Dynamics of invariant solutions of mKdV-ZK arising in a homogeneous magnetised plasma

Classical Lie symmetry analysis is proposed to get a new variety of similarity solutions to a nonlinear (3+1)-modified Korteweg de Vries-Zakharov–Kuznetsov equation. The equation is often used to control the type of weakly nonlinear ion-acoustic waves in a homogeneous magnetised electron-positron plasma. In a magnetised plasma, including some hot and cold ions, such waves exist. By comparing the results reported earlier, new variety of analytical solutions are created and validated. Trigonometric, hyperbolic, rational, and exponential solutions of various types are derived. To prove their physical significance, animation profiles are plotted using MATLAB. Dominated dynamical behaviour of the animation profile is represented in each case. Electrostatic potential dynamics reveal elastic single solitons to multi solitons, elastic multi solitons, kink to stationary, and bell-shaped to asymptotic behaviour. Equations (13) and (21) in this study that were solved trivially can lead to more research in this area.

1 Introduction : Fully ionised gases with particles of equal and opposite charge and mass are referred to as pair plasmas.Such paired (Electron-positron) plasmas play a significant role in the cooling process and creation of elements of the early universe [1].The Big Bang, active galactic nuclei, gamma ray bursts, pulsar magnetospheres, and the Solar atmosphere all contribute to the early universe's slow cooling.High-energy particles are accelerated along the pulsar magnetic field and release curvature photons, which generate new electron-positron couples in the case of pulsars.
It has been also a key concern of many researchers to get the exact solutions of physical systems inherently nonlinear in nature, which are usually governed by nonlinear partial differential equations (NPDEs).It has became a matter of deep attention since these exact solutions are helpful broadly to explain the inner mechanism/behaviour of such complex phenomena governed by these NPDEs arising in numerous fields, like viscoelasticity, solid state, kinematics, magnetized plasma in space, study of coastal waves in ocean solid mechanics, optical fibers, fluid ions, signal processing, astrophysics, electromagnetic waves, biomedical sciences, fluid mechanics, etc. .Pluralistic development of nonlinear science has resulted in the last five decades due to the use of many effective methods and techniques for getting the analytical solutions of NPDEs, taking into account different issues, approximations, assumptions, merits, and demerits of the methods.There is no unique technique to apply to all NPDEs.Therefore, it is important to choose appropriate and established techniques for a problem.
In the ongoing research, we investigate the modified Korteweg de Vries-Zakharov-Kuznetsov (mKdV-ZK) (1) equation to get its analytical solutions.The mKdV-ZK controls the nature of weakly nonlinear ionacoustic waves in magnetized electron-positron plasma, consisting of the some hot and cold ions [2].A complete overview of the historical background of mKdV-ZK (1) is presented in Abdullah et al. [3], and Lazarus et al. [4].The mKdV-ZK can be calculated using the following approximations [5] or conditions [4] in a fourcomponent paired (electron-positron) plasma that is homogeneously magnetised and consists of cold electrons and positrons with identical temperatures and equilibrium densities.(i) The density of hot electrons (positrons) can be represented in the form of the electrostatic potential since hot isothermal species have a Boltzmann distribution.(ii) The dynamics of the cooler adiabatic species, follows the continuity equations, equations of motion, and adiabatic pressure equations.(iii) Such system is closed by the Poisson's equation.(iv) Linearization of (i)-(iii) yields some dispersion relations for paired plasmas.(v) Using nonlinear modes by introducing stretched coordinates and then approximating fluid velocity, density, pressure, and electrical potential with a small parameter, e.g.ǫ, one can get Poisson's equation to order ZK equation: where u is the electrostatic potential or voltage in an x, y, z frame at time t, a is a dispersion coefficient, b and c are real constants that appear due to the finiteamplitude effects of first-order disturbances.The values of them depends upon initial fluid velocity along x-direction, thermal velocity of cool component, common mass of electron/positron, charge at hot species, Debye lengths, phase velocity, gyro-frequency, plasma frequency and phase velocity [4].The mKdV-ZK is renamed as extended KdV-ZK [6] when second term is considered as a uux.In a two-component electron-ion plasma, the ion-acoustic wave, which is an ion timescale event, has been explored, as well as the associated linear [7] and nonlinear behaviour [8,9].The mKdV-ZK ( 1) is obtained by applying the singular perturbation method to a mathematical model whose governing equations explain the dynamics of cooler adiabatic species [10].This model