where Φ is electrostatic potential, c is the light speed in vacuum, ν is the collision frequency for ion-neutral plasma, while $\mathrm{\Omega }={\mathrm{\Omega }}_{0x}\hat{x}+{\mathrm{\Omega }}_{0z}\hat{z}$ is the angular rotation frequency vector with magnitude ${\mathrm{\Omega }}_{0x}={\mathrm{\Omega }}_0\cos \theta$ and ${\mathrm{\Omega }}_{0z}={\mathrm{\Omega }}_0\sin \theta$. In this system, we consider a slowly rotating ($\mathrm{\Omega }<1)$ so the higher terms and centrifugal force $\mathrm{\Omega }\times (\mathrm{\Omega }\times r)$ may be neglected here [63].

The presence of a strong magnetic field in the Fermion gas produces to two magnetic effects, One of them is the Pauli paramagnetism which results from the spin of electrons and another is the Landau diamagnetism which produces a result of the quantization of the orbital motion of electrons [64,65]. These magnetic effects on the waves propagation of is very important in astrophysical plasmas of high-density (supernovae and neutron stars, the convective zone of the sun, the early prestellar period of the evolution of the universe) [1,2] as well as in modern technology (the contemporary problems of laser-matter interaction) [3–6]. According to the astrophysical data, the magnetic field on the surface of a neutron star is $H\sim {10}^{\mathbf{11}}-{10}^{\mathbf{13}}G$, and the internal field can reach $H\sim {10}^{\mathbf{15}}G$ or even higher [66,67]. It was shown in Ref. [17] that the presence of rotation of stars may increase the magnetic field by an additional factor of ${10}^{\mathbf{3}}-{10}^{\mathbf{4}}$.

