Novel Approximate Analytical Solutions to the Nonplanar Modified Kawahara Equation and Modeling Nonlinear Structures in Electronegative Plasmas

Abstract

In this investigation, the nonplanar (spherical and cylindrical) modified fifth-order Korteweg–de Vries (nmKdV5) equation, otherwise known as the nonplanar modified Kawahara equation (nmKE), is solved using the ansatz approach. Two general formulas for the semi-analytical symmetric approximations are derived using the recommended methodology. Using the obtained approximations, the nonplanar modified Kawahara (mK) symmetric solitary waves (SWs) and cnoidal waves (CWs) are obtained. The fluid equations for the electronegative plasmas are reduced to the nmKE as a practical application for the obtained solutions. Using the obtained solutions, the characteristic features of both the cylindrical and spherical mK-SWs and -CWs are studied. All obtained solutions are compared with each other, and the maximum residual errors for these approximations are estimated. Numerous researchers that are interested in studying the complicated nonlinear phenomena in plasma physics can use the obtained approximations to interpret their experimental and observational findings.

1. Introduction

The following original planar KdV equation is understood as coming from the truncation of a series involving different derivatives of higher-order nonlinearities and dispersions [1,2,3]

$$\mathrm{KdV}\left(\Phi \right)\equiv {\partial }_t\Phi +a\Phi {\partial }_x\Phi +c{\partial }_x^3\Phi =0,$$

(1)

where a and c are, respectively, the coefficients of quadratic nonlinear and dispersion terms, and $\Phi \equiv \Phi \left(x,t\right)$. Depending on the physics system involved in a study, some of the coefficients multiplying the lowest-order derivatives may vanish [4]. In these cases, it is necessary to retain the next evolution equation that has non-vanishing higher-order nonlinear/dispersion terms. Depending upon the terms retained, the resulting equation is known under several different names: ((un)damped KdV [2,5], nonplanar KdV [6,7,8], nonplanar damped KdV [4,9], (un)damped modified KdV (mKdV) [1,10,11], nonplanar mKdV [12], (un)damped Gardner/Extended KdV (EKdV) [2], nonplanar EKdV [12], the generalized Korteweg–de Vries (GKdV) [13], Burgers KdV-type (KdVB) [2,14,15], (un)damped Kawahara [16,17,18], nonplanar Kawahara [19], nonplanar damped Kawahara [20], modified Kawahara [21,22,23,24], extended Kawahara [25,26], and many other different forms of Kawahara-type equations [27,28,29,30]). This equation family has several applications in different areas of science, such as the physics of plasmas [31,32], mechanical fluids [26], and so on. In the case of vanishing the coefficient “$a=0$”, Equation (1) becomes unsuitable for describing the solitary waves (SWs); accordingly, the next higher-order nonlinearity evolutionary equation, called the mKdV equation, is considered to describe the phenomenon under study.

$$\mathrm{mKdV}\left(\Phi \right)\equiv {\partial }_t\Phi +b{\Phi }^2{\partial }_x\Phi +c{\partial }_x^3\Phi =0,$$

(2)

where b denotes the coefficients of the cubic nonlinear term. Equation (2) is used for modeling and describing some nonlinear phenomena that arise in different physical systems at some critical values of the physical parameters related to the model under consideration. However, if some noises/perturbation [31,32], as well as the geometrical effect [12], are considered, then, in this case, the following nonplanar (cylindrical and spherical) modified Kawahara equation (nmKE) is obtained

$${\partial }_t\Psi +b{\Psi }^2{\partial }_x\Psi +c{\partial }_x^3\Psi -d{\partial }_x^5\Psi +ϝ\left(t\right)\Psi =0,$$

(3)

The last term in Equation (3) $ϝ\left(t\right)=\frac{v}{2t}$ represents the nonplanar geometrical term where $v=0$ for one-dimensional (1D) planar geometry, $v=1\left(2\right)$ for cylindrical (spherical) geometry, d is a perturbation parameter, and $\Psi \equiv \Psi \left(x,t\right)$. Equation (3) is able to describe many nonplanar nonlinear phenomena, e.g., the nonplanar SWs, nonplanar CWs, nonplanar ShWs, and many others that arise in many different physical systems, such as waves in a water tank, fluid mechanics, and in plasma physics.

