mKP Equation Research Articles

A novel model for the fifth-order mKP equation describing (2+1)-dimensional shallow ocean solitary waves and their properties

In this paper, we develop a (2+1)-dimensional fifth-order modified Kadomtsev-Petviashvili (mKP) equation model using multiscale analysis and perturbation expansion, which is a novel model description of the shallow ocean solitary waves phenomenon, featuring stronger nonlinear terms and a higher-order compared to traditional models. We examine the conservation law within the model to further investigate the properties of solitary waves. Subsequently, we obtain the exact solutions of the model by employing the expansion method. We simulate the generation and propagation of solitary waves through appropriate parameter adjustments, deriving the solitary wave solutions, kink wave solutions, and periodic wave solutions. Additionally, we analyze the effects of different parameters on the propagation of solitary waves. Finally, we explore the evolution of solitary waves before and after generation under a given initial perturbation conditions. This study contributes to a deeper understanding of the intricate dynamics of shallow ocean solitary waves, shedding light on their behaviour and implications.

Soliton interactions and Mach reflection in gas bubbles–liquid mixtures

In this study, we assume that blood is assumed to be a viscoelastic and incompressible homogeneous media in which several uniform sized oxygen bubbles are uniformly distributed. Based on this, we establish a (3 + 1)-dimensional modified Kadomtsev–Petviashvili (mKP) equation to describe the long nonlinear pressure waves in the gas bubbles–liquid mixtures. Using bell polynomials, a new bilinear form of the mKP equation is constructed, and then the one- and two-soliton solutions of the equation are obtained by the Hirota method. Via the one-soliton solutions, parametric conditions of the existence of shock wave, elevation and depression solitons, and the Mach reflection characters in the mixtures are discussed. Soliton interactions have been discussed on the basis of the two-soliton solutions. We find that the (i) parallel elastic interactions can exist between the shock and elevation solitons; (ii) oblique elastic interactions can exist between the (a) shock and depression solitons and (b) the elevation and depression solitons; and (iii) oblique inelastic interactions can exist between the two depression solitons.

Archipelagos, islands, necklaces, and other exotic structures in external force-driven chaotic dusty plasmas

In this paper, the modified Kadomtsev Petviashvili (mKP) equation is derived by considering the dynamics of the dust-acoustic waves (DAWs) with kappa-distributed hot and cold ions and Boltzmannian electrons. The reductive perturbation technique is applied for deriving mKP. The bifurcation theory of the planar dynamical system is used to obtain the phase portrait of the DAWs in the framework of mKP equation. The formation of dust-acoustic solitary waves (DASWs) is analyzed for different plasma parameters in the Saturn’s magnetosphere. In the current plasma model, the external periodic force is introduced to study the quasiperiodic and chaotic behavior of the DAWs. It is noted that periodic initial conditions lead to the emergence of many outlandish features in the system by comparison with the solitary initial conditions. The frequency of the external periodic force ω plays an important role in the transition from quasiperiodic to chaotic behavior. Moreover, the strength of the external periodic force, the value of the nonlinear coefficient of the mKP equation, and the nonlinear acoustic speed are also found to have a significant effect on the chaotic behavior of the system.

Undular bores in the (3+1) dimensional mKP equation

Undular bore (dispersive shock wave) solutions in the (3+1) dimensional modified Kadomtsev-Petviashvili (mKP) equation for step type initial condition along a paraboloid type wavefront are found. Then, using a suitable solution form for the (3+1) dimensional mKP equation, it is reduced to the (1+1) dimensional focusing spherical mKdV (smKdV) and defocusing spherical mKdV (smKdV(d)) equations. Next, the Whitham modulation equations of the smKdV and smKdV(d) equations are found in terms of Riemann variables. Numerical solutions of the derived modulation equations are obtained. Also, an error analysis is performed for direct numerical solutions of both smKdV and smKdV(d).

Some electrostatic structures of the mKP equation for a nonthermal plasma system by a unified solver technique

The reductive perturbation technique is used to obtain the Modified KP equation at critical densities distinguished by the MKPE in plasma ion pair with fast electron positron. The new structures reveal that super-solitary and period waveforms are derived via the mathematical analysis of Jacobi-elliptic function expansions (MJEFE) for MKPE. The electrostatic new structures such as super-soliton, cnoidals, shock-like, super-dispersive and superperiodic plasma waves which existed at critical densities have been introduced. The positrons (electron) nonthermality supports on nonlinear dispersive new structures have been discussed. Also, many of the obtained electrostatic solutions are important and may be applicable in ionosphere plasma observations.