The total electron number density is defined as (see the details elucidated in Refs. [64,65,68–70]) (6) $\begin{array}{rl}{n}_e& =N_0[\frac{3}{2}\eta (1+\mathrm{\Phi }{)}^{1/2}+(1+\mathrm{\Phi }-\eta {)}^{3/2}\\ & -\frac{{\tilde{T}}^2}{2}\eta (1+\mathrm{\Phi }{)}^{-3/2}+{\tilde{T}}^2(1+\mathrm{\Phi }-\eta {)}^{-1/2}],\end{array}$(6) where $\eta (=h{\omega }_{\mathrm{ce}}/2\pi k_B{T}_{\mathrm{Fe}})$ is the coefficient which the effect of the quantizing magnetic field appears and $N_0(=P_{\mathrm{Fe}}^3/3{\pi }^2{\hslash }^3)$ is the number density for fully degenerate plasma. The potential Φ is normalized by factor $\mathrm{e\varphi }/{\epsilon }_{\mathrm{Fe}}$ and the temperature $\tilde{T}$ is normalized by $\mathrm{\pi T}/2^{3/2}{\epsilon }_{\mathrm{Fe}}$ in the above expression. The normalized density of degenerate electrons takes a form: (7) $n_e=\alpha +{\alpha }_1\mathrm{\Phi }+{\alpha }_2{\mathrm{\Phi }}^2+\cdots ,$(7) where α, ${\alpha }_1$, and ${\alpha }_2$, are defined as follows, $\begin{array}{rl}\alpha & =\frac{\eta }{2}(3-{\tilde{T}}^2)+{\tilde{T}}^2{(1-\eta )}^{-1/2}+{(1-\eta )}^{3/2},\\ {\alpha }_1& =\frac{3}{2}[\frac{\eta }{2}(1+{\tilde{T}}^2)+{(1-\eta )}^{1/2}-\frac{{\tilde{T}}^2}{3}{(1-\eta )}^{-3/2}],\\ {\alpha }_2& =\frac{3}{8}[{(1-\eta )}^{-1/2}-\frac{\eta }{2}(1+5{\tilde{T}}^2)\\ & +{\tilde{T}}^2{(1-\eta )}^{-5/2}].\end{array}$ The normalized form of Equations  (3(3) $\begin{array}{rl}{\mathrm{\partial }}_t{n}_i+\mathrm{\nabla }.\left(n_i{\mathbf{v}}_i\right)& =0,\end{array}$(3) )– (5(5) $\begin{array}{rl}\mathrm{\nabla }\mathbf{.}\mathbf{E}& \mathbf{=}4\mathrm{\pi e}(n_i-n_e).\end{array}$(5) ) are given by (8) $\begin{array}{rl}& {\mathrm{\partial }}_{\stackrel{\sim }{t}}{\stackrel{\sim }{n}}_i+{\mathrm{\partial }}_{\stackrel{\sim }{x}}\left({\stackrel{\sim }{n}}_i\stackrel{\sim }{v_{\mathrm{ix}}}\right)+{\mathrm{\partial }}_{\stackrel{\sim }{z}}\left({\stackrel{\sim }{n}}_i\stackrel{\sim }{v_{\mathrm{iz}}}\right)=0,\end{array}$(8) (9) $\begin{array}{rl}& {\mathrm{\partial }}_{\stackrel{\sim }{t}}\stackrel{\sim }{v_{\mathrm{ix}}}+(\stackrel{\sim }{v_{\mathrm{ix}}}{\mathrm{\partial }}_{\stackrel{\sim }{x}}+\stackrel{\sim }{v_{\mathrm{iz}}}{\mathrm{\partial }}_{\stackrel{\sim }{z}})\stackrel{\sim }{v_{\mathrm{ix}}}=-{\mathrm{\partial }}_{\stackrel{\sim }{x}}\mathrm{\Phi }+{\omega }_{\mathrm{ci}}\stackrel{\sim }{v_{\mathrm{iy}}}\\ & +2{\mathrm{\Omega }}_0\cos \theta \stackrel{\sim }{v_{\mathrm{iy}}}-\nu \stackrel{\sim }{v_{\mathrm{ix}}},\end{array}$(9) (10) $\begin{array}{rl}& {\mathrm{\partial }}_{\stackrel{\sim }{t}}\stackrel{\sim }{v_{\mathrm{iy}}}+(\stackrel{\sim }{v_{\mathrm{ix}}}{\mathrm{\partial }}_{\stackrel{\sim }{x}}+\stackrel{\sim }{v_{\mathrm{iz}}}{\mathrm{\partial }}_{\stackrel{\sim }{z}})\stackrel{\sim }{v_{\mathrm{iy}}}=-{\omega }_{\mathrm{ci}}\stackrel{\sim }{v_{\mathrm{ix}}}-2{\mathrm{\Omega }}_0\cos \theta \stackrel{\sim }{v_{\mathrm{ix}}}\\ & +2{\mathrm{\Omega }}_0\sin \theta \stackrel{\sim }{v_{\mathrm{iz}}}-\nu \stackrel{\sim }{v_{\mathrm{iy}}},\end{array}$(10) (11) $\begin{array}{rl}& {\mathrm{\partial }}_{\stackrel{\sim }{t}}\stackrel{\sim }{v_{\mathrm{iz}}}+(\stackrel{\sim }{v_{\mathrm{ix}}}{\mathrm{\partial }}_{\stackrel{\sim }{x}}+\stackrel{\sim }{v_{\mathrm{iz}}}{\mathrm{\partial }}_{\stackrel{\sim }{z}})\stackrel{\sim }{v_{\mathrm{iz}}}=-{\mathrm{\partial }}_{\stackrel{\sim }{z}}\mathrm{\Phi }-2{\mathrm{\Omega }}_0\sin \theta \stackrel{\sim }{v_{\mathrm{iy}}}\\ & -\nu \stackrel{\sim }{v_{\mathrm{iz}}},\end{array}$(11) (12) $\begin{array}{rl}& ({\mathrm{\partial }}_{\stackrel{\sim }{x}}^2+{\mathrm{\partial }}_{\stackrel{\sim }{z}}^2)\stackrel{\sim }{\mathrm{\Phi }}=\stackrel{\sim }{n_e}-{\stackrel{\sim }{n}}_i.\end{array}$(12) To drive the linear dispersion relation for the current plasma model, we assumed the normalized density, potential, and velocity parameters $(\stackrel{\sim }{D}(x,z,t)\equiv [\stackrel{\sim }{n_i}(\stackrel{\sim }{x},\stackrel{\sim }{z},\stackrel{\sim }{t})\stackrel{\sim }{{\mathbf{v}}_i}(\stackrel{\sim }{x},\stackrel{\sim }{z},\stackrel{\sim }{t})\stackrel{\sim }{\mathrm{\Phi }}(\stackrel{\sim }{x},\stackrel{\sim }{z},\stackrel{\sim }{t})])$ are expand as $D=D_0+\stackrel{\sim }{D}(D_0\equiv [100])$, whereas the parameter $\stackrel{\sim }{D}$ are proportional to (13) $\stackrel{\sim }{D}=D_A\exp (i(k_x\stackrel{\sim }{x}+k_z\stackrel{\sim }{z}-\omega \stackrel{\sim }{t})),$(13) where $D_A$ is the perturbation amplitude, $(k,\omega )$ are the normalized wave vector and frequency, respectively. By applying the transformation in Equation (13(13) $\stackrel{\sim }{D}=D_A\exp (i(k_x\stackrel{\sim }{x}+k_z\stackrel{\sim }{z}-\omega \stackrel{\sim }{t})),$(13) ) into Equations  (8(8) $\begin{array}{rl}& {\mathrm{\partial }}_{\stackrel{\sim }{t}}{\stackrel{\sim }{n}}_i+{\mathrm{\partial }}_{\stackrel{\sim }{x}}\left({\stackrel{\sim }{n}}_i\stackrel{\sim }{v_{\mathrm{ix}}}\right)+{\mathrm{\partial }}_{\stackrel{\sim }{z}}\left({\stackrel{\sim }{n}}_i\stackrel{\sim }{v_{\mathrm{iz}}}\right)=0,\end{array}$(8) )– (12(12) $\begin{array}{rl}& ({\mathrm{\partial }}_{\stackrel{\sim }{x}}^2+{\mathrm{\partial }}_{\stackrel{\sim }{z}}^2)\stackrel{\sim }{\mathrm{\Phi }}=\stackrel{\sim }{n_e}-{\stackrel{\sim }{n}}_i.\end{array}$(12) ) we get the following dispersion relation: (14) $A_1{\omega }^4-A_2{\omega }^2+A_3=0,$(14) where $A_1=k^2+{\alpha }_1$, $A_2=({\chi }^2{\sin }^2\theta +{\mathrm{\Omega }}_{\mathrm{eff}}^2){\alpha }_1+k^2$, and $A_3=(k_z{\mathrm{\Omega }}_{\mathrm{eff}}+k_x\mathrm{\chi sin}\theta {)}^2$, and by solving Equation (14(14) $A_1{\omega }^4-A_2{\omega }^2+A_3=0,$(14) ), we get ${\omega }^2=\frac{A_2\pm \sqrt{{\left(A_2\right)}^2-4A_1{A}_3}}{2A_1}.$ We have defined ${\mathrm{\Omega }}_{\mathrm{eff}}={\omega }_{\mathrm{ci}}+\mathrm{\chi cos}\theta$ and $\chi =2{\mathrm{\Omega }}_{{}_0}$ in the above expression. When ${\mathrm{\Omega }}_{{}_0}=0$ (without rotation), the dispersion relation become as: (15) $A_{11}{\omega }^4-A_{21}{\omega }^2+A_{31}=0,$(15) where $A_{11}=k^2+{\alpha }_1$, $A_{21}={\omega }_{\mathrm{ci}}^2{\alpha }_1+k^2$, and $A_{31}=k_z^2{\omega }_{\mathrm{ci}}^2$.

In Figure 1, we have plotted the dispersion relation in the presence Equation (14(14) $A_1{\omega }^4-A_2{\omega }^2+A_3=0,$(14) ) and absence Equation (15(15) $A_{11}{\omega }^4-A_{21}{\omega }^2+A_{31}=0,$(15) ) of rotation. We have observed that in the presence of Landau quantization there are critical points beyond which wave does not propagate. In the absence of rotation, i.e. $\chi =0$, wave start propagating earlier in comparison to the case when the rotation of the plasma is considered.