The planar Kawahara-type equations have been solved and analyzed via several analytical and numerical approaches, such as those featured in [26,27,28,29]. For instance, the Kawahara/fifth-order KdV equation [33], the nonlinear Sawada–Kotera equation [34], as well as the Kaup–Kupershmidt and Ito equations [35] have been analyzed numerically via the septic B-spline collocation method [33,34,35]. There are many analytical and numerical methods that succeeded in finding many exact and numerical solutions for many evolution equations, and we have mentioned some, but not all, of them (see Refs. [36,37,38,39]). However, in the presence of the geometric term ($\frac{v}{2t}$), the nmKE (3) cannot be solved analytically; i.e., it does not support exact solutions. Consequently, the primary goal of this investigation is to find some approximate analytical solutions to the nmKE (3). These approximations can help many researchers explain the mystery around many of the nonlinear phenomena that arise in plasma physics and other branches of science. The main objectives of this investigation are introduced in the following brief points:

• For the first time, we try to find an analytical approximate solution to the nmKE (3) through an ansatz.
• Second, by applying the obtained solutions, we can study all nonplanar traveling waves describe by the nmKE (3), such as the nonplanar SWs, nonplanar CWs, and nonplanar ShWs that can propagate plasma physics, oceans and seas, nonlinear optic, and so on.
• All obtained analytical approximations can be applied to the investigation of the properties of nonplanar SWs and CWs in electronegative plasmas composed of inertialess thermal electrons and negative ions, as well as inertial cold positive ions [12].

The essay is structured as follows: In Section 2, an effective method based on a suitable ansatz is implemented to find some approximations to the nmKE (3). In this section, two general formulas for the analytical approximate solutions are derived. In Section 3, the obtained formulas for analytical approximations are applied for bringing the solitary wave (SW) and cnoidal wave (CW) solutions to the nmKE. In Section 4, the obtained solutions are devoted to analyzing and studying the characteristic behavior of both nonplanar CWs and SWs in electronegative plasma. The most important results are summarized in Section 5.

2. General Approximations to the Evolution Equation (nmKE)

As mentioned above, this equation does not have an exact analytic solution. Therefore, we try to derive a general analytical approximate solution to the nmKE (3) through an ansatz. This technique is discussed in the following brief steps:

• According to our hypothesis/ansatz, the approximate solution to the nmKE (3) is assumed to have the following form

  $$\Psi (x,t)=g_1(t)\Phi \left(x{g}_2(t),g_3(t)\right)\equiv g_1(t)\tilde{\Phi },$$

  (4)

  where $g_1(t)$ and $g_2(t)$ express the temporal amplitude and the inverted width of the nonplanar structures, while the velocity of the temporal structures can be obtained using the function $g_3(t)$. For $g_{1,2}(t)=1$ and $g_3(t)=t$, the planar case ($v=0$) is recovered. The function $\Phi \left(x,t\right)$ represents any exact analytic solution to the planar mKE, i.e., Equation (3) for $v=0$.
• From the planar mKE, the value of ${\partial }_t\Phi$ is obtained

  $${\partial }_t\Phi ={\left.-b{\Phi }^2{\partial }_x\Phi -c{\partial }_x^3\Phi +d{\partial }_x^5\Phi \right|}_{\Phi \to \tilde{\Phi }}.$$

  (5)
• Based on the ansatz (4) and Equation (5), the differential operators ${\partial }_t\Psi$, ${\partial }_x\Psi$, ${\partial }_x^3\Psi$, and ${\partial }_x^5\Psi$ of Equation (3) are calculated as follows

  $$\begin{array}{cc}\hfill {\partial }_t\Psi & =-b{g}_2(t){\tilde{\Phi }}^2{\partial }_x\tilde{\Phi }-c{g}_2{(t)}^3{\partial }_x^3\tilde{\Phi }+d{g}_2{(t)}^5{\partial }_x^5\tilde{\Phi },\hfill \\ \hfill {\partial }_x\Psi & =g_1(t)g_2(t){\partial }_x\tilde{\Phi },\hfill \\ \hfill {\partial }_x^3\Psi & =g_1(t)g_2{(t)}^3{\partial }_x^3\tilde{\Phi },\hfill \\ \hfill {\partial }_x^5\Psi & =g_1(t)g_2{(t)}^5{\partial }_x^5\tilde{\Phi }.\hfill \end{array}$$

