Supernonlinear wave, associated analytical solitons, and sensitivity analysis in a two-component Maxwellian plasma

1 Introduction

Interestingly, depending on the M, a limited exceed (Das et al., 2012) throughout the magnitude towards a solitary wave was discovered. The researchers, on the other hand, proved powerless to characterize the pattern of a newly discovered form of a solitary wave that differed from nonlinear solitary waves. Dubinov et al. (2012) designated such novel forms of waves as SNWs, which have been characterized by the nontrivial topology of the phase plots, for the very first instance in plasmas. A while back, super nonlinear waves in three-, four, or five components plasma systems were investigated for the small magnitude (Chapagai et al., 2020; Tamang and Saha, 2020) as well as arbitrary amplitude (Taha and El-Taibany, 2020; Saha and Tamang, 2019) limitations. Consequently, three-component Maxwellian or non-Maxwellian plasmas might create super nonlinear solitary and super nonlinear periodic waves in plasmas. If a small amplitude nonlinear wave is considered using the reductive perturbation method $(\mathit{RPT})$ and the KdV equation in a two-component electron-ion Maxwellian plasma is obtained, the coefficient of a nonlinear term is 1, i.e. a constant. As a result, higher-order corrections cannot be considered in a two-component electron-ion Maxwellian plasma. Also, in conclusion, small amplitude super nonlinear wave solutions aren’t feasible in a two-component electron-ion Maxwellian plasma. Only solitary and periodic wave solutions are feasible in this instance. In contrast, Cairns et al. (1995) stated where there exists a potential well in its positive direction on a two-component Maxwellian plasma, and therefore there exists a homoclinic loop throughout the phase domain of the differential equation that symbolizes a solitary wave exhibiting positive potential.

In this investigation, we obtained many solutions with the help of generalized auxiliary equation (GAE) method (Akinyemi et al., 2021), for SNWs in Maxwellian plasma with two components (Saha et al., 2020), including hyperbolic trigonometric, trigonometric, exponential, and rational. Therefore the light, darkened, periodic, singular, and exclusive soliton solutions that seem to be truly significant throughout mathematical physics were discovered underneath the variety of restriction constraints. So far as the authors are informed, no previous work on soliton solutions using the GAE technique has been published. An additional advantage of this technique is that it could be applied to other nonlinear models that are connected to it. In fact, the super nonlinear wave (SNW) is characterized by the nontrivial topology of their phase portraits. In this case, the paths contain at least two stable equilibrium points (centers) and one separatrix layer. These waves exist in the plasma system having three or more components. These can be studied by the direct and reductive perturbation techniques. It is good to understand that the phase paths of the dynamical system are the wave solution of the corresponding plasma system (Saha and Banerjee, 2021).

Non-linear differential equations have been commonly used to characterize a comprehensive variety of complex scientific phenomena, such as optics, deeper water, quantum mechanics, biophysics, fluid mechanics, plasma physics, marine engineering, chemistry, as well as physics. However, In literature, the more magnificent technique for finding new exact and explicit solutions for non-linear PDEs (Riaz et al., 2021). Because soliton solutions may be discovered in a variety of mathematical physics models, the theory of optical solitons is highly fascinating. One of the most important elements of nonlinear fiber optics is the observation of optical solitons. Soliton has several applications in applied science and engineering. There are several effective techniques for addressing nonlinear differential equations. Just a few examples include the new extended direct algebraic method (Jhangeer et al., 2020; Munawar et al., 2021; Jhangeer et al., 2021; Hussain et al., 2021; Hussain et al., 2020).

Relying upon the research of Atangana-Baleanu, Abdou et al. (2020) researchers discover novel solutions to the space-time fractional nonlinear Schrödinger equation, where $\frac{G^{\prime }}{G^2}$ –expansion and generalised Kurdyashov techniques are proposed. Fractional calculus having increasingly piqued the attention of many researchers in domains including physics, plasma physics, nonlinear optics and fractional characteristics, control system, signal processing, biology, architecture, fluid mechanics with visco elasticity, and so on (Abdou, 2017; Manafian, 2016). In Khater et al. (2018), researchers use a novel approach for solving the higher order nonlinear Schrödinger equation, which represents the propagation of brief light pulses in monomode optical fibres. The optical soliton to Wu-Zhang set of evolution equations depicts a (1 + 1)-dimensional dispersive long wave (Jawad and Al Azzawi, 2019). Several organizations in space-time fractional nonlinear optics as well as other fields having researched optical soliton solution in the conclusion (Edeki et al., 2019; Abdou and Yildirim, 2012; Mena-Contla et al., 2018).

