Kadomtsev-Petviashvili Burgers Equation Research Articles

Large time behavior and optimal decay estimate for solutions to the generalized Kadomtsev–Petviashvili–Burgers equation in 2D

We consider the Cauchy problem for the generalized Kadomtsev–Petviashvili–Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropic dissipative term. Under some suitable regularity assumptions on the initial data u0, especially the condition ∂x−1u0∈L1(R2), it is known that the solution to this problem decays at the rate of t−74 in the L∞-sense. In this paper, we investigate the more detailed large time behavior of the solution and construct the approximate formula for the solution at t→∞. Moreover, we obtain a lower bound of the L∞-norm of the solution and prove that the decay rate t−74 of the solution given in the previous work to be optimal.

Propagation of dust-ion-acoustic solitary waves for damped modified Kadomtsev–Petviashvili–Burgers equation in dusty plasma with a q-nonextensive nonthermal electron velocity distribution

Propagation of dust-ion-acoustic solitary waves for damped modified Kadomtsev–Petviashvili–Burgers equation in dusty plasma with a q-nonextensive nonthermal electron velocity distribution

Bifurcations of solitary wave solutions for (two and three)-dimensional nonlinear partial differential equation in quantum and magnetized plasma by using two different methods

Abstract In this research, we study new two techniques that called the extended simple equation method and the novel G ′ G -expansion method. The extended simple equation method depend on the auxiliary equation d ϕ d ξ = α + λ ϕ + μ ϕ 2 which has three ways for solving depends on the specific condition on the parameters as follow: When λ = 0 this auxiliary equation reduces to Riccati equation, when α = 0 this auxiliary equation reduces to Bernoulli equation and when α ≠ 0 , λ ≠ 0 , μ ≠ 0 we the general solutions of this auxiliary equation while the novel G ′ G -expansion method depends also on similar auxiliary equation G ′ G ′ = μ + λ G ′ G + ( v - 1 ) G ′ G 2 which depend also on the value of ( λ 2 - 4 ( v - 1 ) μ ) and the specific condition on the parameters as follow: When λ = 0 this auxiliary equation reduces to Riccati equation, when μ = 0 this auxiliary equation reduces to Bernoulli equation and when ( λ 2 ≠ 4 ( v - 1 ) μ ) we the general solutions of this auxiliary equation. This show how both of these auxiliary equation are special cases of Riccati equation. We apply these methods on two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma and three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. We obtain the exact traveling wave solutions of these important models and under special condition on the parameters, we get solitary traveling wave solutions. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions.

Ion acoustic solitary wave solutions of two‐dimensional nonlinear Kadomtsev–Petviashvili–Burgers equation in quantum plasma

Propagation of two‐dimensional nonlinear ion‐acoustic solitary waves and shocks in a dissipative quantum plasma is analyzed. By applying the reductive perturbation theory, the two‐dimensional ion acoustic solitary waves in a dissipative quantum plasma lead to a nonlinear Kadomtsev–Petviashvili–Burgers (KPB) equation. By implementing extended direct algebraic mapping, extended sech‐tanh, and extended direct algebraic sech methods, the ion solitary traveling wave solutions of the two‐dimensional nonlinear KPB equation are investigated. An analytical as well as numerical solution of the two‐dimensional nonlinear KPB equation is obtained and analyzed with the effects of external electric field and ion pressure. Copyright © 2016 John Wiley & Sons, Ltd.

Self-adjointness and conservation laws for Kadomtsev–Petviashvili–Burgers equation

In this work we study the Kadomtsev–Petviashvili–Burgers equation, which is a natural model for the propagation of the two-dimensional damped waves. We show that the equation is nonlinear self-adjoint and it will become strict self-adjoint or weak self-adjoint in some equivalent form. By using Ibragimov’s theorem on conservation laws we find some conservation laws for this equation.