  (6)
• Substituting both the ansatz (4) and the differential operators ${\partial }_t\Psi$, ${\partial }_x\Psi$, ${\partial }_x^3\Psi$, and ${\partial }_x^5\Psi$ given in Equation (6) into Equation (3), we obtain

  $$R_1\tilde{\Phi }+R_2{\partial }_x\tilde{\Phi }+R_3{\partial }_x^3\tilde{\Phi }+R_4\Phi {\partial }_x\tilde{\Phi }+R_5{\partial }_x^5\tilde{\Phi }=0,$$

  (7)

  with

  $$\left\{\begin{array}{c}{R}_1=v{g}_1(t)+2t{\dot{g}}_1(t),\\ R_2=2xt{g}_1(t){\dot{g}}_2(t),\\ R_3=2ct{g}_1(t)\left(g_2{(t)}^3-{\dot{g}}_3(t)\right),\\ R_4=2bt{g}_1(t)\left(g_2(t)g_1{(t)}^2-{\dot{g}}_3(t)\right),\\ R_5=-2dt{g}_1(t)\left(g_2{(t)}^5-{\dot{g}}_3(t)\right),\end{array}\right.$$

  (8)

  where ${\dot{g}}_i(t)\equiv {\partial }_t{g}_i(t)$ for $i=1,2,3.$
• By solving $R_1=0$ with the help of the initial condition (IC) $g_1(t_0)=1$, the value of $g_1(t)$ is obtained as follows

  $$g_1(t)={\left(\frac{t_0}{t}\right)}^{\frac{v}{2}}=T^{\frac{v}{2}},$$

  (9)

  where $T=t_0/t$.
• To find the values of $\left(g_2(t),g_3(t)\right)$, we can solve the system $R_4=0$ with $R_5=0$ and $R_4=0$ with $R_3=0$, i.e.,

  $$\left\{\begin{array}{c}{\dot{g}}_3(t)=g_2(t)g_1{(t)}^2=g_2(t)T^v,\\ {\dot{g}}_3(t)=g_2{(t)}^5,\end{array}\right.$$

  (10)

  and

  $$\left\{\begin{array}{c}{\dot{g}}_3(t)=g_2(t)g_1{(t)}^2=g_2(t)T^v,\\ {\dot{g}}_3(t)=g_2{(t)}^3,\end{array}\right.$$

  (11)

  with the IC ${\dot{g}}_3(t_0)=0$.
• It is clear from systems (10) and (11) that we have two different values for $\left(g_2(t),g_3(t)\right)$ with a different residual error for each case. By eliminating ${\dot{g}}_3(t)$ from system (10), the first value of $g_2(t)$ is obtained as follows

  $$g_2{(t)}^5=g_2(t)T^v\Rightarrow g_2(t)=T^{\frac{v}{4}}.$$

  (12)
  Note here that the value given in Equation (12) can be recovered in both the planar and nonplanar cases.
• Inserting the value of $g_2(t)$ in Equation (12) into the second equation in system (10), we obtain

  $${\dot{g}}_3(t)=T^{\frac{5v}{4}},$$

  (13)

  and solving this using the IC $g_3\left(t_0\right)=0$, the first value of $g_3(t)$ is obtained as follows

  $$g_3(t)=\frac{t_0}{{\rho }_1}\left(1-T^{{\rho }_1}\right),$$

  (14)

  where ${\rho }_1=\frac{5}{4}v-1.$
• By eliminating ${\dot{g}}_3(t)$ from system (11), the second value of $g_2(t)$ is obtained as follows

  $$g_2{(t)}^3=g_2(t)T^v\Rightarrow g_2(t)=T^{\frac{v}{2}}.$$

  (15)
• Using Equation (15) in the second equation in system (11), and solving the obtained results with the help of the IC $g_3\left(t_0\right)=0$, the second value of $g_3(t)$ is estimated as follows