represents the wave propagation in a three-dimensional homogeneous magnetised, electron-positron plasma, containing equal amount of cool and hot ions.Abdullah et al. [3] employed modified extended mapping method to the mKdV-ZK equation ( 1), and obtained periodic, kink and antikink, bright, dark soliton solutions.Solitary wave solutions such as electrostatic field potentials, electric fields, magnetic fields, and quantum statistical pressures were obtained by them [3].Some of the two-dimensional properties of solitary waves arising in the Earth's auroral area at high altitude were explained by Mace and Hellberg [11].An aurora is a natural light that can be seen in the sky and is most commonly seen in the Arctic and Antarctic regions of the Earth.Auroras produce beautiful light patterns that seem like curtains, spirals, or dynamic flickers that cover the entire sky.Auroras are created by solar wind-induced instabilities in the magnetosphere.Charged particles, primarily electrons and protons, whose trajectories are altered by these disturbances, precipitate in the thermosphere and exosphere, i.e. in the magnetospheric plasma.Ionization and excitation of atmospheric components provide a wide range of colour and complexity in light.The amount of acceleration imparted to the precipitating particles determines the shape of the aurora, which occurs in bands around both polar regions.Auroras can be found on most of the planets in the Solar System, as well as some natural satellites, brown dwarfs, and even comets.Many of the observed waveforms for KdV-ZK can be attributed to magnetised [11] and unmagnetized planar and ellipsoidal soliton with high time resolution electric and magnetic field measurements.
An interesting literature overview related to the solutions of mKdV-ZK is presented to know more about it like this equation is a mutual combination of the mKdV and ZK equations.For a = α, and b = c = 1 in (1), the equation governs the oblique propagation of electrostatic modes in magnetized plasmas and has been re-derived in a fully systematic way for general mixtures of hot isothermal, warm adiabatic fluid, and cold immobile background species [5,12].
Khalique and Adeyemo [13] explored the Lie symmetry reductions method to find a non-topological soliton solution by solving generalized KdV-ZK equation of the form Bibi et al. [14] employed the G ′ G 2 -expansion method to the following form of (3+1)-time fractional KdV-ZK equations where D α t is used as a modified Riemann-Liouville derivative [14], and ǫ, η, ν are constants.They [14] have attained trigonometric and rational type solutions, while Jin et al. [15] attained the dark solitons for the following space-time fractional mKdV-ZK equation by using the fractional complex transform with undetermined coefficients.
where d, e, f, g are constants.They [15] have taken Riemann Liouville fractional order derivatives.For α = 1, f = g it becomes the same mKdV-ZK (1).More over, Ali Akbar et al. [16] used rational G ′ Gexpansion method to obtain hyperbolic, trigonometric, and rational form solutions of the following form of space-time fractional KdV-ZK equations.
where fractional derivatives are considered as Riemann-Liouville's fractional order [16] and in the meaning of conformable derivative.Abdelrahman [17] investigated a similar form (5) of fractional KdV-ZK and used the Riccati-Bernoulli Sub-ODE approach to find its trigonometric solutions.Islam et al. [18] used the enhanced G ′ G -expansion method, and Lu et al. [2] used the extended G ′ Gexpansion method to solve mKdV-ZK (1) and derived its trivial solutions, while for a = α, and b = c = 1 in Eq. ( 1), the mKdV-ZK is investigated by Zhang [12] employing Jacobi elliptic function expansion method and derived travelling wave solutions.
Lie group symmetry method is employed by Sahoo et al. [19] to Eq. ( 1) treating b = c = 1 therein, and obtained the trivial form of the solutions without considering the effect of all variables x, y, z and t because they [19] have solved to Eq. ( 1) taking directly the six infinitesimal generators.The modified extended direct algebraic method by Lu et al. [6] and the modified simple equation method is employed by Khan and Ali Akbar [20] for the same form of the mKdV-ZK and claimed to get some new exact solutions, while the auxiliary equation method is used to get traveling wave solutions of (1) by Tariq and Seadawy [21].Khalique and Adeyemo [13] solved extended form of KdV-ZK via Lie symmetry reductions and Kudryashov's method and non-topological soliton, cnoidal and snoidal periodic solutions were found to be possible.
The rest of the framework of this article is arranged as follows: In section 2, the authors recall some basic steps to generate Lie symmetries.The analytical solutions are derived by using the Lie symmetry in sections 3. Comparison with existing analytical solutions are performed in section 4, while in section 5, the solutions are depicted physically via their animation profiles.Conclusions with future scope of the present research appear in section 6.

