An analytical study on the existence of solitary wave and double layer solution of the well-known energy integral at M= Mc

A general theory for the existence of solitary wave and double layer at M= Mc has been discussed, where Mc is the lower bound of the Mach number M, i.e., solitary wave and/or double layer solutions of the well-known energy integral start to exist for M> Mc. Ten important theorems have been proved to confirm the existence of solitary wave and double layer at M = Mc. If V({\phi})({\equiv}V(M,{\phi})) denotes the Sagdeev potential with {\phi} is the perturbed field or perturbed dependent variable associated with the specific problem, V(M,{\phi}) is well defined as a real number for all M {\in} \mathcal{M} and for all {\phi} {\in} {\Phi}, and V(M,0)=V'(M,0)=V"(Mc,0)=0, V"'(Mc,0)0), \deltaV/\deltaM 0 and for all {\phi}({\in} {\Phi}) > 0 ({\phi}({\in}{\Phi}) < 0), where " '{\equiv} \delta/\delta{\phi} ", the main analytical results for the existence of solitary wave and double layer solution of the energy integral at M= Mc are as follows. Result-1: If there exists at least one value M0 of M such that the system supports positive (negative) potential solitary waves for all Mc<M<M0, then there exist either a positive (negative) potential solitary wave or a positive (negative) potential double layer at M= Mc. Result-2: If the system supports only negative (positive) potential solitary waves for M> Mc, then there does not exist positive (negative) potential solitary wave at M= Mc. Result-3: It is not possible to have coexistence of both positive and negative potential solitary structures (including double layers) at M= Mc. Apart from the conditions of Result-1, the double layer solution at M= Mc is possible only when there exists a double layer solution in any right neighborhood of Mc. Finally these analytical results have been applied to a specific problem on dust acoustic waves in nonthermal plasma in search of new results.Comment: 40 pages, 7 figures, communicate

Effect of Landau damping on ion acoustic solitary waves in a multi-species collisionless unmagnetized plasma consisting of nonthermal and isothermal electrons

A Korteweg-de Vries (KdV) equation including the effect of Landau damping is derived to study the propagation of weakly nonlinear and weakly dispersive ion acoustic waves in a collisionless unmagnetized plasma consisting of warm adiabatic ions and two different species of electrons at different temperatures. The hotter energetic electron species follows the nonthermal velocity distribution of Cairns et al. [Geophys. Res. Lett. 22, 2709 (1995)] whereas the cooler electron species obeys the Boltzmann distribution. It is found that the coefficient of the nonlinear term of this KdV like evolution equation vanishes along different family of curves in different parameter planes. In this context, a modified KdV (MKdV) equation including the effect of Landau damping effectively describes the nonlinear behaviour of ion acoustic waves. It has also been observed that the coefficients of the nonlinear terms of the KdV and MKdV like evolution equations including the effect of Landau damping, are simultaneously equal to zero along a family of curves in the parameter plane. In this situation, we have derived a further modified KdV (FMKdV) equation including the effect of Landau damping to describe the nonlinear behaviour of ion acoustic waves. In fact, different modified KdV like evolution equations including the effect of Landau damping have been derived to describe the nonlinear behaviour of ion acoustic waves in different region of parameter space. The method of Ott & Sudan [Phys. Fluids 12, 2388 (1969)] has been applied to obtain the solitary wave solution of the evolution equation having the nonlinear term $(\phi^{(1)})^{r}\frac{\partial \phi^{(1)}}{\partial \xi}$, where $\phi^{(1)}$ is the first order perturbed electrostatic potential and $r =1,2,3$. We have found that the amplitude of the solitary wave solution decreases with time for all $r =1,2,3$.Comment: 36 pages, 9 figure

Dust acoustic solitary structures in presence of nonthermal ions, isothermally distributed electrons and positrons