  $$g_3(t)=\frac{t_0}{{\rho }_2}\left(1-T^{{\rho }_2}\right),$$

  (16)

  where ${\rho }_2=\frac{3}{2}v-1$.
• By substituting all obtained values for $g_i(t)$ into the ansatz (4), we finally obtain the following two generic formulations of the analytical symmetric approximations for the nmKE (3)

  $${\Psi }_1(x,t)=T^{\frac{v}{2}}\Phi \left[x{T}^{\frac{v}{4}},\frac{t_0}{{\rho }_1}\left(1-T^{{\rho }_1}\right)\right],$$

  (17)

  and

  $${\Psi }_2(x,t)=T^{\frac{v}{2}}\Phi \left[x{T}^{\frac{v}{2}},\frac{t_0}{{\rho }_2}\left(1-T^{{\rho }_2}\right)\right].$$

  (18)
  The two general approximations (17) and (18) may be applied to the study of the properties of all nonlinear structures (e.g., SWs and CWs) described by the nmKE (3). In the following section, we will try to employ the obtained approximations (17) and (18) for studying different nonlinear waves that arise in many different plasma models.

3. Nonplanar SW and CW Solutions to nmKE

To investigate the unique characteristics of the nonplanar SWs and CWs that are described by the nmKE (3), in any physical medium, whether plasma, optical fibers, or seas and oceans, the analytic solution to the planar mKE should be known. Many analytical and symmetric solutions to the planar mKE have been derived by many researchers. In our study, we can take only one of the analytical solutions from the literature in order to study the properties of the distinctive features of the nonplanar SWs and CWs.

In Ref. [25], the following planar SW solution to the planar mKE was derivedl

$${\Phi }_{sol}(x,t)={\Phi }_{max}\frac{1}{1+cosh[W_{sol}(x-\frac{4c^2}{25d}t)]},$$

(19)

where ${\Phi }_{max}=\frac{3c}{5b}\sqrt{\frac{10b}{d}}$ indicates the planar SW maximum amplitude, and $W_{sol}=\sqrt{\frac{c}{5d}}$ is the planar SW width.

Inserting the planar SW solution (19) into the general symmetric approximations (17) and (18), the following two nonplanar SW solutions to the nmKE (3) are obtained

$${\Psi }_1(x,t)={\Phi }_{max}{T}^{\frac{v}{2}}\frac{1}{1+cosh[W_{sol}(x{T}^{\frac{v}{4}}-\frac{8b{c}^2}{50bd}\frac{t_0}{{\rho }_1}\left(1-T^{{\rho }_1}\right)]},$$

(20)

and

$${\Psi }_2(x,t)={\Phi }_{max}{T}^{\frac{v}{2}}\frac{1}{1+cosh[W_{sol}(x{T}^{\frac{v}{2}}-\frac{8b{c}^2}{50bd}\frac{t_0}{{\rho }_2}\left(1-T^{{\rho }_2}\right)]}.$$

(21)

To obtain the nonplanar CW solution to the nmKE (3), the followingplanar exact CW solution to the planar mKE is introduced [25]

$$\Phi (x,t)={\Phi }_{c}c{n}^2\left[W_c\left(x-u_{c}t+x_0\right),m\right]$$

(22)

with

$$\begin{array}{cc}\hfill {\Phi }_c& =\frac{3cm}{\sqrt{10bd}(2m-1)},\hfill \\ \hfill W_c& =\sqrt{\frac{c}{20d(2m-1)}},\hfill \\ \hfill u_c& =\frac{c^2(23m^2-23m+8)}{50d{(2m-1)}^2},\hfill \end{array}$$

where the elliptic modulus is m ($m>\frac{1}{2}$) for the CW solution, and $x_0$ represents the initial position. Remember that solution (22) can recover the SW solution for $m=1$.