Lie symmetries
This section depicts the steps to generate Lie symmetries of the (3 + 1)-dimensional mKdV-ZK equation (1).The first step of this process is to construct the invariant condition.Secondly, the invariance reduces the number of independent variables in the existing mKdV-ZK.Repeated use of such a process finally gives an ODE.Authors have obtained the solution to the mKdV-ZK equation after obtaining its solution.These solutions are appealing in terms of explaining physical nature and depicting its behaviour with respect to changes in space and time.Due to its nonlinear nature, the mKdV-ZK equation is not easily solvable to get analytical solutions.Thus, the generation of similarity forms via Lie symmetries leads to the reduction of an equivalent PDE with one fewer number of independent variables.With repeated use of Lie symmetry reduction, the (3 + 1)-mKdV-ZK reduces to an ODE, which is much easier to solve.The brief description and applications of the method can be studied from some text books [22,23] and references therein [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].Assuming one-parameter Lie group of the following transformations where the notation (χ, t) denotes (x, y, z, u, t) and ξ (1) , ξ (2) , ξ (3) , τ and ηu are the infinitesimals among which the each one is a function of (χ, t).
The generator V associated with the one-parameter transformations can be explored as Lie symmetry keeps the Eq. ( 1) invariant satisfying the condition: where the third prolongation P r (3) (termed in [23]), can be expressed as .
After using Eq. ( 1) into ( 8), and eliminating one of the partial derivative, from the invariance condition (8), it turns into a PDE which provides the following determining system of the equations: x = ξ x = ξ After solving them, the authors obtained the following infinitesimals: The constants a ′ i s, (i = 1, 2, 3, 4, 5, 6) are arbitrary.The infinitesimals derived here are same as calculated in Sahoo et al. [19] but they did not get even a single solution to Eq. ( 1) involving the effect of all variables x, y, z and t on u.In the following section, the authors have taken care of it and used classical Lie symmetry approach considering with more general choices of arbitrary constants a ′ i s to get more variety of analytical solutions.Now, the six dimensional Lie-algebra L 6 can be generated by The commutator table for the Lie-symmetry algebra L 6 is given in table 1.
The generators derived here are same as calculated in Sahoo et al. [19].They [19] have solved to Eq. ( 1) taking directly using the six infinitesimal generators to start with Lagrange's characteristic equations.Consequently, they obtained the very particular form of the solutions.