Rational sine-Gordon expansion method which is a generalized form of the sine-Gordon expansion method is a newly developed method for analytical solutions (Yel et al., 2022). The generalized variable separation method and the extended homoclinic test the approach is implemented for the construction of the non-traveling wave solutions of $(2+1)$ -dimensional breaking soliton equation (Shang, 2022). Jhangeer et al. (2021), closely scrutinized the traveling wave solutions of a family of long-wave unstable lubrication models, a study of the perturbed Fokas-Lenells model’s traveling, periodic, quasiperiodic, and chaotic structures, as well as the dynamical analysis and phase pictures of two-mode waves in various mediums. Exact solutions regarding blood flow via a circular tube underneath magnetic field impact employing fractional Caputo-Fabrizio derivatives (Rihan et al., 2016) as well as Synchronization for tumor-immune framework involving fractional-order (Riaz and Zafar, 2018; Jhangeer et al., 2012; Jhangeer, 2018; Jhangeer et al., 2020) were evaluated together a systematic investigation to find and analyze the evidence to determine the reality by M.B. Riaz. Fractional calculus has made a major advancement in the development of solving models for implementations, and it attempts to illustrate that it can create more realistic models, Ghalib et al. (2020), Riaz et al. (2016), Ghalib et al. (2020), Riaz et al. (2020), Riaz et al. (2020), Riaz (2018), and Riaz et al. (2021).

The following is how paper is made: Section 2 is devoted to demonstrating the idea of the proposed method, i.e, GAE method. Section 3, deals with the quest of SNWs in Maxwellian plasma with two components. A simple electron-ion plasma composed of cold mobile ions as well as Maxwell distribution electrons is defined by the dimensionless fundamental equations in the exploration of SNWs. The wave frame is used to transfer the fundamental equations into the nonlinear ordinary differential equation. In Section 4 there is the application of GAE method to find soliton solutions. Furthermore, we have seen various textures of waves in the 2D graphical representation of the solutions, for different values of M. Visualization of study findings in a graphical format is discussed in Section 5. In Section 6, there is the sensitivity assessment for the supersonic, subsonic, and sonic wave profiles. Section 7 deals with the discussions and results analysis. We assessed various dynamic performances, structures, and texturing of wave solutions by analyzing graphs relying on their physical ranges. Associated numerical values for parameters were utilized as an aspect of our investigation to explore distinct dynamic behavior, patterns, and textures of wave solutions. In subSection 7.1 there is a comparative examination of the current study with the previously published literature. Section 8 includes the study’s conclusion.

2 Recapitulation for the GAE method

Throughout this segment, readers analyze the core notion of the GAE methodology (Akinyemi et al., 2021) through investigating the nonlinear PDE of the manner:

Assume Eq. (3) seems to have the following solutions:

Family:1 For $m_1>0$ , we have;

Family:2 For $m_1>0$ and $\mathrm{\Theta }>0$ , we have;

Family:3 For $m_1>0$ and $m_3>0$ , we have;

Family:4 For $m_1>0$ and $\mathrm{\Theta }=0$ , we have;

Family:5 For $m_1<0$ and $\mathrm{\Theta }>0$ , we have;

Family:6 For $m_1<0$ and $m_3>0$ , we have;

Family:7 For $m_1>0$ , we have;

Family:8 For $m_1>0$ and $m_2=0$ , we have;

Family:9 For $m_1=0$ , we have;

Family:10 For $m_1=0$ and $m_2=0$ , we have;

During which $\mathrm{\Theta }=m_2^2-4m_1{m}_3$ . There are four basic types of solutions: hyperbolic trigonometric, trigonometric, exponential, and rational. However many polynomial equations in ${\mathrm{\Psi }}^j(\zeta )$ of indeterminate variables are gathered into Eqs. (4) and (5) are replaced into Eq. (3) and the coefficients of ${\mathrm{\Psi }}^j(\zeta ),(j=0,1,2\dots )$ being equal to zero. Consequently, simply solving the preceding nonlinear polynomial equations, inserting the resulting constants in Eq. (4) with such a known N, and considering the aforementioned solutions set on Eq. (4), we rapidly find the suitable solutions of Eq. (1).

