## Abstract

The nonlinear propagation and interaction of dust acoustic multi-solitons in a four component dusty plasma consisting of negatively and positively charged cold dust fluids, non-thermal electrons, and ions were investigated. By employing reductive perturbation technique (RPT), we obtained Korteweged–de Vries (KdV) equation for our system. With the help of Hirota’s bilinear method, we derived two-soliton and three-soliton solutions of the KdV equation. Phase shifts of two solitons and three solitons after collision are discussed. It was observed that the parameters *α*, *β*, *β*_{1}, *μ*_{e}, *μ*_{i}, and *σ* play a significant role in the formation of two-soliton and three-soliton solutions. The effect of the parameter *β*_{1} on the profiles of two soliton and three soliton is shown in detail.

## 1 Introduction

The nonlinearities cannot be ignored when the amplitudes of the waves are sufficiently large. The nonlinearities come from the harmonic generation involving fluid advection, the nonlinear Lorentz force, trapping of particles in the wave potential, etc. The nonlinearities in plasmas contribute to the localisation of waves, leading to different types of interesting coherent structures (namely solitary structures, shock waves, vortices, etc.) which are important from both theoretical and experimental points of view. Research on the nonlinear propagation of dust acoustic (DA) waves has been developed rapidly in the last few decades, both in laboratory and space plasmas [1–10]. It is well known that Rao et al. [11] first theoretically worked on the DA waves in an unmagnetised dusty plasmas. Solitary waves are omnipresent in dusty plasma, and due to this reason, investigations of nonlinear solitary waves in dusty plasma have received considerable attention [12–14].

But to study DA solitons and shocks [15–17] most of the researchers have considered the negatively charged dust only. The consideration of only negatively charged dust in a plasma is valid only when dust charging process by collection of plasma particles (viz. electron and ions) is much more important than the other charging processes. There are important charging processes by which the dust grains can be positively charged [3, 18, 19]. Dust grains become positively charged by three mechanisms. These are (i) photo emission in the presence of a flux of ultraviolet photons, (ii) thermionic emission induced by radiative heating, and (iii) a secondary emission of electrons from the surface of the dust grains.

Examples of the existence of plasma with both positively and negatively charged dust particle can be found in several astrophysical plasmas such as the Earth’s mesosphere [20], cometary tails [3, 21], magnetosphere of Jupiter [3, 21, 22]. Chow et al. [23] explained the situations under which smaller dust particles become positively charged and larger particles become negatively charged. Sayed and Mamun [24] investigated solitary waves in four component plasmas where they considered both positively and negatively charged dust particles. Chatterjee and Roy [25] studied the nonthermal electron effect on the four component dusty plasma. Mandal et al. [26] also studied the double layers in four component dusty plasma. El-Taibany and Sabry [27] studied the DASWs and double layers in a magnetised dusty plasma with nonthermal ions and dust charged variation. Roy et al. [28] studied the dressed soliton in four component dusty plasma. Also Roy et al. [29] investigated double layers in a four component dusty plasma with kappa distributed electrons.

Zabusky and Kruskal [30] were first to remark that when solitary waves undergo a collision then they preserve their shape and velocities after the collision. For the collision of solitary waves, some phenomena have been observed in the laboratory [31–33] and the same can be explained in the solution of two solitary waves of Korteweg–de Vries (KdV) equations. In a one- (or quasi-one-) dimensional system, the solitons may interact between them in two different ways. One is the overtaking collision and the other is the head-on collision [34]. Because of the multi-solitons solutions of the KdV equation, where waves travel in the same direction, the overtaking collision of solitary waves can be studied by the inverse scattering transformation method [35]. The collision of solitary waves phase shifts [36] show the effect of the collisions.

The role of non-thermal electron distribution on characterization of solitary waves has been reported, and attempts have been made to explain the observation of solitary wave structures with density depression [37, 38]. The non-thermal distributions associated with the particle flows resulting from the force fields present in the space and astrophysical plasma are of relevance for super-thermal particles. It is known that electron and ion distributions play important roles in the formation of nonlinear structures. Thus, it is significant to study the coherent nonlinear wave structures of electrons or ions, which do not follow the Boltzmann distribution.

In this study, we have considered a four component dusty plasma comprising ions, electrons, and positively and negatively charged dust grains, where ions follow the Boltzmann distribution and electrons are non-thermal. Standard reductive perturbation technique (RPT) [39] has been used to derive the KdV equation, and the solutions of KdV equation are analysed in terms of different plasma parameters. We are particularly interested in the observation of the phase shifts due to the overtaking collision of solitary waves.

