Kadomtsev Petviashvili Burgers 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.

Nonlinear density waves on graphene electron fluids

The hydrodynamic behavior of charged carriers leads to nonlinear phenomena such as solitary waves and shocks, among others. As an application, such waves might be exploited on plasmonic devices either for modulation or signal propagation along graphene waveguides. We study the nature of nonlinear perturbations following an approach similar to Sagdeev potential analysis and also by performing the reductive perturbation method on the hydrodynamic description of graphene electrons, taking into consideration the effect of Bohm quantum potential and odd viscosity. Thus, deriving a dissipative Kadomtsev-Petviashvili-Burgers (KPB) equation for the bidimensional flow as well as its unidimensional limit in the form of Korteweg--de Vries--Burgers (KdVB). The stability analysis of these equations unveils the existence of unstable modes that can be excited and launched through graphene plasmonic devices.

Nonplanar ion-acoustic solitons and shocks with superthermal trapped electrons and transverse perturbations

The formation and propagation of nonlinear ion-acoustic (IA) waves are studied in an unmagnetized cold viscous plasma, comprising the inertial ions and superthermal trapped electrons in cylindrical geometry with transverse perturbations. The well-known reductive perturbation method is employed to derive cylindrical trapped Kadomtsev Petviashvili Burgers (CTKPB) and cylindrical trapped Kadomtsev Petviashvili (CTKP) equations with and without dissipation effects, respectively. The exact solutions of CTKPB and CTKP equations are obtained for the first time by utilizing the dependent variable transformation. The soliton and shock structures are found to be significantly affected by the plasma parameters including the trapping efficiency parameter β, the superthermality parameter κ, viscosity µ 0 and the geometrical effects. Importantly, the geometrical effects and transverse perturbations alter the shape of solitons and shocks, resulting into the parabolic nonlinear structures. These structures become more parabolic at later times due to an interplay of transverse and time coordinates (η, τ). The results of the present study might be helpful to understand the characteristics of IA structures in space plasmas, such as auroral regions, where energetic trapped electrons have been observed.

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.

Two-dimensional dissipative structures in spatially nonuniform dusty plasmas in planetary magnetospheres with kappa distributed ions

Linear and nonlinear propagation of coupled dust drift acoustic waves are investigated in the presence of a kappa distributed ion population and Boltzmannian electrons. It is observed that the frequency of the dust drift acoustic wave decreases by increasing the kappa value. Using the drift approximation, a nonlinear equation is derived for the dust drift acoustic shock wave that reduces to a Kadomtsev–Petviashvili–Burgers (KPB)-like equation in the comoving frame of reference. It is found that the nonlinear structure of dust drift acoustic shock waves modifies in the presence of increasing values of kappa, dust number density, ambient magnetic field, and the dust neutral collision frequency. Interestingly, it is found that the non-Maxwellian ion population modifies the scale lengths over which the nonlinear structures are formed. The results obtained are applied in Saturnʼs magnetosphere, where the non-Maxwellian distribution of particles has been reported by several satellite missions.

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.

Two dimensional ion acoustic shocks in electron-positron-ion plasmas with warm ions, and q-nonextensive distributed electrons and positrons

Two dimensional ion acoustic shocks in electron-positron-ion (e-p-i) plasma with warm ions, and nonthermal electrons and positrons following the q-nonextensive velocity distribution are studied in the presence of weak transverse perturbations. The kinematic viscosity of warm ions is included for the dissipation in the plasma system. Kadomtsev–Petviashvili–Burgers (KPB) equation is derived by using reductive perturbation method in small amplitude limit and its analytical solution is also presented. The effects of variations of positrons concentration, q-indices of electrons and positrons, ion temperature and kinematic viscosity of ions, on the propagation characteristic of two dimensional shock profile are also discussed.