3 The quest of SNWs in Maxwellian plasma with two components

We consider a simple electron-ion plasma composed of cold mobile ions as well as Maxwell distribution electrons defined by the dimensionless fundamental equations in exploration of SNWs (Saha et al., 2020):

n is the number density for ions, along with $(n_e)$ is the number density for electrons, which is normalized to $n_0$ is the number density for ions at unperturbed condition, along with $(n_{e0})$ is the number density for electrons at unperturbed condition. Since, $u(\phi )$ where u is ion’s velocity and $\phi$ is electrostatic potential) is normalized to ion-acoustic speed $c_s=\sqrt{(K_B{T}_e/m)}(K_B{T}_e/e)$ , where e is electron charge and m is mass of ions. The length of $\mathit{Debyelength}$ , i.e, $\lambda =\sqrt{(K_B{T}_e/4\pi e^2{n}_0)}$ and inverse of ion plasma frequency, i.e, ${\omega }^{-1}=\sqrt{(m/4\pi e^2{n}_0)}$ is normalized to the space variable i.e, time denoted by $x(t)$ .

To analyze supernonlinear and other nonlinear waves in the given plasma system, we use the wave frame below.

When Eqs. (25) and (26) are applied to Eq. (24) and up to third degree terms are included, one may get:

4 Application of GAE method

Within that segment, the GAE method is used to solve Eq. (22). We achieve $N=1$ , by balancing ${\phi }^{″}$ and ${\phi }^3$ . As a result, the solution corresponding to Eq. (4) may be written as:

Placing Eqs. (5) as well as (28) inside (22) and setting the coefficient of ${\mathrm{\Psi }}^j(\zeta ),(j=0,1,2,3)$ to zero yields the foregoing:

When we solve the above equations for the parameters $M,m_1$ , and $m_2$ , we get the following solutions:

Set 1.

Set 2.

Moreover, the solutions of Eq. (27) are stated as follows, using Eqs. (28) and (30) in relation towards the solution sets established in Section 2:

• For $m_1>0$ ,

  (32)
  $$\begin{array}{cc}{\phi }_1(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{-m_1{m}_2{\mathrm{sech}}^2\left(\frac{\sqrt{m_1}}{2}\zeta \right)}{m_2^2-m_1{m}_3{\left(1\pm \tanh \left(\frac{\sqrt{m_1}}{2}\zeta \right)\right)}^2}\right),\hfill \end{array}$$
  (33)
  $$\begin{array}{cc}{\phi }_2(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{m_1{m}_2{\mathrm{csch}}^2\left(\frac{\sqrt{m_1}}{2}\zeta \right)}{m_2^2-m_1{m}_3{\left(1\pm \coth \left(\frac{\sqrt{m_1}}{2}\zeta \right)\right)}^2}\right)\text{.}\hfill \end{array}$$

• For $m_1>0$ and $\mathrm{\Theta }>0$ ,

  (34)
  $$\begin{array}{cc}{\phi }_3(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{2m_1{\mathrm{sech}}^2\left(\sqrt{m_1}\zeta \right)}{\pm \sqrt{\mathrm{\Theta }}-m_2\mathrm{sech}\left(\sqrt{m_1}\zeta \right)}\right),\hfill \end{array}$$
  (35)
  $$\begin{array}{cc}{\phi }_4(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{2m_1{\mathrm{csch}}^2\left(\sqrt{m_1}\zeta \right)}{\pm \sqrt{-\mathrm{\Theta }}-m_2\mathrm{csch}\left(\sqrt{m_1}\zeta \right)}\right)\text{.}\hfill \end{array}$$

• For $m_1>0$ and $m_3>0$ ,

  (36)
  $$\begin{array}{cc}{\phi }_5(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{-m_1{\mathrm{sech}}^2\left(\frac{\sqrt{m_1}}{2}\zeta \right)}{m_2\pm 2\sqrt{m_1{m}_3}\tanh \left(\frac{\sqrt{m_1}}{2}\zeta \right)}\right),\hfill \end{array}$$
  (37)
  $$\begin{array}{cc}{\phi }_6(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{m_1{\mathrm{csch}}^2\left(\frac{\sqrt{m_1}}{2}\zeta \right)}{m_2\pm 2\sqrt{m_1{m}_3}\coth \left(\frac{\sqrt{m_1}}{2}\zeta \right)}\right)\text{.}\hfill \end{array}$$