This article is organized as follows. The basic governing equations describing the dusty plasma model are stated and KdV equation is derived by using reductive perturbation technique in Section 2. In Section 3, results are shown and discussed in some detail. Finally, Section 4 covers our conclusions.

## 2 Basic Equations and Derivation of KdV Equation

We consider a four component dusty plasma consisting of Boltzmann distributed ions, non-thermal electrons, and negatively and positively charged dust grains. The basic equations are given by

where *n*_{1} and *n*_{2} are the number densities of negatively and positively charged dust particles, respectively. At equilibrium, we have *n*_{i0} + *Z*_{2}*n*_{20}=*n*_{e0} + *Z*_{1}*n*_{10}. Equations (1) and (2) are respectively the continuity equation and momentum equation for the negatively charged dust particles, and (3) and (4) are counterparts for the positively charged dust particles. Equation (5) is the Poission equation. Here, *u*_{1} and *u*_{2} are negative and positive dust field speed normalized to *ϕ* is the electric potential normalised to *k*_{B}*T*_{i}/*e*. The space variable *x* and time variable *t* are normalised to *α*=*Z*_{2}/*Z*_{1}, *β*=*m*_{1}/*m*_{2}, *μ*_{e}=*n*_{e0} + *Z*_{1}*n*_{10}, *μ*_{i}=*n*_{i0}/*Z*_{1}*n*_{10}, *σ*=*T*_{i}/*T*_{e}, and *Z*_{1} and *Z*_{2} are the number of electrons and protons residing on a negative and positive dust particle, respectively. *β*_{1}=4*γ*_{1}/(1 + 3*γ*_{1}), where *γ*_{1} determines the proportion of fast electrons and *m*_{1} and *m*_{2} are masses of the negative and positive dust particles, respectively. *T*_{i} and *T*_{e} are ion and electron temperatures, respectively, *k*_{B} is the Boltzmann constant, and *e* is the charge of the electrons.

Now, we derive the KdV equation from (1) to (5) employing the RPT. The independent variables are the stretched variables given by Schamel [40, 41] *ξ*=*ϵ*^{1/2} (*x* − *v*_{o}*t*), *τ*=*ϵ*^{3/2}*t*. The dependent variables are expanded as

where *ϵ* is a small non-zero parameter proportional to the amplitude of the perturbation. Now, substituting (6)–(10) into (1)–(5) and considering the lowest order of *ϵ*, we obtain the dispersion relation as

In the next higher order of *ϵ*, we eliminate the second order perturbed quantities from a set of equations to obtain the required KdV equation

where the nonlinear coefficient *A* and the dispersion coefficient *B* are given by the following relations:

Let us replace *ξ* by *ξ*̅*B*^{1/3}, *ϕ*^{(1)} by −6*ϕ*̅^{(1)}*A*^{−1}*B*^{1/3}, and *τ* by *τ*̅, then (12) is transformed to the following standard KdV equation:

## 3 Results and Discussion

Now our aim is to obtain the two-solitons and three-solitons solutions of (15) and to study the interaction between them. To do so, we employ Hirota’s bilinear method [42]. Though Hirota’s method is well known, for the sake of completness we give here a gist of the method for KdV equation. Using the transformation *ϕ*̅^{(1)}=−2(log *f*)*ξ*̅*ξ*̅ in the standard KdV (15), one can obtain the bilinearized form of (15) as

By using the Hirota-D operator [43], we get

Using (17) and (18) in (16), we get the Hirota bilinear form (for details see [42, 44])

To construct two-solitons solution, we use the Hirota’s perturbation technique and we insert *θ*̅_{i}=*k*_{i}*ξ*̅ + *ω*_{i}*τ*̅ + *α*_{i}, *i*=1, 2. The coefficient of different powers of *ϵ* will give *a*_{12}=(*k*_{1} − *k*_{2})^{2}/(*k*_{1} + *k*_{2})^{2}; *a*_{12} determines the phase shifts of the respective solitons after overtaking takes place.