Figure 1. Plot of ω versus k, for the absence and presence of rotation for parameters, at $\chi =0$ for solid (black) curve, $\chi =0.7$ for dotdashed (red) curve with $\tilde{T}=0.3$, $\theta =2^{\circ }$, $\eta =0.2$, and $l_z=0.3$

3. Small amplitude electrostatic excitations

For studying the dissipative IASWs in quantum e−i plasma, the RP scheme could be applied to derive the governing equation [71]. Accordingly, the following stretching to the autonomous variables are considered (16) $\begin{array}{rl}\xi & ={ϵ}^{1/2}(l_x\stackrel{\sim }{x}+l_z\stackrel{\sim }{z}-\lambda \stackrel{\sim }{t}),\tau ={ϵ}^{3/2}\stackrel{\sim }{t}\mathrm{and}\\ \nu & ={ϵ}^{3/2}{\nu }_0,\end{array}$(16) where $l_x$ $(l_z)$ indicates the wave vector's directional cosines along $x-$ $(z)-$axes, such that $l_x^2+l_z^2=1$, λ is the linearized phase velocity of the IASWs, and ϵ is a small and real parameter $(0<ϵ<<1)$ which measures the strength of the perturbed amplitude to the corresponding equilibrium quantity.

The different perturbed variables in term of small expansion parameter ϵ are expanded as: (17) $\begin{array}{rl}{\stackrel{\sim }{n}}_i& =1+ϵ\stackrel{\sim }{n_{i1}}+{ϵ}^2\stackrel{\sim }{n_{i2}}+\cdots ,\\ \stackrel{\sim }{v_{\mathrm{ix}}}& ={ϵ}^2\stackrel{\sim }{v_{\mathrm{ix}1}}+{ϵ}^3\stackrel{\sim }{v_{\mathrm{ix}2}}+\cdots ,\\ \stackrel{\sim }{v_{\mathrm{iy}}}& ={ϵ}^{\frac{3}{2}}\stackrel{\sim }{v_{\mathrm{iy}1}}+{ϵ}^{\frac{5}{2}}\stackrel{\sim }{v_{\mathrm{iy}2}}+\cdots ,\\ \stackrel{\sim }{v_{\mathrm{iz}}}& =ϵ\stackrel{\sim }{v_{\mathrm{iz}1}}+{ϵ}^2\stackrel{\sim }{v_{\mathrm{iz}2}}+\cdots ,\\ \stackrel{\sim }{\mathrm{\Phi }}& =ϵ{\stackrel{\sim }{\mathrm{\Phi }}}_1+{ϵ}^2{\stackrel{\sim }{\mathrm{\Phi }}}_2+\cdots .\end{array}$(17) Using the stretching given in Equation (16(16) $\begin{array}{rl}\xi & ={ϵ}^{1/2}(l_x\stackrel{\sim }{x}+l_z\stackrel{\sim }{z}-\lambda \stackrel{\sim }{t}),\tau ={ϵ}^{3/2}\stackrel{\sim }{t}\mathrm{and}\\ \nu & ={ϵ}^{3/2}{\nu }_0,\end{array}$(16) ) and expansions given in Equation (17(17) $\begin{array}{rl}{\stackrel{\sim }{n}}_i& =1+ϵ\stackrel{\sim }{n_{i1}}+{ϵ}^2\stackrel{\sim }{n_{i2}}+\cdots ,\\ \stackrel{\sim }{v_{\mathrm{ix}}}& ={ϵ}^2\stackrel{\sim }{v_{\mathrm{ix}1}}+{ϵ}^3\stackrel{\sim }{v_{\mathrm{ix}2}}+\cdots ,\\ \stackrel{\sim }{v_{\mathrm{iy}}}& ={ϵ}^{\frac{3}{2}}\stackrel{\sim }{v_{\mathrm{iy}1}}+{ϵ}^{\frac{5}{2}}\stackrel{\sim }{v_{\mathrm{iy}2}}+\cdots ,\\ \stackrel{\sim }{v_{\mathrm{iz}}}& =ϵ\stackrel{\sim }{v_{\mathrm{iz}1}}+{ϵ}^2\stackrel{\sim }{v_{\mathrm{iz}2}}+\cdots ,\\ \stackrel{\sim }{\mathrm{\Phi }}& =ϵ{\stackrel{\sim }{\mathrm{\Phi }}}_1+{ϵ}^2{\stackrel{\sim }{\mathrm{\Phi }}}_2+\cdots .\end{array}$(17) ) into the Equations  (8(8) $\begin{array}{rl}& {\mathrm{\partial }}_{\stackrel{\sim }{t}}{\stackrel{\sim }{n}}_i+{\mathrm{\partial }}_{\stackrel{\sim }{x}}\left({\stackrel{\sim }{n}}_i\stackrel{\sim }{v_{\mathrm{ix}}}\right)+{\mathrm{\partial }}_{\stackrel{\sim }{z}}\left({\stackrel{\sim }{n}}_i\stackrel{\sim }{v_{\mathrm{iz}}}\right)=0,\end{array}$(8) )–(12(12) $\begin{array}{rl}& ({\mathrm{\partial }}_{\stackrel{\sim }{x}}^2+{\mathrm{\partial }}_{\stackrel{\sim }{z}}^2)\stackrel{\sim }{\mathrm{\Phi }}=\stackrel{\sim }{n_e}-{\stackrel{\sim }{n}}_i.\end{array}$(12) ), so we can get the lowest-order equations of ϵ as: (18) $\begin{array}{rl}\stackrel{\sim }{v_{\mathrm{iz}1}}& =\frac{\lambda }{l_z}\stackrel{\sim }{n_{i1}},\\ \stackrel{\sim }{v_{\mathrm{iy}1}}& =\frac{l_x}{{\mathrm{\Omega }}_{\mathrm{eff}}}{\mathrm{\partial }}_{\xi }\stackrel{\sim }{{\mathrm{\Phi }}_1},\\ \stackrel{\sim }{v_{\mathrm{ix}1}}& =\frac{\lambda }{{\mathrm{\Omega }}_{\mathrm{eff}}}{\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{iy}1}},\\ \stackrel{\sim }{n_{i1}}& ={\alpha }_0\stackrel{\sim }{{\mathrm{\Phi }}_1}.\end{array}$(18) where ${\alpha }_0=\frac{1}{{\lambda }^2}(l_z^2+\frac{l_z{l}_x\mathrm{\chi sin}\theta }{{\mathrm{\Omega }}_{\mathrm{eff}}})$ and λ can be expressed as (19) $\lambda =\sqrt{(l_z^2+\frac{l_z{l}_x\mathrm{\chi sin}\theta }{{\mathrm{\Omega }}_{\mathrm{eff}}})\frac{1}{{\alpha }_1}}.