The attributes of the dust-acoustic solitary and periodic structures in the Saturn's inner magnetosphere

Nonlinear characteristics of dust-acoustic (DA) structures including the localized and periodic waves in a plasma having Maxwellian ions and superthermal two-temperature electrons are investigated. The wave equations, including both Kadomtsev–Petviashvili (KP) and modified KP (mKP) equations, are derived using the reductive perturbation technique (RPT). The quantitative and qualitative characteristics of both compressive and rarefactive structures are studied. The Jacobi elliptic function expansion method (JEFEM) is employed for the purpose of quantitative analysis, while the qualitative behavior is studied by the dint of the dynamical system approach. The solutions to the mKP equation hold under a critical condition where the quadratic nonlinearity ceases to exist. It is noticed that the KP equation admits only rarefactive solitary waves (SWs), whereas the mKP equation admits both compressive and rarefactive SWs. It is found that the profile (amplitude and width) of both DA solitary and periodic structures are different at different radii of Saturn's inner magnetosphere. The effect of the kappa spectral index is studied, and it is found that when the population of energetic cold electrons is decreased, the solitary structure gets energized. Our study is applied to Saturn's inner magnetosphere where kappa distributed two-temperature electrons and dust grains with negative charge are observed by various satellite missions.

Ion-acoustic wave dynamics and sensitivity study in a magnetized Auroral phase plasma

The dynamic behavior of ion-acoustic waves (IAWs) inside a magnetized auroral zone plasma including ions as well as two-temperature electrons is investigated. The Kadomtsev–Petviashvili (KP) and modified KP (mKP) equations are provided using the reductive perturbation method (RPT). In fluid dynamics, plasma physics, and electrodynamics, mKP equation can be utilized to characterize wave phenomena. It increases our interest due to its significant physical and mechanical value. The new extended direct algebraic approach is used to compute several traveling wave patterns for the nonlinear mKP equation. It is a simple, concise and general method in symbolic computations for solitary wave solutions for nonlinear problems. It yields dark solitary solutions, the community for semi-dark soliton results, dark singular soliton effects, kink soliton results, and soliton consequences encompassing trigonometric function, mixture hyperbolic function, as well as rational functions. Some arbitrarily chosen solutions are displayed visually for physical understanding. Moreover, the sensitivity of the perturbed dynamical structure equivalent to the mKP equation has been explored. Furthermore, the effect of IAWs spectral index on the parameters of the solitary-wave solutions and the sensitivity of the perturbed dynamical system of mKP equations is investigated in detail.

Formation and interaction of multi-dimensional electrostatic ion-acoustic solitons in two-electron temperature plasmas

Ion-acoustic waves are investigated in an unmagnetized collisionless plasma comprising dynamical ions and inertialess cold and hot (C–H) electrons expressed by Maxwellian, kappa, and (r, q) distributions. The reductive perturbation theory is applied for deriving a modified Kadomtsev–Petviashvili (mKP) equation to examine the characteristics of ion-acoustic structures (solitary waves, IASWs). The Hirota bilinear formalism is used to investigate the propagation of a single soliton and the interaction of two solitons with special reference to space plasmas. It is found that ratio of number density of cold to hot electrons and non-Maxwellian nature of cold electrons not only affect the propagation characteristics of single mKP solitons but also alter the interaction time of the IASWs. It is found that the swiftness of the interaction of solitons for flat-topped distribution surpasses both kappa and Maxwellian distributions. It is also found that the bipolar and tripolar structures depend on the ratio of the propagation vectors. The ranges of electric field amplitude for the IASWs are calculated for C–H electron distributions corresponding to Saturn's B-ring and the region just beyond terrestrial magnetopause, and are shown to agree with Cassini wideband receiver observational data and wideband plasma wave instrument's waveform data.

Rational, semi-rational solution and self-consistent sources extension of the variable-coefficient extended modified Kadomtsev-Petviashvili equation

In this paper, we apply Hirota bilinear method and determinant technique to derive the Nth-order rational solution expressed compactly in terms of Matsuno determinants for the variable-coefficient extended modified Kadomtsev-Petviashvili (mKP) equation. As a special case, we obtain the M-lump solution expressed in terms of 2M × 2M determinants for the mKPI equation and investigate the dynamical behaviors of 1- and 2-lump solutions. Furthermore, we present the Wronskian and Grammian solution for the variable-coefficient extended mKP equation. Based on the Grammian solution, we construct the line soliton, the line breather and the semi-rational solution on constant and periodic backgrounds for the mKPI equation. Through the asymptotic analysis, we show that the semi-rational solutions describe the fission and fusion of lumps and line solitons. In addition, we construct the variable-coefficient extended mKP equation with self-consistent sources via the source generation procedure and derive its N-soliton solution in the compact form of Grammian and Wronskian.