Multidimensional ion-acoustic solitary waves and shocks in quantum plasmas

The nonlinear theory of two-dimensional ion-acoustic (IA) solitary waves and shocks (SWS) is revisited in a dissipative quantum plasma. The effects of dispersion, caused by the charge separation of electrons and ions and the quantum force associated with the Bohm potential for degenerate electrons, as well as, the dissipation due to the ion kinematic viscosity are considered. Using the reductive perturbation technique, a Kadomtsev–Petviashvili–Burgers (KPB)-type equation, which governs the evolution of small-amplitude SWS in quantum plasmas, is derived, and its different solutions are obtained and analyzed. It is shown that the KPB equation can admit either compressive or rarefactive SWS according to when H≶2/3, or the particle number density satisfies n0≷1.3×1031cm−3, where H is the ratio of the electron plasmon energy to the Fermi energy densities. Furthermore, the properties of large-amplitude stationary shocks are studied numerically in the case when the wave dispersion due to charge separation is negligible. It is also shown that a transition from monotonic to oscillatory shocks occurs by the effects of the quantum parameter H.

Ion acoustic shock waves in plasmas with warm ions and kappa distributed electrons and positrons

The monotonic and oscillatory ion acoustic shock waves are investigated in electron-positron-ion plasmas (e-p-i) with warm ions (adiabatically heated) and nonthermal kappa distributed electrons and positrons. The dissipation effects are included in the model due to kinematic viscosity of the ions. Using reductive perturbation technique, the Kadomtsev-Petviashvili-Burgers (KPB) equation is derived containing dispersion, dissipation, and diffraction effects (due to perturbation in the transverse direction) in e-p-i plasmas. The analytical solution of KPB equation is obtained by employing tangent hyperbolic (Tanh) method. The analytical condition for the propagation of oscillatory and monotonic shock structures are also discussed in detail. The numerical results of two dimensional monotonic shock structures are obtained for graphical representation. The dependence of shock structures on positron equilibrium density, ion temperature, nonthermal spectral index kappa, and the kinematic viscosity of ions are also discussed.

Nonlinear structure of coupled drift ion acoustic waves in dissipative pair-ion-electron plasmas

The Kadomtsev-Petviashvili-Burgers (KPB) equation is derived for coupled drift acoustic shock waves in a partially ionized non-uniform pair-ion-electron (PIE) plasma in the presence of both density and temperature gradients respectively. Both linear and nonlinear studies are presented. The nonlinear KPB equation is derived in the small amplitude approximation method and its solution is found using the tanh method. The numerical calculations also presented for PIE plasmas of fullerene plasma for illustration keeping in view the recent experiments. The effect of density and temperature inhomogeneities on the nature of the shock is also highlighted. The role of the velocity of the nonlinear structure with regard to the density and temperature gradients driven drift velocities is also pointed out and the effect of ion-neutral collision frequency is also investigated. This work may be useful for future laboratory experimental investigations on pair-ion-electron plasmas.

Two dimensional electromagnetic shock structures in dense electron-positron-ion magnetoplasmas

Linear and nonlinear analysis of low frequency magnetoacoustic waves propagating at an angle θ with the ambient magnetic field are investigated in dense electron-positron-ion (e-p-i) plasmas using the quantum magnetohydrodynamic (QMHD) model. In this regard, a quantum Kadomtsev-Petviashvili-Burgers (KPB) equation is derived in the small amplitude limit. The stability of KPB equation is also presented. The variation of the nonlinear fast and slow magnetoacoustic shock waves with the positron concentration, kinematic viscosity, obliqueness parameter θ, and the magnetic field, are also investigated. It is observed that the aforementioned plasma parameters significantly modify the propagation characteristics of two dimensional nonlinear magnetoacoustic shock waves in dissipative quantum magnetoplasmas. The relevance of the present investigation with regard to dense astrophysical environments is also pointed out.