$(19) In an analogous way, the second-order equations are obtained as: (20) $\begin{array}{rl}& -\lambda {\mathrm{\partial }}_{\xi }\stackrel{\sim }{n_{i2}}-l_z{\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{iz}2}}={\mathrm{\partial }}_{\tau }\stackrel{\sim }{n_{i1}}+l_x{\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{ix}1}}+l_z{\mathrm{\partial }}_{\xi }\left(\stackrel{\sim }{n_{i1}}\stackrel{\sim }{v_{\mathrm{iz}1}}\right),\\ & l_x{\mathrm{\partial }}_{\xi }\stackrel{\sim }{{\mathrm{\Phi }}_2}=\lambda {\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{ix}1}}+{\mathrm{\Omega }}_{\mathrm{eff}}\stackrel{\sim }{v_{\mathrm{iy}2}},\\ & \lambda {\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{iz}2}}-l_z{\mathrm{\partial }}_{\xi }\stackrel{\sim }{{\mathrm{\Phi }}_2}-\mathrm{\chi sin}\theta \stackrel{\sim }{v_{\mathrm{iy}2}}\\ & ={\mathrm{\partial }}_{\tau }\stackrel{\sim }{v_{\mathrm{iz}1}}+l_z{v}_{\mathrm{iz}}^{(1)}{\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{iz}1}}+{\nu }_0{v}_{\mathrm{iz}}^{(1)},\\ & \stackrel{\sim }{v_{\mathrm{iy}2}}=\frac{-\lambda }{{\mathrm{\Omega }}_{\mathrm{eff}}}{\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{ix}1}}+\frac{l_x}{{\mathrm{\Omega }}_{\mathrm{eff}}}{\mathrm{\partial }}_{\xi }\stackrel{\sim }{{\mathrm{\Phi }}_2},\\ & {\mathrm{\partial }}_{\xi }^2{\stackrel{\sim }{\mathrm{\Phi }}}_1={\alpha }_1\stackrel{\sim }{{\mathrm{\Phi }}_2}+{\alpha }_2{\left({\stackrel{\sim }{\mathrm{\Phi }}}_1\right)}^2-n_{i2}.\end{array}$(20) The damped KdV equation is obtained by using some simple but prolonged algebra on Equations (18(18) $\begin{array}{rl}\stackrel{\sim }{v_{\mathrm{iz}1}}& =\frac{\lambda }{l_z}\stackrel{\sim }{n_{i1}},\\ \stackrel{\sim }{v_{\mathrm{iy}1}}& =\frac{l_x}{{\mathrm{\Omega }}_{\mathrm{eff}}}{\mathrm{\partial }}_{\xi }\stackrel{\sim }{{\mathrm{\Phi }}_1},\\ \stackrel{\sim }{v_{\mathrm{ix}1}}& =\frac{\lambda }{{\mathrm{\Omega }}_{\mathrm{eff}}}{\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{iy}1}},\\ \stackrel{\sim }{n_{i1}}& ={\alpha }_0\stackrel{\sim }{{\mathrm{\Phi }}_1}.\end{array}$(18) )) and ((20(20) $\begin{array}{rl}& -\lambda {\mathrm{\partial }}_{\xi }\stackrel{\sim }{n_{i2}}-l_z{\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{iz}2}}={\mathrm{\partial }}_{\tau }\stackrel{\sim }{n_{i1}}+l_x{\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{ix}1}}+l_z{\mathrm{\partial }}_{\xi }\left(\stackrel{\sim }{n_{i1}}\stackrel{\sim }{v_{\mathrm{iz}1}}\right),\\ & l_x{\mathrm{\partial }}_{\xi }\stackrel{\sim }{{\mathrm{\Phi }}_2}=\lambda {\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{ix}1}}+{\mathrm{\Omega }}_{\mathrm{eff}}\stackrel{\sim }{v_{\mathrm{iy}2}},\\ & \lambda {\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{iz}2}}-l_z{\mathrm{\partial }}_{\xi }\stackrel{\sim }{{\mathrm{\Phi }}_2}-\mathrm{\chi sin}\theta \stackrel{\sim }{v_{\mathrm{iy}2}}\\ & ={\mathrm{\partial }}_{\tau }\stackrel{\sim }{v_{\mathrm{iz}1}}+l_z{v}_{\mathrm{iz}}^{(1)}{\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{iz}1}}+{\nu }_0{v}_{\mathrm{iz}}^{(1)},\\ & \stackrel{\sim }{v_{\mathrm{iy}2}}=\frac{-\lambda }{{\mathrm{\Omega }}_{\mathrm{eff}}}{\mathrm{\partial }}_{\xi }\stackrel{\sim }{v_{\mathrm{ix}1}}+\frac{l_x}{{\mathrm{\Omega }}_{\mathrm{eff}}}{\mathrm{\partial }}_{\xi }\stackrel{\sim }{{\mathrm{\Phi }}_2},\\ & {\mathrm{\partial }}_{\xi }^2{\stackrel{\sim }{\mathrm{\Phi }}}_1={\alpha }_1\stackrel{\sim }{{\mathrm{\Phi }}_2}+{\alpha }_2{\left({\stackrel{\sim }{\mathrm{\Phi }}}_1\right)}^2-n_{i2}.\end{array}$(20) ), (21) ${\mathrm{\partial }}_{\tau }{\stackrel{\sim }{\mathrm{\Phi }}}_1+F{\stackrel{\sim }{\mathrm{\Phi }}}_1{\mathrm{\partial }}_{\xi }{\stackrel{\sim }{\mathrm{\Phi }}}_1+G{\mathrm{\partial }}_{\xi }^3{\stackrel{\sim }{\mathrm{\Phi }}}_1+R{\stackrel{\sim }{\mathrm{\Phi }}}_1=0,$(21) where $F=(\lambda /{\alpha }_0)(\frac{3}{2}{\alpha }_0^2-{\alpha }_2)$, $G=(l_x/{\alpha }_1)({\lambda }^4-{\alpha }_1{\lambda }^2+1)$, and $R={\nu }_0/2$.