Now, inserting solution (22) into the obtained approximations (17) and (18), the following two nonplanar CW solutions to the nmKE (3) are obtained

$${\Psi }_1(x,t)={\Phi }_c{T}^{\frac{v}{2}}c{n}^2\left[W_c\left(\left(x+x_0\right)T^{\frac{v}{4}}-u_c\frac{t_0}{{\rho }_1}\left(1-T^{{\rho }_1}\right)\right),m\right],$$

(23)

and

$${\Psi }_2(x,t)={\Phi }_c{T}^{\frac{v}{2}}c{n}^2\left[W_c\left(\left(x+x_0\right)T^{\frac{v}{2}}-u_c\frac{t_0}{{\rho }_2}\left(1-T^{{\rho }_2}\right)\right),m\right].$$

(24)

4. Nonplanar mK-SWs and -CWs in Electronegative Plasmas

Now, let us use the obtained approximations to investigate the nonplanar mK-SWs and -CWs in a collisionless, unmagnetized, electronegative plasma composed of inertialess Maxwellian negative ions and electrons as well as inertial positive ions [12]. The normalized fluid governing the equations described in the present plasma model read as follows

$$\left\{\begin{array}{c}{\partial }_{\tau }n+\frac{1}{r^{\upsilon }}{\partial }_r\left(r^{\upsilon }nu\right)=0,\\ {\partial }_{\tau }u+u{\partial }_{r}u+{\partial }_r\phi =0,\\ {\partial }_r\left(r^{\upsilon }{\partial }_r\phi \right)+r^{\upsilon }\left(n-n_e-n_n\right)=0.\end{array}\right.$$

(25)

The density of both electrons and negative ions species are, respectively, given by

$$\begin{array}{cc}\hfill n_e+n_n& ={\mu }_e{e}^{\phi }+{\mu }_n{e}^{\sigma \phi }\hfill \\ \hfill & \approx {\alpha }_0+{\alpha }_1\phi +{\alpha }_2{\phi }^2+{\alpha }_3{\phi }^3,\hfill \end{array}$$

(26)

with ${\alpha }_i=\left({\mu }_e+{\mu }_n{\sigma }^i\right)/i!$ where $i=0,1,2,3.$

In Equations (25) and (26) above, the quantities n and u denote the normalized number density and velocity of positive ions, respectively; $n_n$ and $n_e$ indicate, respectively, the normalized number densities of the negative ions and electrons; $\phi$ indicates the dimensionless electrostatic wave potential; and t and r are, respectively, dimensionless independent variables. The parameter ${\sigma }_n=T_e/T_n$ denotes the temperature ratio in which $T_n/T_e$ (pointing to, respectively, the temperature of the electrons and negative ions). The parameters ${\mu }_e=n_e^{(0)}/n^{(0)}$ and ${\mu }_n=n_{n\left(0\right)}/n_{\left(0\right)}$ where $n_{e\left(0\right)}$, $n_{n\left(0\right)}$ and $n_{\left(0\right)}$ are, respectively, the unperturbed densities of the electrons, negative ions, and positive ions. Thus, the neutrality condition for this model reads as follows

$${\mu }_e+{\mu }_n=1,$$

(27)

with ${\mu }_e=1/\left(1+\alpha \right)$ and ${\mu }_n=\alpha /\left(1+\alpha \right)$ where $\alpha =n_{n\left(0\right)}/n_{e\left(0\right)}$ expresses the negative ions concentration.

Now, the RPT is introduced for deriving the evolution equation. Based on this method, the independent variables $\left(r,\tau \right)$ are stretched as $x=\epsilon \left(r-{\lambda }_{ph}\tau \right)$ and $t={\epsilon }^3\tau$ where ${\lambda }_{ph}$ indicates the phase velocity, and $\epsilon$ is a small parameter ($\epsilon <<1$). The dependent quantities $\left(n,u,\phi \right)$ are expanded as: $\left(n,u,\phi \right)=\left(1,0,0\right)+\epsilon \left(n_{\left(1\right)},u_{\left(1\right)},{\phi }_{\left(1\right)}\right)+{\epsilon }^2\left(n_{\left(2\right)},u_{\left(2\right)},{\phi }_{\left(2\right)}\right)+\cdots .$ By inserting both the stretching of the independent variables and the expansion of the dependent quantities into the basic Equations (25) and (26), by following the identical steps outlined in Ref. [12], and by taking the fifth-order of derivatives into consideration, we finally obtain the nmKE

$$\frac{\partial \Psi }{\partial t}+\frac{b}{3}\frac{\partial {\Psi }^3}{\partial x}+c\frac{{\partial }^3\Psi }{\partial x^3}-d\frac{{\partial }^5\Psi }{\partial x^5}+ϝ\left(t\right)\Psi =0,$$