KP Burgers shocks in a warm electronegative plasma with q-nonextensive distributed electrons

Nonlinear properties of ion-acoustic (IA) shock are studied by incorporating the effects of electron nonextensivity in a warm electronegative plasma, whose constituents are the inertial positive/negative ions and q-distributed electrons. For this purpose, the evolution equations are solved to obtain Kadomtsev-Petviashvili (KP) Burgers equation by using the reductive perturbation technique and its solution by the tanh method. Furthermore, the conditions for the existence of oscillatory and monotonic shocks are discussed. Numerically, it is found that IA shock propagation characteristics are significantly modified by the variation of plasma parameters, such as, the effects of electron nonextensivity, the positive and negative ion-to-electron temperature ratios (θi,θn), respectively. The former also affect the dispersion, dissipation and nonlinearity coefficients of the KPB equation involving the IA shocks. The present analyses could be useful for understanding the nonlinear shock wave excitations in space and laboratory plasmas, where two distinct groups of ions are present.

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.

On the well-posedness for Kadomtsev–Petviashvili–Burgers I equation

We prove local and global well-posedness in Hs,0(R2), s>−12, for the Cauchy problem associated with the Kadomtsev–Petviashvili–Burgers I equation (KPBI) by working in Bourgainʼs type spaces. This result is almost sharp if one requires the flow-map to be smooth.

Two dimensional nonplanar evolution of electrostatic shock waves in pair-ion plasmas

Electrostatic waves in a two dimensional nonplanar geometry are studied in an unmagnetized, dissipative pair-ion plasma in the presence of weak transverse perturbations. The dissipation in the system is taken into account by incorporating the kinematic viscosity of both positive and negative ions in plasmas. The nonplanar Kadomtsev-Petviashvili-Burgers (KPB) as well as the Burgers Kadomtsev-Petviashvili (Burgers KP) equations are derived using the small amplitude expansion method and the range of applicability of both the equations are discussed. The system under consideration is observed to admit compressive rarefactive shocks. The present study may have relevance to understand the formation of two dimensional nonplanar electrostatic shocks in laboratory plasmas.

Interesting features of nonlinear shock equations in dissipative pair-ion-electron plasmas

Two dimensional nonlinear electrostatic waves are studied in unmagnetized, dissipative pair-ion-electron plasmas in the presence of weak transverse perturbation. The dissipation in the system is taken into account by incorporating the kinematic viscosity of both positive and negative ions. In the linear case, a biquadratic dispersion relation is obtained, which yields the fast and slow modes in a pair-ion-electron plasma. It is shown that the limiting cases of electron-ion and pair-ion can be retrieved from the general biquadratic dispersion relation, and the differences in the characters of the waves propagating in both the cases are also highlighted. Using the small amplitude approximation method, the nonlinear Kadomtsev Petviashvili Burgers as well as Burgers-Kadomtsev Petviashvili equations are derived and their applicability for pair-ion-electron plasma is explained in detail. The present study may have relevance to understand the formation of two dimensional electrostatic shocks in laboratory produced pair-ion-electron plasmas.

The 3+1 dimensional Kadomtsev–Petviashvili Burgers' equation in non-uniform dusty plasmas

Using the perturbation method, the 3+1 dimensional modified variable coefficient Kadomtsev–Petviashvili (MKP) Burgers' equation governing nonlinear dust acoustic shock waves is derived with the combined effects of non-adiabatic dust charge fluctuation, higher-order transverse perturbation and the non-thermally distributed ions in inhomogeneous dusty plasmas due to spatial gradients of dust charge and plasma densities for the first time. The MKP Burgers' equation is reduced to the standard variable coefficient Kadomtsev–Petviashvili (KP) Burgers' equation. A particular solution of this KP Burgers' equation is also obtained. It is shown that the dust acoustic shock waves can exist in the KP Burgers' equation.

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.

Nonlinear electrostatic shock waves in inhomogeneous dense dusty magnetoplasmas

The Linear and nonlinear propagation characteristics of drift dust acoustic waves were studied in an inhomogeneous electron–ion-dust (e–i–d) quantum magnetoplasma with neutrals in the background using the well-known quantum hydrodynamic (QHD) model. The Korteweg–de Vries–Burgers (KdVB) and Kadomtsev–Petviashvili–Burgers (KPB) equations are obtained. Furthermore, the solutions of the KdVB and KPB equations are presented using the tangent hyperbolic (tanh) method. The variation of the shock profile with the quantum Bohm potential, collision frequency, ratio of drift to shock velocity in the co-moving frame and effect of magnetic field is also investigated. The relevance of the present investigation to dense astrophysical environments is also highlighted.

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.