Equation (21(21) ${\mathrm{\partial }}_{\tau }{\stackrel{\sim }{\mathrm{\Phi }}}_1+F{\stackrel{\sim }{\mathrm{\Phi }}}_1{\mathrm{\partial }}_{\xi }{\stackrel{\sim }{\mathrm{\Phi }}}_1+G{\mathrm{\partial }}_{\xi }^3{\stackrel{\sim }{\mathrm{\Phi }}}_1+R{\stackrel{\sim }{\mathrm{\Phi }}}_1=0,$(21) ) cannot supply an elaborate characterization of many nonlinear waves observed in laboratory experiments [38] for small or zero values to the dispersion coefficient (G), so it is necessary to investigate the effect of higher-order which leads to the following damped KE (22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) Here, $\phi \equiv {\stackrel{\sim }{\mathrm{\Phi }}}_1$ and this equation is a not integrable system and its solutions are not yet obtained analytically due to the existence of damping term $\mathrm{R\phi }$. Thus for investigating the characteristics of the non-stationary nonlinear waves that can propagate in the present model, some semi-analytically or numerically solution to the damped KE (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ) should be obtained.

3.1. An approximate solution to the damped KE

To find some approximations to the damped KE (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ), the following ansatz is introduced [49–54] (23) $\phi =W_1\tilde{\mathrm{\Psi }},$(23) where $\tilde{\mathrm{\Psi }}\equiv \tilde{\mathrm{\Psi }}(\xi W_2(\tau ),W_3(\tau ))$ and $\mathrm{\Psi }(\xi ,\tau )$ indicates any analytic solution to the damped KE (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ) when the damping term is absent (R = 0), $W_1\equiv W_1(\tau )$ expresses the amplitude of the dissipative nonlinear structures including any nonlinear solutions to the undamped KE, Equation (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ) for R = 0, e.g. SWs and CWs, periodic waves, etc., $W_2\equiv W_2(\tau )$ symbolizes the width inverse of the non-stationary wave and $W_3\equiv W_3(\tau )$.

Inserting the ansatz (23(23) $\phi =W_1\tilde{\mathrm{\Psi }},$(23) ) into the damped KE (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ), we can get [49–54] (24) $\begin{array}{rl}& [F{W}_1{W}_2(W_1-W_2^4)\mathrm{\Psi }+\xi W_1{\mathrm{\partial }}_{\tau }{W}_2]{\mathrm{\partial }}_{\xi }\tilde{\mathrm{\Psi }}\\ & +G{W}_1(W_2^3-W_2^5){\mathrm{\partial }}_{\xi }^3\tilde{\mathrm{\Psi }}+(R{W}_1+{\mathrm{\partial }}_{\tau }{W}_1)\tilde{\mathrm{\Psi }}\\ & +W_1({\mathrm{\partial }}_{\tau }{W}_3-W_2^5){\mathrm{\partial }}_{\tau }\tilde{\mathrm{\Psi }}=0.\end{array}$(24) From Equation (24(24) $\begin{array}{rl}& [F{W}_1{W}_2(W_1-W_2^4)\mathrm{\Psi }+\xi W_1{\mathrm{\partial }}_{\tau }{W}_2]{\mathrm{\partial }}_{\xi }\tilde{\mathrm{\Psi }}\\ & +G{W}_1(W_2^3-W_2^5){\mathrm{\partial }}_{\xi }^3\tilde{\mathrm{\Psi }}+(R{W}_1+{\mathrm{\partial }}_{\tau }{W}_1)\tilde{\mathrm{\Psi }}\\ & +W_1({\mathrm{\partial }}_{\tau }{W}_3-W_2^5){\mathrm{\partial }}_{\tau }\tilde{\mathrm{\Psi }}=0.\end{array}$(24) ), we have (25) $\begin{array}{r}\{\begin{array}{l}{W}_1-W_2^4=0,\\ R{W}_1+{\mathrm{\partial }}_{\tau }{W}_1=0,\\ {\mathrm{\partial }}_{\tau }{W}_3-W_2^5=0.\end{array}\end{array}$(25) The values $W_1$, $W_2$, and $W_3$ can be obtained by solving system (25(25) $\begin{array}{r}\{\begin{array}{l}{W}_1-W_2^4=0,\\ R{W}_1+{\mathrm{\partial }}_{\tau }{W}_1=0,\\ {\mathrm{\partial }}_{\tau }{W}_3-W_2^5=0.\end{array}\end{array}$(25) ).

