A three-dimensional KP (Kadomtsev Petviashvili) equation is derived here describing the propagation of weakly nonlinear and weakly dispersive dust ion acoustic wave in a collisionless unmagnetized plasma consisting of warm adiabatic ions, static negatively charged dust grains, nonthermal electrons, and isothermal positrons. When the coefficient of the nonlinear term of the KP-equation vanishes an appropriate modified KP (MKP) equation describing the propagation of dust ion acoustic wave is derived. Again when the coefficient of the nonlinear term of this MKP equation vanishes, a further modified KP equation is derived. Finally, the stability of the solitary wave solutions of the KP and the different modified KP equations are investigated by the small-k perturbation expansion method of Rowlands and Infeld [J. Plasma Phys. 3, 567 (1969); 8, 105 (1972); 10, 293 (1973); 33, 171 (1985); 41, 139 (1989); Sov. Phys. - JETP 38, 494 (1974)] at the lowest order of k, where k is the wave number of a long-wavelength plane-wave perturbation. The solitary wave solutions of the different evolution equations are found to be stable at this order.

The aim of this paper is to extend the recent work of Sardar et al. [Phys. Plasmas 23, 073703 (2016)] on the stability of the small amplitude dust ion acoustic solitary wave in a collisionless unmagnetized nonthermal plasma in the presence of isothermal positrons. Sardar et al. [Phys. Plasmas 23, 073703 (2016)] have derived a KP (Kadomtsev Petviashvili) equation to study the stability of the dust ion acoustic solitary wave when the weak dependence of the spatial coordinates perpendicular to the direction of propagation of the wave is taken into account. They have also derived a modified KP (MKP) equation to investigate the stability of the dust ion acoustic solitary wave when the coefficient of the nonlinear term of the KP equation vanishes. When the coefficient of the nonlinear term of the KP equation is close to zero, a combined MKP-KP equation more efficiently describes the nonlinear behaviour of the dust ion acoustic wave. This equation is derived in the present paper. The alternative solitary wave solution of the combined MKP-KP equation having profile different from sech2 or sech is obtained. This alternative solitary wave solution of the combined MKP-KP equation is stable at the lowest order of the wave number. It is found that this alternative solitary wave solution of the combined MKP-KP equation and its lowest order stability analysis are exactly same as those of the solitary wave solution of the MKP equation when the coefficient of the nonlinear term of the KP equation tends to zero.

The recent work of Sardar et al. [Phys. Plasmas 23, 073703 (2016)] on the existence and stability of the small amplitude dust ion acoustic solitary waves in a collisionless unmagnetized plasma consisting of warm adiabatic ions, static negatively charged dust grains, isothermal positrons, and nonthermal electrons due to Cairns et al. [Geophys. Res. Lett. 22, 2709 (1995)] has been extended by considering nonthermal electrons having a vortex-like velocity distribution due to Schamel [Plasma Phys. 13, 491 (1971); 14, 905 (1972)] instead of taking nonthermal electrons. This distribution takes care of both free and trapped electrons. A Schamel's modified Kadomtsev Petviashvili (SKP) equation describes the nonlinear behaviour of dust ion acoustic waves in this plasma system. The nonlinear behaviour of the dust ion acoustic wave is described by the same Kadomtsev Petviashvili (KP) equation of Sardar et al. [Phys. Plasmas 23, 073703 (2016)] when B = 0, where B is the coefficient of nonlinear term of the SKP equation. A combined SKP-KP equation more efficiently describes the nonlinear behaviour of dust ion acoustic waves when B → 0. The solitary wave solution of the SKP equation and the alternative solitary wave solution of the combined SKP-KP equation having profile different from both sech4 and sech2 are stable at the lowest order of the wave number. It is found that this alternative solitary wave solution of the combined SKP-KP equation and its lowest order stability analysis are exactly the same as those of the solitary wave solution of the KP equation when B → 0.

The purpose of this paper is to expand the recent work of Sardar et al. [Phys. Plasmas 23, 123706 (2016)] on the existence and stability of alternative dust ion acoustic solitary wave solution of the combined modified Kadomtsev Petviashvili -Kadomtsev Petviashvili (MKP-KP) equation in a nonthermal plasma. Sardar et al. [Phys. Plasmas 23, 123706 (2016)] have derived a combined MKP-KP equation to describe the nonlinear behaviour of the dust ion acoustic wave when the coefficient of the nonlinear term of the KP equation tends to zero. Sardar et al. [Phys. Plasmas 23, 123706 (2016)] have used this combined MKP-KP equation to investigate the existence and stability of the alternative solitary wave solution having a profile different from sech 2 or sech when L > 0, where L is a function of the parameters of the present plasma system. In the present paper, we have considered the same combined MKP-KP equation to study the existence and stability of the double layer solution and it is shown that double layer solution of this combined MKP-KP equation exists if L = 0. Finally, the lowest order stability of the double layer solution of this combined MKP-KP equation has been investigated with the help of multiple scale perturbation expansion method of Allen and Rowlands [ J. Plasma Phys. 50, 413 (1993)]. It is found that the double layer solution of the combined MKP-KP equation is stable at the lowest order of the wave number for long-wavelength plane-wave perturbation.

We have derived a Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation to study the nonlinear behavior of dust–ion acoustic waves in a collisionless magnetized five components dusty plasma consisting of warm adiabatic ions, nonthermal hot electrons, isothermal cold electrons, nonthermal positrons and static negatively charged dust particulates. It is found that the coefficient of the nonlinear term of the KdV-ZK equation vanishes along different family of curves in different compositional parameter planes. In this situation, to describe the nonlinear behavior of dust–ion acoustic waves, we have derived a modified KdV-ZK (MKdV-ZK) equation. When the coefficients of the nonlinear terms of both KdV-ZK and MKdV-ZK equations are simultaneously equal to zero, then we have derived a further modified KdV-ZK (FMKdV-ZK) equation which effectively describes the nonlinear behavior of dust–ion acoustic waves. Analytically and numerically, we have investigated the solitary wave solutions of different evolution equations propagating obliquely to the direction of the external static uniform magnetic field. We have seen that the amplitude of the KdV soliton strictly increases with increasing β e, whereas the amplitude of the MKdV soliton strictly decreases with increasing β e, where β e is the nonthermal parameter associated with the hot electron species. Also, there exists a critical value β r ( c ) ${\beta }_{\text{r}}^{(\text{c})}$ of β e such that the FMKdV soliton exists within the interval β r ( c ) < β e ≤ 4 7 ${\beta }_{\text{r}}^{(\text{c})}< {\beta }_{\text{e}}\le \frac{4}{7}$ , whereas the FMKdV soliton does not exist within the interval 0 < β e < β r ( c ) $0< {\beta }_{\text{e}}< {\beta }_{\text{r}}^{(\text{c})}$ . We have also discussed the effect of different parameters of the system on solitary waves obtained from the different evolution equations.