Arbitrary amplitude dust acoustic solitary structures have been investigated in a four component multi-species plasma consisting of negatively charged dust grains, nonthermal ions, isothermally distributed electrons and positrons including the effect of dust temperature. We have used the Sagdeev pseudo-potential method to discuss the arbitrary amplitude steady state dust acoustic solitary structures in the present plasma system. We have designed a computational scheme to draw the existence domains of different dust acoustic solitary structures. We have observed only negative potential solitary waves for isothermal ions. But for strong nonthermality of ions the system supports positive potential solitary waves, positive potential double layers and coexistence of solitary waves of both polarities. The positive potential solitary waves are restricted by the positive potential double layers but negative potential double layer has not been found for any parameter regime. The system does not support dust acoustic supersoliton of any polarity. The concentration of positrons plays an important role in the formation of positive potential double layers. Finally, the phase portraits of the dynamical system have been presented to confirm the existence of different dust acoustic solitary structures.Comment: 24 pages, 12 figures. arXiv admin note: text overlap with arXiv:1801.07581, arXiv:1610.0955

Ion acoustic solitary structures in a collisionless unmagnetized plasma consisting of nonthermal electrons and isothermal positrons

Employing the Sagdeev pseudo-potential technique the ion acoustic solitary structures have been investigated in an unmagnetized collisionless plasma consisting of adiabatic warm ions, nonthermal electrons and isothermal positrons. The qualitatively different compositional parameter spaces clearly indicate the existence domains of solitons and double layers with respect to any parameter of the present plasma system. The present system supports the negative potential double layer which always restricts the occurrence of negative potential solitons. The system also supports positive potential double layers when the ratio of the average thermal velocity of positrons to that of electrons is less than a critical value. However, there exists a parameter regime for which the positive potential double layer is unable to restrict the occurrence of positive potential solitary waves and in this region of the parameter space, there exist positive potential solitary waves after the formation of a positive potential double layer. Consequently, positive potential supersolitons have been observed. The nonthermality of electrons plays an important role in the formation of positive potential double layers as well as positive potential supersolitons. The formation of positive potential supersoliton is analysed with the help of phase portraits of the dynamical system corresponding to the ion acoustic solitary structures of the present plasma system.Comment: 12 pages, 12 figures, Formation of positive potential supersoliton is analysed with the help of phase portraits of the dynamical system corresponding to the ion acoustic solitary structures of the present plasma syste

Dust ion acoustic solitary structures in presence of isothermal positrons

The Sagdeev potential technique has been employed to study the dust ion acoustic solitary waves and double layers in an unmagnetized collisionless dusty plasma consisting of negatively charged static dust grains, adiabatic warm ions, and isothermally distributed electrons and positrons. A computational scheme has been developed to draw the qualitatively different compositional parameter spaces or solution spaces showing the nature of existence of different solitary structures with respect to any parameter of the present plasma system. The qualitatively distinct solution spaces give the overall scenario regarding the existence of different solitary structures. The present system supports both positive and negative potential double layers. The negative potential double layer always restricts the occurrence of negative potential solitary waves, i.e., any sequence of negative potential solitary waves having monotonically increasing amplitude converges to a negative potential double layer. However, there exists a parameter regime for which the positive potential double layer is unable to restrict the occurrence of positive potential solitary waves. As a result, in this region of the parameter space, there exist solitary waves after the formation of positive potential double layer, i.e., positive potential supersolitons have been observed. But the amplitudes of these supersolitons are bounded. A general theory for the existence of bounded supersolitons has been discussed analytically by imposing the restrictions on the Mach number. For any small value of positron concentration, there is no effect of very hot positrons on the dust ion acoustic solitary structures. The qualitatively different solution spaces are capable of producing new results for the formation of solitary structures

Existence of dust ion acoustic solitary wave and double layer solution at M = Mc

The Sagdeev potential technique has been used to investigate the existence and the polarity of dust ion acoustic solitary structures in an unmagnetized collisionless nonthermal dusty plasma consisting of negatively charged static dust grains, adiabatic warm ions and nonthermal electrons when the velocity of the wave frame is equal to the linearized velocity of the dust ion acoustic wave for long wave length plane wave perturbation, i.e., when the velocity of the solitary structure is equal to the acoustic speed. A compositional parameter space has been drawn which shows the nature of existence and the polarity of dust ion acoustic solitary structures at the acoustic speed. This compositional parameter space clearly indicates the regions for the existence of positive and negative potential dust ion acoustic solitary structures. Again, this compositional parameter space shows that the present system supports the negative potential double layer at the acoustic speed along a particular curve in the parametric plane. However, the negative potential double layer is unable to restrict the occurrence of all negative potential solitary waves. As a result, in a particular region of the parameter space, there exist negative potential solitary waves after the formation of negative potential double layer. But the amplitudes of these supersolitons are bounded. A finite jump between amplitudes of negative potential solitons separated by the negative potential double layer has been observed, and consequently, the present system supports the supersolitons at the acoustic speed in a neighbourhood of the curve along which negative potential double layer exist. The effects of the parameters on the amplitude of the solitary structures at the acoustic speed have been discussed.Comment: 46 pages 14 figures, communicated. arXiv admin note: text overlap with arXiv:1108.1777, arXiv:1505.0500