Solving the ODE: $R{W}_1+{\mathrm{\partial }}_{\tau }{W}_1=0$, under the condition $W_1({\tau }_{\mathrm{in}})=1$, in we get (26) $W_1=\exp (-\mathrm{RT}).$(26) From Equations  (26(26) $W_1=\exp (-\mathrm{RT}).$(26) )) and ( (25(25) $\begin{array}{r}\{\begin{array}{l}{W}_1-W_2^4=0,\\ R{W}_1+{\mathrm{\partial }}_{\tau }{W}_1=0,\\ {\mathrm{\partial }}_{\tau }{W}_3-W_2^5=0.\end{array}\end{array}$(25) ), the value of $W_2$ can obtain as follows (27) $W_2=\exp (-\frac{1}{4}\mathrm{RT}).$(27) Finally, the value of $W_3$ can be obtained by solving the ODE: ${\mathrm{\partial }}_{\tau }{W}_3-W_2^5=0$ with the help of ${\mathrm{\partial }}_{\tau }{W}_3({\tau }_{\mathrm{in}})=0$, (28) $W_3=\frac{4}{5R}[1-\exp (-\frac{5}{4}\mathrm{RT})],$(28) with $T=(\tau -{\tau }_{\mathrm{in}})$, where ${\tau }_{\mathrm{in}}$ denotes the initial value of the propagation time.

Inserting the values of $W_1$, $W_2$, and $W_3$ into Equation (28(28) $W_3=\frac{4}{5R}[1-\exp (-\frac{5}{4}\mathrm{RT})],$(28) ), the approximate analytical solution to Equation (26(26) $W_1=\exp (-\mathrm{RT}).$(26) ) is obtained (29) $\phi =e^{-\mathrm{RT}}\tilde{\mathrm{\Psi }}(\xi e^{-\frac{1}{4}\mathrm{RT}},\frac{4}{5R}(1-e^{-\frac{5}{4}\mathrm{RT}})).$(29) This is one of the general solutions to the damped KE (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ) and to study the dissipative SWs, CWs, etc. we can use the analytical solutions to the undamped (R = 0) KE (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ) for this purpose. For example, the stationary solitary wave solution of the undamped (R = 0) KE (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ) is given by applying the transformation $\mathrm{\Psi }(\varsigma )=\phi (\xi -\mathrm{s\tau }),$ where s is the velocity of travelling waves in the ξ direction at time τ. Now, we can get a fifth order ordinary differential equation in the variable ς (30) $-s{\mathrm{\partial }}_{\varsigma }\mathrm{\Psi }+F\mathrm{\Psi }{\mathrm{\partial }}_{\varsigma }\mathrm{\Psi }+G{\mathrm{\partial }}_{\varsigma }^3\mathrm{\Psi }-K{\mathrm{\partial }}_{\varsigma }^5\mathrm{\Psi }=0,$(30) By integration Equation (30(30) $-s{\mathrm{\partial }}_{\varsigma }\mathrm{\Psi }+F\mathrm{\Psi }{\mathrm{\partial }}_{\varsigma }\mathrm{\Psi }+G{\mathrm{\partial }}_{\varsigma }^3\mathrm{\Psi }-K{\mathrm{\partial }}_{\varsigma }^5\mathrm{\Psi }=0,$(30) ) once: (31) $-s\mathrm{\Psi }+\frac{F}{2}{\mathrm{\Psi }}^2+G{\mathrm{\partial }}_{\varsigma }\mathrm{\Psi }+G{\mathrm{\partial }}_{\varsigma }^3\mathrm{\Psi }-K{\mathrm{\partial }}_{\varsigma }^4\mathrm{\Psi }=Q,$(31) where Q is an arbitary integration constant. By multipling Equation (31(31) $-s\mathrm{\Psi }+\frac{F}{2}{\mathrm{\Psi }}^2+G{\mathrm{\partial }}_{\varsigma }\mathrm{\Psi }+G{\mathrm{\partial }}_{\varsigma }^3\mathrm{\Psi }-K{\mathrm{\partial }}_{\varsigma }^4\mathrm{\Psi }=Q,$(31) ) by ${\mathrm{\partial }}_{\varsigma }\mathrm{\Psi }$ and integrating we get a conserved quantity for Equation (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ) in the travelling wave variable ς (32) $\begin{array}{rl}\varkappa & =-2Q\mathrm{\Psi }-s{\mathrm{\Psi }}^2+\frac{F}{3}{\mathrm{\Psi }}^3+G{\mathrm{\partial }}_{\varsigma }{\mathrm{\Psi }}^2\\ & -K[2{\mathrm{\partial }}_{\varsigma }\mathrm{\Psi }{\mathrm{\partial }}_{\varsigma }^3\mathrm{\Psi }-{\left({\mathrm{\partial }}_{\varsigma }^2\mathrm{\Psi }\right)}^2]\equiv \mathrm{const}.\end{array}$(32) Thus, we can get the stationary solitary wave solution as (See for details Ref. [72]) (33) $\mathrm{\Psi }={\mathrm{\Psi }}_0\sec h^4\left[\frac{1}{W_d}(\xi -\frac{36G^2}{169K}\tau )\right],$(33) where ${\mathrm{\Psi }}_0=(105G^2/169\mathrm{FK})$ and $W_d=\sqrt{52K/G}$ give the maximum amplitude and the width of the undamped Kawahara soliton.

According to the stationary solitary wave solution in Equation (33(33) $\mathrm{\Psi }={\mathrm{\Psi }}_0\sec h^4\left[\frac{1}{W_d}(\xi -\frac{36G^2}{169K}\tau )\right],$(33) ) and the relation in Equation (29(29) $\phi =e^{-\mathrm{RT}}\tilde{\mathrm{\Psi }}(\xi e^{-\frac{1}{4}\mathrm{RT}},\frac{4}{5R}(1-e^{-\frac{5}{4}\mathrm{RT}})).$(29) ) the dissipative solitary wave solution of the damped KE (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ) reads (34) $\begin{array}{rl}\phi & ={\mathrm{\Psi }}_0{e}^{-\mathrm{RT}}\sec h^4[\frac{1}{W_d}(\xi e^{-\frac{1}{4}\mathrm{RT}}\\ & -\frac{36G^2}{169K}\frac{4}{5R}(1-e^{-\frac{5}{4}\mathrm{RT}}))].\end{array}$(34) The energy of Kawahara soliton for both the undamped and damped cases, are, respectively, given by (35) $E_{\mathrm{undam}}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{\Psi }}^2\mathrm{d}\xi =\frac{32}{35}{\mathrm{\Psi }}_0^2{W}_d,$(35) and (36) $E_{\mathrm{dam}}=E_{\mathrm{undam}}{e}^{-2\mathrm{Rt}}.$(36) It is clear from Equation (33(33) $\mathrm{\Psi }={\mathrm{\Psi }}_0\sec h^4\left[\frac{1}{W_d}(\xi -\frac{36G^2}{169K}\tau )\right],$(33) ) that the stationary Kawahara soliton energy $E_{\mathrm{undam}}$ is conserved and does not depend on the wave time propagation, i.e. ${\mathrm{\partial }}_{\tau }{E}_{\mathrm{undam}}=0$ while the non-stationary Kawahara soliton energy $E_{\mathrm{dam}}$ is not conserved and its value depends on the wave time propagation, i.e. ${\mathrm{\partial }}_{\tau }{E}_{\mathrm{dam}}\ne 0$.

