Nonlinearity contributions on critical MKP equation

The mathematical new plasma wave solutions are specified in the compose of trigonometric, rational, hyperbolic, periodic and explosive kinds that are realistic for Modified-Kadomtsev-Petviashvili (MKP) equation. Also, numeral studies for the acquired solutions have been reveals that periodic, shock and explosive new forms may applicable in D-F Earth's ionosphere plasma. The used method is influential and robust in comparison applications in plasma fluids. To depict the propagating soliton profiles in a plasma medium, it is needful to solve MKP equation at a critical mass ratio. The Riccati–Bernoulli sub-ODE technique has been utilized to introduce some new important and applicable solutions. The number of these MKP solutions give a leading deed in applied ion acoustics in ionosphere.

Consider the nonlinear partial differential equation for an unknown function ψ(x, t). Utilizing the wave transformation Equation (1) converted to the following ODE: There are so many models in physics, fluid mechanics and engineering fields in forms of partial differential Equation (1) are transformed into the following ODE: see for example [27][28][29][30][31][32][33][34][35][36][37][38][39]. This observation gives this equation special and important feature. Due to the importance of Equation (3), we pose the robust and unified solver for the widely used NPDEs, utilizing RB sub-ODE method [40]. Namely, RB sub-ODE method [40] is the basic ingredient for this solver. This solver can be used as a box solver for solving so many equations arising in applied science. This solver will be so helpful for engineers, physicists and mathematicians in order to emphasis some interesting phenomena in real-life problems.

Unified solver
In this section, we will see how the concept of a unified solver is in practice implemented.

Mathematical model
where is a very small value and λ is the IA speed. Sabry et al. [17] examined the propagating IAWs in plasma having negative and positive fluids in addition to electrons. In the case of Maxwillian electrons, Poisson's equation reads, Where n −,+ is the number density of heavy negative ions, light positive ions normalized by its equilibrium value n −0,+0 , u − , v − (u + , v + ) is the negative (positive) ion fluid velocity, the electrostatic potential φ. With μ − = Z − n −0 /Z + n +0 and μ e = n e0 /Z + n +0 are the unperturbed negative ion and electron to positive ion ratio, respectively. Thus, the equilibrium condition implies μ − + μ e = 1, Q = m + /m − is the mass ratio, where m − (m + ) is the heavy (light) ion fluid mass, α = Z − /Z + where Z ∓ is charge numbers. The obtained results support that the system becomes at critical at Q = Q c , the modified KP equation was given: By using a similarity transformation given in the form: where L and M are the directional cosines of xand y -axes. The modified KP form transformed into the ODE in the form: Equation (40) gives stationary soliton in the form where u and v are the travelling speed in the two directions.

Results and discussion
Comparing Equation ( Trigonometric function solution: (When υ − s/θ > 0)  The trigonometric solutions of Equation (40) are and (46) Hyperbolic function solution : (When υ − s/θ < 0) The hyperbolic solutions of Equation (41) are (47) and (48) Two-dimensional propagation of MKP solitary nonlinear IAs have been examined in a plasma mode using parameters related to the plasmas of Earth's ionosphere [16,17]. At certain mass ratio value called the criticality value, the obtained equation cannot describe mode. So, new stretching produced MKP equation which describes critical system under investigation. Equation (42) represents soliton with stationary behaviour at different directionalcosine in x-axis (L) as shown in Figure 1. At a critical point, many solitary forms were expected to discuss the IAs behaviour using Riccati-Bernoulli solver for MKP equation. Solution (45) is solitonic wave type called explosive type with rapid increasing amplitude as depicted in Figure 2. Solution (46) is blow-up periodic shape as in Figure 3. On the other hand, the dissipative behaviours are also produced in Figures 4 and 5. In the solution of (48), the shock wave is propagated in the medium as shown in Figure 4. Finally, the explosive shock profile is obtained for solution (49) as shown in Figure 5.

Conclusions
Riccati-Bernoulli solver gives new solitary excitations for MKP equation such as periodic, explosive, shock and new explosive shocks which represent the pictures of wave motion of plasma solitons. It was reported that the obtained forms can be used in verify the broadband and magnetotail electrostatic waves observations.