Finally considering *ϵ*=1, we have the two-soliton solution of the KdV (15) as

Hence, the two-soliton solution of the KdV equation (12) is given by

with

When

Using the result *e*^{−x}/(1 + *e*^{−x})^{2}=sech^{2}(*x*/2)/4 and writing *a*_{12}=*e*^{ln∣a12∣}, we get the asymptotic solution of (12)

where

It is to be noted that the phase shifts Δ_{1} and Δ_{2} are of the same sign, and both of them are proportional to *B*^{1/3} and the amplitude of the the solitons, a result consistent with those obtained in the study of head-on collision [45, 46].

Similarly the three-soliton solution of (12) has the form

where

For *τ*>> 1 this solution is asymptotically transformed into a superposition of three single-soliton solutions as

where

Per soliton theory, the shape and velocities of the solitons do not change after collision. It means that their shapes in the remote past are same as those in future. But in finite time they may collide and merge together and eventually form a single soliton at *τ*=0, say, and after *τ* >> 1 they regain their original shapes. This behaviour is clearly shown in this article.

In Figures 1 and 2, time evaluation of the interaction of compressive two-solitons *ϕ*^{(1)} vs. *ξ* are plotted for the several values of *τ*. In Figure 1a, we show that for *τ*=−5 the larger amplitude soliton is behind the smaller amplitude soliton. Then, in Figure 1b at *τ*=−1, the two solitons merge and become one soliton at *τ*=0, shown in Figure 1c. But at *τ*=1, they separate from each other, which is shown in Figure 1d, and then finally they depart from each other when *τ*=5, shown in Figure 1e. The combined profile of the two soliton is shown in Figure 2. It can be clearly seen from the exact two-soliton solution and asymptotical solution that the amplitude of the merge soliton is greater than the amplitude of the shorter soliton but less than the amplitude of the taller soliton (e.g. see the solution given by Drazin et al. [44, p. 76]), which are obtained using the inverse scattering method).

In Figures 3–6, the effect of *β*_{1} on compressive two solitons *ϕ*^{(1)} vs. *ξ* is shown keeping other parameters fixed. In Figure 3, we show the profile of two solitons for *β*_{1}=0.3 with *τ*=−5, *k*_{1}=1, *k*_{2}=2, *α*=1, *β*=4, *μ*_{e}=0.3, *μ*_{i}=0.7, *σ*=0.15, *α*_{1}=1, *α*_{2}=1. Then, in Figure 4, we show the profile of two solitons for *β*_{1}=0.6 with other parameters the same as Figure 1. In Figure 5, we show the profile of two soliton for *β*_{1}=0.9 with other parameters the same as Figure 1. A variation of the compressive two-solitons profiles for different values of *β*_{1} is shown in Figure 6.

Figure 7 shows time evaluation of the interaction of compressive three-solitons solution *ϕ*^{(1)} vs. *ξ* for different values of *τ*. At *τ*=−10 the larger amplitude soliton is behind the smaller amplitude solitary wave. Then, two solitons merge and become one soliton at *τ*=0. But at *τ*=10 they separate from each other and then finally each appears as a separate soliton acquiring their original speed and shape. In Figure 8, we show the variation of the combined compressive three-soliton profiles for different values of *τ*.

In Figures 9–12, the effect of *β*_{1} on compressive three solitons *ϕ*^{(1)} vs. *ξ* has been plotted with fixed values of the other parameters. In Figure 9, we show the profile of three soliton for *β*_{1}=0.3 with *τ*=−5, *k*_{1}=1, *k*_{2}=2, *k*_{3}=3, *α*=1, *β*=4, *μ*_{e}=0.3, *μ*_{i}=0.7, *σ*=0.15, *α*_{1}=1, *α*_{2}=1. Then, in Figure 10, we show the profile of three soliton for *β*_{1}=0.6 with other parameters as in Figure 9. In Figure 11, we show the profile of three soliton for *β*_{1}=0.9 with other parameters as in Figure 9. A variation of the compressive three-soliton profiles for different values of *β*_{1} is shown in Figure 12.

In Figure 13, we present the variation of phase shift for two solitons against *β*_{1} with *k*_{1}=1, *k*_{2}=2, *α*=1, *β*=4, *μ*_{e}=0.5, *μ*_{i}=0.5, *σ*=0.15. The phase shift increases with the increase of *β*_{1}. We plotted the variation of phase shift with *μ*_{i} for *β*_{1}=0.3 (solid line), 0.6 (dotted line), and 0.9 (dashed line) in Figure 14. The phase shift is monotonically decreasing for each *β*_{1} with the increase of the parameter *μ*_{i}.