Higher order stability of dust ion acoustic solitary wave solution described by the KP equation in a collisionless unmagnetized nonthermal plasma in presence of isothermal positrons

Sardar et al. [Phys. Plasmas 23, 073703 (2016)] have studied the stability of small amplitude dust ion acoustic solitary waves in a collisionless unmagnetized electron - positron - ion - dust plasma. They have derived a Kadomtsev Petviashvili (KP) equation to investigate the lowest - order stability of the solitary wave solution of the Korteweg-de Vries (KdV) equation for long-wavelength plane-wave transverse perturbation when the weak dependence of the spatial coordinates perpendicular to the direction of propagation of the wave is taken into account. In the present paper, we have extended the lowest - order stability analysis of KdV solitons given in the paper of Sardar et al. [Phys. Plasmas 23, 073703 (2016)] to higher order with the help of multiple-scale perturbation expansion method of Allen and Rowlands [J. Plasma Phys. 50, 413 (1993); 53, 63 (1995)]. It is found that solitary wave solution of the KdV equation is stable at the order k^2, where k is the wave number for long-wavelength plane-wave perturbation.Comment: 15 page

Effect of Landau damping on alternative ion-acoustic solitary waves in a magnetized plasma consisting of warm adiabatic ions and non-thermal electrons

Bandyopadhyay and Das [Phys. Plasmas, 9, 465-473, 2002] have derived a nonlinear macroscopic evolution equation for ion acoustic wave in a magnetized plasma consisting of warm adiabatic ions and non-thermal electrons including the effect of Landau damping. In that paper they have also derived the corresponding nonlinear evolution equation when coefficient of the nonlinear term of the above mentioned macroscopic evolution equation vanishes, the nonlinear behaviour of the ion acoustic wave is described by a modified macroscopic evolution equation. But they have not considered the case when the coefficient is very near to zero. This is the case we consider in this paper and we derive the corresponding evolution equation including the effect of Landau damping. Finally, a solitary wave solution of this macroscopic evolution is obtained, whose amplitude is found to decay slowly with time.Comment: 24 pages, 2 figure

Existence and stability of dust ion acoustic double layers described by the combined MKP-KP equation

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.Comment: 21 pages, 3 figure

Modulation instability of obliquely propagating ion acoustic waves in a collisionless magnetized plasma consisting of nonthermal and isothermal electrons

We have studied the modulation instability of obliquely propagating ion acoustic waves in a collisionless magnetized warm plasma consisting of warm adiabatic ions and two different species of electrons at different temperatures. We have derived a nonlinear Schr{\"o}dinger equation using the standard reductive perturbation method to describe the nonlinear amplitude modulation of ion acoustic wave satisfying the dispersion relation of ion acoustic wave propagating at an arbitrary angle to the direction of the external uniform static magnetic field. We have investigated the correspondence between two nonlinear Schr{\"o}dinger equations $-$ one describes the amplitude modulation of ion acoustic waves propagating along any arbitrary direction to the direction of the magnetic field and other describes the amplitude modulation of ion acoustic waves propagating along the direction of the magnetic field. We have derived the instability condition and the maximum growth rate of instability of the modulated ion acoustic wave. We have seen that the region of existence of maximum growth rate of instability decreases with increasing values of the magnetic field intensity whereas the region of existence of the maximum growth rate of instability increases with increasing $\cos \theta$, where $\theta$ is the angle of propagation of the ion acoustic wave with the external uniform static magnetic field. Again, the maximum growth rate of instability increases with increasing $\cos \theta$ and also this maximum growth rate of instability increases with increasing $\beta_{e}$ upto a critical value of the wave number, where $\beta_{e}$ is the parameter associated with the nonthermal distribution of hotter electron species.Comment: 22 pages, 10 figure