Formation of electrostatic solitary and periodic waves in dusty plasmas in the light of Voyager 1 and 2 spacecraft and Freja satellite observations

Motivated by the observations of Voyager 1 and 2 spacecraft and Freja satellite observations in Saturn’s magnetosphere, the formation of dust-acoustic (DA) localized and periodic waves in a complex plasma having superthermal electrons and ions are reported. In this regard, a modified Kadomtsev–Petviashvili (mKP) equation is derived by employing the weak turbulence theory for studying the characteristics of the nonlinear dust-acoustic waves (DAWs) in the model under consideration. The localized and periodic wave solutions to the mKP equation are derived using ansatz method in terms of Jacobi elliptic functions (JEFs). It is reported that the phase velocity of the DAWs in the Saturn’s magnetosphere is lower for kappa distributed ions and electrons by comparison with regions of space plasmas where the electrons and ions follow the Maxwellian distribution. The conditions for the existence of both localized and periodic waves are also presented. Estimates are also given of the spatial scales over which the dust-acoustic solitary/periodic structures form in Saturn’s magnetosphere.

The nonextensive effects on the supersoliton structure in critical plasma state

The support of the nonextensive positrons on novel nonlinear structures like super-solitons, cnoidal, shock profiles, super-shocklikes and super-periodic wave propagation with regard to critical behavior depicted by the MKP equation in plasma pairs with nonextensive electrons and positrons have been studied. Also, some newly solutions as periodical and super-solitary solutions are given by expansion method of Jacobi-elliptic functions (EMJEFs) for MKP form. These formal structures may use to investigate the ionosphere plasmas observations. The method suggested here is explicit, durable and efficacious in the plasma fluids. To discuss the nonextensive effects on the propagating nonlinear profiles, many of the obtained solutions are examined by the nonextensive index in the ionosphere observations.

On the dynamics of nonlinear propagation and interaction of the modified KP solitons in multicomponent complex plasmas

Dust-acoustic waves (DAWs) are analyzed in the small amplitude limit in a collisionless unmagnetized dusty plasma whose constituents are inertial dust grains, massless ions expressed by the generalized (r,q) distribution and inertialess Maxwellian electrons using the fluid theory of plasmas. The modified Kadomtsev-Petviashvili (mKP) equation is derived at a critical plasma condition for which the quadratic nonlinearity vanishes. The propagation of single soliton and interaction of two solitons are analyzed for the mKP equation in the context of plasma physics by employing Hirota bilinear formalism. The effects of the flatness parameter r and tail parameter q of the ions on the frequency of the DAWs are studied and the comparison with Maxwellian and kappa distributions is drawn. Using the plasma parameters corresponding to the Saturn’s E-ring, the range of electric field amplitude for dust-acoustic solitary waves (DASWs) for different ion distributions is calculated and is shown to agree very well with the Cassini Wideband Receiver (WBR) observations. The interaction time of two DASWs for non-Maxwellian ion distributions is estimated and shown to be fastest for the (r,q) distributed ions. The interesting feature of the interaction between compressive solitons with their rarefactive counterparts is also discussed in detail.

Dynamical behavior and multistability of ion-acoustic waves in a magnetized Auroral zone plasma

Dynamical behavior of ion-acoustic waves (IAWs) is explored in magnetized Auroral zone plasma comprising ions and two-temperature electrons. Employing reductive perturbation technique (RPT), Kadomtsev–Petviashvili (KP) and modified KP (mKP) equations are derived. By considering a typical set of parameters for the Auroral zone, all probable phase plots for unperturbed and perturbed dynamical systems are analyzed. Numerical and analytical wave solutions corresponding to distinct phase trajectories of the unperturbed system are presented. Effectiveness of spectral index ( $$\kappa $$ ) and velocity of IAWs on the properties of solitary-wave solutions of KP and mKP equations are also examined. For Maxwellian limit, the system has no superperiodic wave structures in the subsonic region. Numerical computation reveals that the influence of an external perturbation results in existence of only quasiperiodic orbits for the system corresponding to the KP equation whereas various coexisting orbits exist for the system analogous to the mKP equation.

Positron nonextensivity contributions on the rational solitonic, periodic, dissipative structures for MKP equation described critical plasmas

New closed forms have been exposed that rational, trigonometric, periodical, explosive, hyperbolic and shock solutions can be usable in the ionosphere plasma of earth. To explore the nonextensive impacts on the features of nonlinear waves in this plasma model, using Riccati-Bernoulli sub-ODE process, the MKP equation has been solved. This method is very important in the study of dynamics and motion in fluids. Some of the obtained new potential solutions are prefect achievements in plasma observations and applications in ionosphere.