Invariant solutions
Authors can utilize the characteristic equation with the following Lagrange system for the proposed Eq. ( 1) Now, to proceed further, authors can consider the following cases: Case (I) For most general case, one should take a 1 = 0 in Eq. ( 12) to reduce it via Lie symmetry, then the corresponding Lagrange's characteristic equation recasts as For the similarity variables X = (3t , it gives the similarity function as u = (3t + α 5 ) − 1 3 F (X, Y, Z), and Then, the first similarity reduction for mKdV-ZK (1) Employing Lie symmetry to (13), all infinitesimals are zero, and a trivial solution F = 0 is obtained for it, and hence u = u 1 = 0 is not beneficially.The authors are intended to consider another case as: Case (II) Taking a 1 = a 3 = 0, a 6 = 0 in Eq. ( 12), one can get It gives the following similarity form and , Therefore, similarity reduction of the system (1) yields Again, applying the similarity reduction, the infinitesimals are as follows: The corresponding characteristic equation for ( 15) is Therefore, further similarity reduction of Eq. ( 14) provides Again, applying the similarity which yields Thus, Lagrange's characteristic equation for ( 17) is Case (IIa) : For a 11 = 0 in Eq. ( 18), then gives Lie similarity reduction of the system (16) as where Its integration gives Table where and are integration constants.To solve it further, it is imperative to split the cases as: , the solutions for mKdV-ZK (1) are Case (IIa 2 ) : If B 4 = C 3 = C 4 = 0, then solution for Eq. ( 1) is Case (IIa 3 ) : , C 3 = C 4 = 0, the solutions can be expressed as Case (IIa 4 ) : , the solutions are , and Case (IIa 5 ) : Case (IIa 6 ) : Case (IIb) : a 10 = 0 in Eq. ( 18) , where E 2 = a 11 a 10 .
Therefore, similarity reduction of the system (16) provides where
Other solutions are diametrically opposed to previous findings.

Analysis and discussions
The physical behaviour of analytical solutions represented by Eqs Eqs. ( 22)-( 31), ( 35)-( 44), (48)-( 57) is described in this section.Trigonometric, hyperbolic, rational, and exponential forms are investigated in solutions.The animation is produced using symbolic computations in MATLAB to provide a solution profile.With respect to space and time, there is a variation in u (electrostatic potential).As a result, the authors have captured the dominating behaviour of a frame of animation and displayed it in Figs.1-4    The shape of the profile changes from kink to stationary at time 0 ≤ t ≤ 282 and with constant C 10 = 0.9706, which is shown in figure 3 and the variation in potential u 7 .The nature of the other profiles, u 17 and u 27 , is the same as u 7 .

Figure 1 :
Figure 1: The behavior of electrostatic potential for u 2 varies from single soliton to multi solitons during 0 ≤ t ≤ 40.The arbitrary constant C 5 = 0.4218 is used to simulate the animation of u 2 .The physical behaviours of others u 3 , u 5 , u 6 , u 12 , u 13 , u 15 , u 16 , u 22 , u 23 , u 25 , and u 26 , are analogues to u 2 .

Figure 2 :
Figure 2: Elastic multi solitons can be seen in Fig.2, for the solution u 4 .As the space range for x and y expands, the potential function u 4 decreases.C 7 = 0.6324 is held constant in the simulation.Potential behaviour remains the same when the time range is extended.The profiles for u 8 , u 9 , u 11 , u 14 , u 18 , u 19 , u 21 , u 24 , u 28 , u 29 , and u 31 are similar to u 4 .

Figure 3 :
Figure 3:  The shape of the profile changes from kink to stationary at time 0 ≤ t ≤ 282 and with constant C 10 = 0.9706, which is shown in figure3and the variation in potential u 7 .The nature of the other profiles, u 17 and u 27 , is the same as u 7 .

Figure 4 :Fig. 1 .Fig. 2 .
Figure 4: The shape of the figure changes from bell-shaped to the asymptotic profile of the solutions u 10 .The potential increases at 0 ≤ t ≤ 182 with C 13 = 0.9722.Other profiles, u 20 and u 30 , are of the same nature as u 10 .