Figure 15 shows the variation of the phase shift for respective solitons against *β*_{1} when the values of the other parameters are kept fixed. As before, the phase shift increases with increased *β*_{1}, as the value of *B* increases with increased *β*_{1}.

## 4 Conclusions

In this work, we presented the nature of the nonlinear propagation and interaction of dust acoustic two solitons and three solitons in a four component dusty plasma consisting of negatively and positively charged cold dust fluids, non-thermal electrons, and Boltzmann distributed ions. The KdV equation is derived by using RPT, and it is transformed to the standard KdV equation with the help of suitable transformation. Using the Hirota direct method, we obtained two-soliton and three-soliton solutions to the KdV equation. Propagations of two solitons and three solitons have been discussed in detail. It has been observed that the larger soliton moves faster, approaches the smaller one, and, after the overtaking collision, both resume their original shapes and speeds. However, it should be noted that the KdV equation describes multi-soliton solutions depending on the initial conditions. Hirota’s method is an innovative, powerful method by which we can obtain, in principle, any number of solutions for many nonlinear partial differential equations. With the two-soliton solution, it was found that *k*_{i}Δ_{i}, (*i*=1, 2) are the same values. However, in the three-soliton case, phase shift of any soliton is different from the others. Our present study may be helpful in understanding the nonlinear features of the two-soliton and three-soliton solutions in Earth’s mesosphere [20], cometary tails [1, 47], and Jupiter’s magnetosphere [3, 22], where non-thermal electrons and Boltzmann distributed ions are present.

## References

[1] D. A. Mendis and M. Rosenberg, Annu. Rev. Astron. Astrophys. **32**, 419 (1994).Search in Google Scholar

[2] M. Horianyi and D. A. Mendis, J. Geophys. Res. **91**, 355 (1986).Search in Google Scholar

[3] M. Horanyi, Annu. Rev. Astron. Astrophys. **34**, 383 (1996).Search in Google Scholar

[4] P. K. Shukla, Phys. Plasmas **8**, 1791 (2001).10.1063/1.1343087Search in Google Scholar

[5] P. K. Shukla and A. A. Mamum, Introduction to Dusty Plasma Physics, Institute of Physics Publishing, Bristol, UK 2002.10.1887/075030653XSearch in Google Scholar

[6] F. Verheest, Waves in Dusty Plasmas, Kluwer Academic, Dordrecht, The Netherlands 2000.10.1007/978-94-010-9945-5Search in Google Scholar

[7] A. Barkan, R. L. Merlino, and N. D’Angelo, Phys. Plasmas **2**, 3563 (1995).10.1063/1.871121Search in Google Scholar

[8] A. Barkan, N. D’Angelo, and R. L. Merlino, Planet. Space Sci. **44**, 239 (1996).Search in Google Scholar

[9] R. L. Merlino, A. Barkan, C. Thompson, and N. D’Angelo, Phys. Plasmas **5**, 1607 (1998).10.1063/1.872828Search in Google Scholar

[10] A. Homann, A. Melzer, S. Peters, and A. Piel, Phys. Rev. E **56**, 7138 (1997).10.1103/PhysRevE.56.7138Search in Google Scholar

[11] N. N. Rao, P. K. Shukla, and M. Y. Yu, Planet Space Sci. **38**, 543 (1990).Search in Google Scholar

[12] W. S. Duan, Chin. Phys. **13**, 598 (2004).Search in Google Scholar

[13] J. F. Zhang and Y. Y. Wang, Phys. Plasmas **13**, 022304 (2006).10.1063/1.2167916Search in Google Scholar

[14] Y. T. Gao and B. Tian, Phys. Lett. A **361**, 523 (2007).10.1016/j.physleta.2006.11.019Search in Google Scholar

[15] H. R. Pakzad, Chaos Solitons Fract. **42**, 874 (2009).Search in Google Scholar

[16] A. P. Misra and A. Roychowdhury, EPJD **39**, 49 (2006).10.1140/epjd/e2006-00079-1Search in Google Scholar

[17] M. Rosenberg and D. A. Mendis, IEEE Trans. Plasma Sci. **23**, 177 (1995).Search in Google Scholar

[18] M. Rosenberg, D. A. Mendis, and D. P. Sheehan, IEEE Trans. Plasma Sci. **27**, 239 (1999).Search in Google Scholar

[19] V. E. Fortov, J. Exp. Theor. Theor. Phys. **87**, 1087 (1998).Search in Google Scholar