(28)

with

$$\begin{array}{cc}\hfill b& =\frac{3}{4}{\lambda }_{ph}^3\left(\frac{5}{{\lambda }_{ph}^6}-2{\alpha }_3\right),c=\frac{1}{2}{\lambda }_{ph}^3,\hfill \\ \hfill {\lambda }_{ph}& =\sqrt{\frac{1}{{\alpha }_1}},\mathrm{and}ϝ\left(t\right)=\frac{\upsilon }{2t},\hfill \end{array}$$

where $\Psi \equiv {\phi }_{\left(1\right)}$.

Based on the experimental and theoretical studies [40,41], the following values of the plasma parameters are considered: $T_e\approx 0.69$ eV, $T_n$≈$\left(0.06\pm 0.02\right)$ eV, and $n_e=3.8\times {10}^9$ cm${}^{-3},$ which lead to ${\sigma }_n=8.7$, $11.5$, and $17.25$, while $\alpha =0\to 1$. Figure 1a–d demonstrate the profile of both the cylindrical and spherical mK-SWs for the first symmetric approximation (20) and second approximation (21). It is clear that the spherical mK (dressed)-SWs are faster than the cylindrical ones, i.e., the spherical SW amplitude is larger than the cylindrical SW amplitude. Moreover, it is observed that the nonplanar mK-CWs have the same qualitative behavior as the nonplanar mK-SWs, as illustrated in Figure 2a–d, for the first approximation (20) and the second symmetric approximation (21). Furthermore, the residual maximum error $L_{MRE}$ for all obtained approximations for both the SWs and CWs along the study domain is estimated as elucidated in

Table 1. The residual maximum $L_{MRE}$ for all obtained approximations is estimated.

Approximation	${\left.{\mathit{L}}_{\mathit{MRDE}}\right|}_{\mathit{Cyl}.}$	${\left.{\mathit{L}}_{\mathit{MRDE}}\right|}_{\mathit{Sph}}$
Approx. (20)	0.0243681	0.0462795
Approx. (21)	0.0509174	0.0979829

.

It is observed that the first nonplanar modified Kawahara soliton approximation (20) is better than the second one (21). We remind the reader that this is not the only application for the two approximations (17) and (18), but these approximations can be used in the study of many nonlinear waves in various fields of science.

5. Conclusions

In this investigation, some symmetric approximations to the nonplanar (cylindrical and spherical) modified Kawahara equation (nmKE) have been derived for the first time using a suitable ansatz. According to the proposed technique, two general formulas for the analytical approximations have been obtained. Numerous researchers can benefit from all obtained approximations for modeling many nonplanar structures, such as the nonplanar solitary waves (SWs), nonplanar cnoidal waves (CWs), and nonplanar shock waves, that can arise and propagate in different plasma models and many other areas of science. As a realistic application for the obtained results, all obtained approximations were applied to the study of both the cylindrical and spherical modified Kawahara solitary and cnoidal waves in unmagnetized electronegative plasmas that have Maxwellian electrons and negative ions. It was found that both spherical modified Kawahara structures, including the spherical SWs and CWs are faster than the cylindrical SWs and CWs, i.e., the amplitude of the spherical structures is larger than the amplitude of the cylindrical structures. Additionally, the global residual error along the whole study domain has been estimated to assess the accuracy of the obtained approximations. At last, the obtained solutions can help many researchers understand the propagation mechanism of the several nonplanar structures that arise and propagate in various branches of science, e.g., mechanical fluid, plasma physics, oceans and seas, and many others.

6. Future Work

Considering some frictional/collisional forces, we finally obtain a damped modified Kawahara equation (dmKE). Additionally, this equation does not support exact solutions and needs to be solved numerically or using some approximate methods.