• For $m_1>0$ and $\mathrm{\Theta }=0$ ,

  (38)
  $$\begin{array}{cc}{\phi }_7(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(-\frac{m_1}{m_2}\left(1\pm \tanh \left(\frac{\sqrt{m_1}}{2}\zeta \right)\right)\right),\hfill \end{array}$$
  (39)
  $$\begin{array}{cc}{\phi }_8(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(-\frac{m_1}{m_2}\left(1\pm \coth \left(\frac{\sqrt{m_1}}{2}\zeta \right)\right)\right)\text{.}\hfill \end{array}$$

• For $m_1<0$ and $\mathrm{\Theta }>0$ ,

  (40)
  $$\begin{array}{cc}{\phi }_9(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{2m_1{\sec }^2\left(\sqrt{-m_1}\zeta \right)}{\pm \sqrt{\mathrm{\Theta }}-m_2\sec \left(\sqrt{-m_1}\zeta \right)}\right),\hfill \end{array}$$
  (41)
  $$\begin{array}{cc}{\phi }_{10}(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{2m_1{\csc }^2\left(\sqrt{-m_1}\zeta \right)}{\pm \sqrt{\mathrm{\Theta }}-m_2\csc \left(\sqrt{-m_1}\zeta \right)}\right)\text{.}\hfill \end{array}$$

• For $m_1<0$ and $m_3>0$ ,

  (42)
  $$\begin{array}{cc}{\phi }_{11}(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{-m_1{\sec }^2\left(\frac{\sqrt{-m_1}}{2}\zeta \right)}{m_2\pm 2\sqrt{-m_1{m}_3}\tan \left(\frac{\sqrt{-m_1}}{2}\zeta \right)}\right),\hfill \end{array}$$
  (43)
  $$\begin{array}{cc}{\phi }_{12}(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{-m_1{\csc }^2\left(\frac{\sqrt{-m_1}}{2}\zeta \right)}{m_2\pm 2\sqrt{-m_1{m}_3}\cot \left(\frac{\sqrt{-m_1}}{2}\zeta \right)}\right),\hfill \end{array}$$

• For $m_1>0$ ,

  (44)
  $$\begin{array}{cc}{\phi }_{13}(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{4m_1{e}^{\pm \sqrt{m_1}\zeta }}{{(e^{\pm \sqrt{m_1}\zeta }-m_2)}^2-4m_1{m}_3}\right)\text{.}\hfill \end{array}$$

• For $m_1>0$ and $m_2=0$ ,

  (45)
  $$\begin{array}{cc}{\phi }_{14}(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{\pm 4m_1{e}^{\pm \sqrt{m_1}\zeta }}{1-4m_1{m}_3{e}^{\pm 2\sqrt{m_1}\zeta }}\right)\text{.}\hfill \end{array}$$

• For $m_1=0$ ,

  (46)
  $$\begin{array}{cc}{\phi }_{15}(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{\pm m_1{m}_2}{m_2^2{\zeta }^2-m_1{m}_3}\right)\text{.}\hfill \end{array}$$

• For $m_1=0$ and $m_2=0$ ,

  (47)
  $$\begin{array}{cc}{\phi }_{16}(\zeta )=\hfill & -\frac{(3M^4\pm \sqrt{-15M^8+24M^6-54M^4+360M^2-279)}-9)M^2}{2(M^6-15)}\hfill \\ \hfill & \pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\pm \frac{1}{\sqrt{m_3}\zeta }\right)\text{.}\hfill \end{array}$$

  Consequently, the solutions for Eq. (27) are stated here as pursues, employing Eqs. (28) and (31) with respect towards the solution sets specified in Section 2:

• For $m_1>0$ ,

  (48)
  $${\phi }_{17}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{-m_1{m}_2{\mathrm{sech}}^2\left(\frac{\sqrt{m_1}}{2}\zeta \right)}{m_2^2-m_1{m}_3{\left(1\pm \tanh \left(\frac{\sqrt{m_1}}{2}\zeta \right)\right)}^2}\right),$$
  (49)
  $${\phi }_{18}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{m_1{m}_2{\mathrm{csch}}^2\left(\frac{\sqrt{m_1}}{2}\zeta \right)}{m_2^2-m_1{m}_3{\left(1\pm \coth \left(\frac{\sqrt{m_1}}{2}\zeta \right)\right)}^2}\right)\text{.}$$