Two dimensional electrostatic shock waves in relativistic electron positron ion plasmas

Ion-acoustic shock waves (IASWs) are studied in an unmagnetized plasma consisting of electrons, positrons and hot ions. In this regard, Kadomtsev–Petviashvili–Burgers (KPB) equation is derived using the small amplitude perturbation expansion method. The dependence of the IASWs on various plasma parameters is numerically investigated. It is observed that ratio of ion to electron temperature, kinematic viscosity, positron concentration, and the relativistic ion streaming velocity affect the structure of the IASW. Limiting case of the KPB equation is also discussed. Stability of KPB equation is also presented. The present investigation may have relevance in the study of electrostatic shock waves in relativistic electron-positron-ion plasmas.

Obliquely propagating low frequency electromagnetic shock waves in two dimensional quantum magnetoplasmas

Linear and nonlinear propagation characteristics of low frequency magnetoacoustic waves in quantum magnetoplasmas are studied employing the quantum magnetohydrodynamic model. In this regard, a quantum Kadomtsev–Petviashvili–Burgers (KPB) equation is derived using the small amplitude expansion method. The dissipation is introduced by taking into account the kinematic viscosity among the plasma constituents. Furthermore, the solution of KPB equation is presented using the tangent hyperbolic (tanh) method. The variation in the fast and slow magnetoacoustic shock profiles with the quantum Bohm potential via increasing number density, obliqueness angle θ, magnetic field, and the resistivity are also investigated. It is observed that the aforementioned plasma parameters significantly modify the propagation characteristics of nonlinear magnetoacoustic shock waves in quantum magnetoplasmas. The relevance of the present investigation with regard to dense astrophysical environments is also pointed out.

Drift ion acoustic shock waves in an inhomogeneous two-dimensional quantum magnetoplasma

Linear and nonlinear propagation characteristics of drift ion acoustic waves are investigated in an inhomogeneous quantum plasma with neutrals in the background employing the quantum hydrodynamics (QHD) model. In this regard, a quantum Kadomtsev–Petviashvili–Burgers (KPB) equation is derived for the first time. It is shown that the ion acoustic wave couples with the drift wave if the parallel motion of ions is taken into account. Discrepancies in the earlier works on drift solitons and shocks in inhomogeneous plasmas are also pointed out and a correct theoretical framework is presented to study the one-dimensional as well as the two-dimensional propagation of shock waves in an inhomogeneous quantum plasma. Furthermore, the solution of KPB equation is presented using the tangent hyperbolic (tanh) method. The variation of the shock profile with the quantum Bohm potential, collision frequency, and ratio of drift to shock velocity in the comoving frame, v∗/u, are also investigated. It is found that increasing the number density and collision frequency enhances the strength of the shock. It is also shown that the fast drift shock (i.e., v∗/u>0) increases, whereas the slow drift shock (i.e., v∗/u<0) decreases the strength of the shock. The relevance of the present investigation with regard to dense astrophysical environments is also pointed out.

Nonplanar dust-ion acoustic shock waves with transverse perturbation

The nonlinear dust-ion acoustic shock waves in dusty plasmas with the combined effects of bounded cylindrical/spherical geometry, the transverse perturbation, the dust charge fluctuation, and the nonthermal electrons are studied. Using the perturbation method, a cylindrical/spherical Kadomtsev–Petviashvili Burgers equation that describes the dust-ion acoustic shock waves is deduced. A particular solution of the cylindrical/spherical Kadomtsev–Petviashvili Burgers equation is also obtained. It is shown that the dust-ion acoustic shock wave propagating in cylindrical/spherical geometry with transverse perturbation will be slightly deformed as time goes on.

Kadomtsev-Petviashvili (KP) Burgers equation in a dusty plasmas with non-adiabatic dust charge fluctuation

The nonlinear dust acoustic waves in a dusty plasmas with the combined effects of non-adiabatic dust charge fluctuation and higher-order transverse perturbation are studied. Using the perturbation method, a Kadomtsev-Petviashvili (KP) Burgers equation that governing the dust acoustic waves is deduced for the first time. A particular solution of this KP Burgers equation is also obtained. It is show that the dust acoustic shock waves can exist in the KP Burgers equation.