Moreover, the undamped (R = 0) KE (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ) supports the analytical CW solution as mentioned in Ref. [72] (37) $\begin{array}{rl}\mathrm{\Psi }& =\frac{5G^2}{3\times 7\times {13}^2\mathrm{FK}}\{7+\sqrt{7\times 31}c{n}^4\\ & {[\sqrt[4]{\frac{31}{7}}\sqrt{\frac{G}{78K}}(x-\frac{2^7{G}^2}{3\times {13}^{2}K}t);\frac{1}{\sqrt{2}}]\}}^2.\end{array}$(37) By substituting the analytical CW solution (37(37) $\begin{array}{rl}\mathrm{\Psi }& =\frac{5G^2}{3\times 7\times {13}^2\mathrm{FK}}\{7+\sqrt{7\times 31}c{n}^4\\ & {[\sqrt[4]{\frac{31}{7}}\sqrt{\frac{G}{78K}}(x-\frac{2^7{G}^2}{3\times {13}^{2}K}t);\frac{1}{\sqrt{2}}]\}}^2.\end{array}$(37) ) into the general solution of the dissipative nonlinear structures (29(29) $\phi =e^{-\mathrm{RT}}\tilde{\mathrm{\Psi }}(\xi e^{-\frac{1}{4}\mathrm{RT}},\frac{4}{5R}(1-e^{-\frac{5}{4}\mathrm{RT}})).$(29) ), we finally get the dissipative CW solution of the damped KE (22(22) ${\mathrm{\partial }}_{\tau }\phi +\mathrm{F\phi }{\mathrm{\partial }}_{\xi }\phi +G{\mathrm{\partial }}_{\xi }^3\phi -K{\mathrm{\partial }}_{\xi }^5\phi +\mathrm{R\phi }=0.$(22) ) as follows (38) $\begin{array}{rl}\phi & =\frac{5G^2{e}^{-\mathrm{RT}}}{3\times 7\times {13}^2\mathrm{FK}}\{7+\sqrt{7\times 31}c{n}^4\\ & [\sqrt[4]{\frac{31}{7}}\sqrt{\frac{G}{78K}}(\xi e^{-\frac{1}{4}\mathrm{RT}}\\ & {-\frac{2^9{G}^2}{15\times {13}^2\mathrm{KR}}(1-e^{-\frac{5}{4}\mathrm{RT}}));\frac{1}{\sqrt{2}}]\}}^2.\end{array}$(38)

4. Results and discussion

In this study, we consider the astrophysical plasma parameters, i.e. plasma density ${10}^{26}-{10}^{29}{\mathrm{cm}}^{-3}$, magnetic field strength ${10}^9-{10}^{11}\mathrm{Gauss}$, respectively, to investigate the effects of Coriolis force and quantized magnetic field on the dissipative Kawahara SWs and CWs [73]. For magnetic field ${10}^{10}\mathrm{Gauss}$ and plasma density ${10}^{26}{\mathrm{cm}}^{-3}$, the value of ${\omega }_{\mathrm{ci}}$ becomes $1.4\times {10}^{17}{\sec }^{-1}$ while ${\omega }_{\mathrm{pi}}$ comes out to be $1.1\times {10}^{18}{\sec }^{-1}$ and the condition ${\omega }_{\mathrm{ci}}/{\omega }_{\mathrm{pi}}<1$ is satisfied. The effects of the physical parameters, namely, the angle of rotation θ, the normalized temperature $\tilde{T}$, the rotational frequency ${\mathrm{\Omega }}_{{}_0}$, the parameter η, and the directional cosine $l_z$, on the profiles of both dissipative SWs and CWs are summarized in the following points: Figure 2 shows the behaviour of the dissipative Kawahara soliton potential φ for various values of the angle of rotation θ ($=2^{\circ },{10}^{\circ }$ and ${20}^{\circ }$). It is evident that both amplitude and width of the dissipative pulses increase with the increase of θ in the presence of Landau quantization. This behaviour can be explained physically based on the soliton energy given in Equation (35(35) $E_{\mathrm{undam}}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{\Psi }}^2\mathrm{d}\xi =\frac{32}{35}{\mathrm{\Psi }}_0^2{W}_d,$(35) ) which both the width and amplitude of the Kawahara solitons increase with increasing the angle of rotation. The normalized temperature $\tilde{T}$ depends on the Fermi energy which is a function of the plasma density. Therefore, increasing the plasma density of fermions would lead to the enhancement of the Fermi energy, and accordingly, the value of $\tilde{T}$ decreases. It is observed that both the amplitude and width of the Kawahara solitary structure grow with increasing the normalized temperature $\tilde{T}$ as shown in Figure 3. Physical speaking, the increase of $\tilde{T}$ leads to an increase of the nonlinearity of the structures which leads to the enhancement of the pulse amplitude. Because of the fact that soliton can propagate when the balance between the nonlinearity and the dispersion is existed, thus increasing the nonlinearity leads to the increase of the dispersion to maintain the balance between them. But as is observed in Figure 4, increasing the value of rotational frequency ${\mathrm{\Omega }}_{{}_0}$ or $\chi =2{\mathrm{\Omega }}_{{}_0}$ causes to increase in the amplitude with a slight change in the width.