Positron superthermality effects on the solitonic, dissipative, periodic waveforms for M-Kadomstev-Petviashvili-plasma-equation

The index of positron superthermality supports on nonlinearly new structures such as solitons, rational, explosive, shocklikes, and periodical propagating waves that exist for critical behavior characterized by the MKP equation in D-F earth ion plasma pairs with superthermal electrons-positrons have been deliberated. Also, the new solutions have been revealed that periodic, trigonometric, explosive, rational and hyperbolic solutions may applicable in ionosphere plasmas. The proposed method is robust in the dynamical flow of fluids. To investigate the superthermal index effects on the nonlinear features in this plasmas model, the MKP equation been solved by Riccati-Bernoulli (RB) sub-ODE method. The enforcement of our model solutions might be especially interesting in the new earth's ionosphere zone environments observations. The proposed method can easily used to solve many other model arising in applied science.

Criteria of the electron pumping in electron-hole quantum plasma

The degradation of quantum semiconductor plasma during electron pumping is an important problem since it has several applications in the technological industry. Moreover, the identification of such a problem cannot be accomplished by experiment alone and a further theoretical investigation is required. Owing to the energy gap of semiconductors, electrons should acquire energy in order to move from the valence band to the conduction band. Therefore, to employ the quantum fluid model that includes the exchange-correlation potentials, Bohm potential, and the degenerate pressure, the electron-hole lifetime should be greater than the plasma oscillation time. If the latter condition is not satisfied, the quantum fluid model failed to describe the system and it is convenient to use the density functional theory. As a result of that, we apply the quantum fluid model to GaN where the plasma oscillation has a time of 0.002 ps while the electron-hole lifetime is 1 ps. The modified Kadomtsev-Petviashvili (mKP) equation had been obtained using a new set of stretched coordinates as the Kadomtsev-Petviashvili (KP) equation had been derived in (Afify et al 2019 Chaos Soliton Fract 124, 18–25). Our findings determine whether the quantum fluid model is valid or not in the case of semiconductor plasma. Studying exchange effects in a dynamic context has certain limitations. Thus, we stressed that the more accurate approaches introduced in the literature are hard to apply for nonlinear issues. Further, the applicability of our results is discussed, but not this much. Moreover, the numerical results showed that the electron beam parameters and quantum effects have significant effects on the growth of the solitary waves.

New Soliton Applications in Earth's Magnetotail Plasma at Critical Densities

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

Super electron acoustic propagations in critical plasma density

ABSTRACT The propagation of some new electrostatic cnoidal, super-solitons, super-periodic and shocklike waves in Earth magneto-tail electron acoustic (EA) plasma at critical density has been investigated via MKP equation. These MKP solutions are obtained by the method of expansion Jacobian elliptic function (EMJEF). Numerous of the given solutions are significant on the observations of broadband electrostatic nonlinear noise forms in layers of magneto-tail.

Conservation laws and line soliton solutions of a family of modified KP equations

A family of modified Kadomtsev-Petviashvili equations (mKP) in 2+1 dimensions is studied. This family includes the integrable mKP equation when the coefficients of the nonlinear terms and the transverse dispersion term satisfy an algebraic condition. The explicit line soliton solution and all conservation laws of low order are derived for all equations in the family and compared to their counterparts in the integrable case.

Rational and semi-rational solutions of the modified Kadomtsev–Petviashvili equation and the $$\varvec{(2 + 1)}$$-dimensional Konopelchenko–Dubrovsky equation

General rational and semi-rational solutions of the modified Kadomtsev–Petviashvili (mKP) equation and the Konopelchenko–Dubrovsky equation are obtained based on the bilinear method and the KP hierarchy reduction technique. These solutions are expressed in terms of $$N \times N$$ determinants. The dynamics of the solutions, which exhibit various patterns, are thoroughly analyzed. It is shown that the rational solutions may describe the elastic interaction of a single-peak wave with either a double-peak (M-shape) wave or another single-peak wave for $$N=1$$ . Depending on the choice of parameters, the semi-rational solutions are found to depict the inelastic interaction between two (Y-shape) or three waves for $$N=1$$ . The second-order ( $$N=2$$ ) rational solutions exhibit the elastic interaction of three single-peak waves with either one double-peak wave or another single-peak wave. Inelastic interaction is displayed by proper choices of the parameters for semi-rational solutions. When $$N > 2$$ , similar local dynamical behaviors of the rational and semi-rational solutions have been observed.