[20] O. Havnes, J. Trøim, T. Blix, W. Mortensen, L. I. Næsheim, et al., J. Geophys. Res. **101**, 10839 (1996).Search in Google Scholar

[21] T. A. Ellis and J. S. Neff, Icarus **91**, 280 (1991).10.1016/0019-1035(91)90025-OSearch in Google Scholar

[22] M. Horanyi, G. E. Morfil, and E. Grun, Nature **363**, 144 (1993).10.1038/363144a0Search in Google Scholar

[23] V. W. Chow, D. A. Mendis, and M. Rosenberg, J. Geophys. Res. **98**, 19065 (1993).Search in Google Scholar

[24] F. Sayed and A. A. Mamun, Phys. Plasmas **14**, 014501 (2007).10.1063/1.2408401Search in Google Scholar

[25] P. Chatterjee and K. Roy, Z. Naturforsch. A **63**, 393 (2008).10.1515/zna-2008-7-802Search in Google Scholar

[26] G. Mandal, K. Roy, and P. Chatterjee, Ind. J. Phys. **83**, 365 (2009).Search in Google Scholar

[27] W. F. El-Taibany and R. Sabr, Phys. Plasmas **12**, 082302 (2005).10.1063/1.1985987Search in Google Scholar

[28] K. Roy, P. Chatterjee, and S. K. Kundu, Adv. Space Res. **50**, 1288 (2012).Search in Google Scholar

[29] K. Roy, K. T. Saha, and P. Chatterjee, Astrophys. Space Sci. **342**, 125 (2012).Search in Google Scholar

[30] N. J. Zabusky and M. D. Zabusky, Phys. Rev. Lett. **15**, 240 (1965).Search in Google Scholar

[31] T. Maxworthy, J. Fluid Mech. **96**, 47 (1980).Search in Google Scholar

[32] K. E. Lonngren, Opt. Quantum Electron. **30**, 615 (1998).Search in Google Scholar

[33] Y. Nakamura, H. Bailung, and K. E. Lonngren, Phys. Plasmas **6**, 3466 (1999).10.1063/1.873607Search in Google Scholar

[34] C. H. Su and R. M. Mirie, J. Fluid Mech. **98**, 509 (1980).Search in Google Scholar

[35] C. S. Gardner, J. M. Greener, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett. **19**, 1095 (1967).Search in Google Scholar

[36] G. Z. Liang, J. N. Han, M. M. Lin, J. N. Wei, and W. S. Duan, Phys. Plasmas **16**, 07370 (2009).10.1063/1.3184822Search in Google Scholar

[37] R. A. Cairns, A. A. Mamum, R. Bingham, R. Boström, R. O. Dendy, et al., Geophys. Rev. Lett. (USA) **22**, 2709 (1995).10.1029/95GL02781Search in Google Scholar

[38] R. A. Cairns, R. Bingham, R. O. Dendy, C. M. C. Nairn, P. K. Shukla, et al., J. Phys. (France) IV **5**, C6-43 (1995).Search in Google Scholar

[39] H. Washimi and T. Sarma, Phys. Rev. Lett. **17**, 996 (1996).Search in Google Scholar

[40] H. Schamel, J. Plasma Phys. **14**, 905 (1972).Search in Google Scholar

[41] H. Schamel, J. Plasma Phys. **9**, 377 (1973).Search in Google Scholar

[42] R. Hirota, The Direct Method in the Soliton Theory, Cambridge University Press, Cambridge, UK 2004.10.1017/CBO9780511543043Search in Google Scholar

[43] R. Hirota, Phys. Rev. Lett. **27**, 1192 (1971).Search in Google Scholar

[44] P.G. Drazin and R.S. Johnson, Solitons, An Introduction, Cambridge University Press, Cambridge, UK 1993.Search in Google Scholar

[45] M.K. Ghorui, P. Chatterjee, and C.S. Wong, Astrophys. Space Sci. **343**, 639 (2013).Search in Google Scholar

[46] U.N. Ghosh, P. Chatterjee, and R. Roychoudhury, Phys. Plasmas **19**, 012113 (2012).10.1063/1.3675603Search in Google Scholar

[47] T.A. Ellis and J.S. Neff, Icarus **91**, 280 (1991).10.1016/0019-1035(91)90025-OSearch in Google Scholar

**Received:**2015-3-3

**Accepted:**2015-6-22

**Published Online:**2015-7-14

**Published in Print:**2015-9-1

©2015 by De Gruyter