• For $m_1>0$ and $\mathrm{\Theta }>0$ ,

  (50)
  $${\phi }_{19}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{2m_1{\mathrm{sech}}^2\left(\sqrt{m_1}\zeta \right)}{\pm \sqrt{\mathrm{\Theta }}-m_2\mathrm{sech}\left(\sqrt{m_1}\zeta \right)}\right),$$
  (51)
  $${\phi }_{20}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{2m_1{\mathrm{csch}}^2\left(\sqrt{m_1}\zeta \right)}{\pm \sqrt{-\mathrm{\Theta }}-m_2\mathrm{csch}\left(\sqrt{m_1}\zeta \right)}\right)\text{.}$$

• For $m_1>0$ and $m_3>0$ ,

  (52)
  $${\phi }_{21}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{-m_1{\mathrm{sech}}^2\left(\frac{\sqrt{m_1}}{2}\zeta \right)}{m_2\pm 2\sqrt{m_1{m}_3}\tanh \left(\frac{\sqrt{m_1}}{2}\zeta \right)}\right),$$
  (53)
  $${\phi }_{22}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{m_1{\mathrm{csch}}^2\left(\frac{\sqrt{m_1}}{2}\zeta \right)}{m_2\pm 2\sqrt{m_1{m}_3}\coth \left(\frac{\sqrt{m_1}}{2}\zeta \right)}\right)\text{.}$$

• For $m_1>0$ and $\mathrm{\Theta }=0$ ,

  (54)
  $${\phi }_{23}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(-\frac{m_1}{m_2}\left(1\pm \tanh \left(\frac{\sqrt{m_1}}{2}\zeta \right)\right)\right),$$
  (55)
  $${\phi }_{24}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(-\frac{m_1}{m_2}\left(1\pm \coth \left(\frac{\sqrt{m_1}}{2}\zeta \right)\right)\right)\text{.}$$

• For $m_1<0$ and $\mathrm{\Theta }>0$ ,

  (56)
  $${\phi }_{25}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{2m_1{\sec }^2\left(\sqrt{-m_1}\zeta \right)}{\pm \sqrt{\mathrm{\Theta }}-m_2\sec \left(\sqrt{-m_1}\zeta \right)}\right),$$
  (57)
  $${\phi }_{26}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{2m_1{\csc }^2\left(\sqrt{-m_1}\zeta \right)}{\pm \sqrt{\mathrm{\Theta }}-m_2\csc \left(\sqrt{-m_1}\zeta \right)}\right)\text{.}$$

• For $m_1<0$ and $m_3>0$ ,

  (58)
  $${\phi }_{27}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{-m_1{\sec }^2\left(\frac{\sqrt{-m_1}}{2}\zeta \right)}{m_2\pm 2\sqrt{-m_1{m}_3}\tan \left(\frac{\sqrt{-m_1}}{2}\zeta \right)}\right),$$
  (59)
  $${\phi }_{28}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{-m_1{\csc }^2\left(\frac{\sqrt{-m_1}}{2}\zeta \right)}{m_2\pm 2\sqrt{-m_1{m}_3}\cot \left(\frac{\sqrt{-m_1}}{2}\zeta \right)}\right),$$

• For $m_1>0$ ,

  (60)
  $${\phi }_{29}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{4m_1{e}^{\pm \sqrt{m_1}\zeta }}{{(e^{\pm \sqrt{m_1}\zeta }-m_2)}^2-4m_1{m}_3}\right)\text{.}$$

• For $m_1>0$ and $m_2=0$ ,

  (61)
  $${\phi }_{30}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{\pm 4m_1{e}^{\pm \sqrt{m_1}\zeta }}{1-4m_1{m}_3{e}^{\pm 2\sqrt{m_1}\zeta }}\right)\text{.}$$

• For $m_1=0$ ,

  (62)
  $${\phi }_{31}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\frac{\pm m_1{m}_2}{m_2^2{\zeta }^2-m_1{m}_3}\right)\text{.}$$

• For $m_1=0$ and $m_2=0$ ,

  (63)
  $${\phi }_{32}(\zeta )=\pm 2\sqrt{3}\sqrt{\frac{m_3}{(M^6-15)}}{M}^3\left(\pm \frac{1}{\sqrt{m_3}\zeta }\right)\text{.}$$