Figure 2. The pulse-shaped profile of the electrostatic dissipative Kawahara solitons is depicted against ξ for different values of the angle of rotation θ. Here, $\tilde{T}=0.2$, $\chi =0.2$, $\eta =0.1$, and $l_z=0.1$

Figure 3. The pulse-shaped profile of the electrostatic dissipative Kawahara solitons is depicted against ξ for different values of the normalized temperature $\tilde{T}$. Here, $\theta =2^{\circ }$, $\chi =0.2$, $\eta =0.1$, and $l_z=0.1$.

Figure 4. The pulse-shaped profile of the electrostatic dissipative Kawahara solitons is depicted against ξ for different values of rotational frequency ${\mathrm{\Omega }}_{{}_0}$ or χ. Here, $\theta =2^{\circ }$, T = 0.2, $\eta =0.1$, and $l_z=0.1$.

The effect of the Landau quantization $\eta =\hslash {\omega }_{\mathrm{ce}}/k_B{T}_{\mathrm{Fe}}$ on the Kawahara soliton profile is studied as demonstrated in Figure 5. It is clear that increasing the parameter η increases both the amplitude and width of the Kawahara solitons. On the other hand, increasing the obliqueness angle $l_z$ makes the wave sharp and taller as illustrated in Figure 6.

Figure 5. The pulse-shaped profile of the electrostatic dissipative Kawahara solitons is depicted against ξ for different values of the Landau quantization η. Here, $\theta =2^{\circ }$, $\stackrel{\sim }{T}=0.2$, $\chi =0.2$, and $l_z=0.1$.

Figure 6. The pulse-shaped profile of the electrostatic dissipative Kawahara solitons is depicted against ξ for different values of the obliqueness angle $l_z$. Here, $\eta =0.1$, $\chi =0.2$, $\tilde{T}=0.2$, and $\theta =2^{\circ }$.

Finally, the impact of the collisional frequency R and the wave propagation time τ on the dissipative Kawahara SWs and CWs is elucidated in Figures 7 and 8, respectively. One can see that the amplitude of the dissipative Kawahara pulses including SWs and CWs shrinks with increasing both R and τ. This behaviour can be explained physically based on the energy of the dissipative Kawahara soliton as seen In Figure 9 which the dissipative Kawahara soliton energy becomes non-conserved and decays as the collisional frequency R and the wave propagation time τ increase. Note that the influence of the physical parameters on the CW profile has the same qualitative behaviour of the soliton (the figures did not insert in the text).

Figure 7. The pulse-shaped profile of the dissipative Kawahara soliton is depicted against ξ for different values of (a) the wave propagation time τ and (b) the collisional frequency R. Here, $\tilde{T}=0.2$, $\chi =0.2$, $\eta =0.1$, and $\theta =2^{\circ }$.

Figure 8. The pulse-shaped profile of the dissipative Kawahara cnoidal wave is depicted against ξ for different values of (a) the wave propagation time τ and (b) the collisional frequency R. Here, $\tilde{T}=0.2$, $\chi =0.2$, $\eta =0.1$, and $\theta =2^{\circ }$.

Figure 9. The energy of the dissipative Kawahara soliton is plotted against the wave propagation time τ for different values of the collisional frequency R. Here, $\tilde{T}=0.2$, $\chi =0.2$, $\eta =0.1$, $\theta =2^{\circ }$, and $l_z=0.1$.

5. Summary

We have investigated the dissipative Kawahara IAWs including dissipative Kawahara SWs and CWs in dense e−i plasmas by taking the Landau quantization of magnetic field and the Coriolis force in addition to the ion-neutral collision into consideration. The linear dispersion relation is derived by applying Fourier analysis and observed that in the absence of rotation, wave starts propagating earlier in comparison to the case when rotation exists. the damped KdV equation has been derived which cannot describe the waves in our system at very small or even zero of the dispersion coefficient G. Therefore, the highest-order perturbation has been taken into account to which leads to the damped Kawahara equation (the damping fifth-order KdV equation). The energy of the dissipative Kawahara solitary wave has been obtained. Moreover, a new approximate solution to the non-integrable damped Kawahara equation is derived in detail. The obtained solution could be applied for investigating dissipative SWs, dissipative CWs, a dissipative periodic waves, etc. As an example, the obtained solution has been applied for studying the effect of different plasma parameters on dissipative Kawahara solitary and cnoidal wave profiles. Also, the impact of the damping parameters (the collisional frequency R and the wave propagation time τ) on the dissipative SWs and CWs has been examined.

The obtained results are useful to understand the features of quantum dissipative higher-order IAWs in dense and slowly rotating astrophysical plasmas where a strong magnetic field exists. These findings may also be helpful to analyse the influence of Coriolis force in different phenomena like the quantum Hall effect of novel materials like graphene and topological insulators [74].

Future work: Malik et al. [75] and Malik [76] studied the propagation of ion-acoustic KP soliton in a collisionless unmagnetized inhomogeneous plasma. The authors used the RPM to derive both KP and modified KP equations. This family of differential equations with third derivative dispersion is suitable for investigating the propagation of acoustic waves in two-dimensions perturbation when the balance is achieved between nonlinearity and dispersion or for small and finite amplitude. However, this family cannot supply an elaborate characterization of many nonlinear waves observed in laboratory experiments [38] for small or zero values to the dispersion coefficient. Thus, to overcome this problem, it is necessary to investigate the effect of higher-order dispersion which leads to a new family with fifth-order derivative dispersion like one-dimensional KE as we studied in this manuscript. In the future work, we will try to derive the KE in two-dimensional in order to study the impact of fifth-order dispersion on the characteristics of the multidimensional dressed solitons.

Declarations

Availability of data and material Data sharing not applicable to this article as no data sets were generated or analysed during the current study.

Acknowledgements

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting project number (PNURSP2023R157), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Authors' contributions

All authors contributed equally and significantly to this paper. All authors have read and approved the final version of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).