The charged dusts in plasma exist in space^[1–5] (e.g. lower and upper mesosphere, cometary tails, planetary rings, planetary magnetospheres, inter-planetary spaces, interstellar media, etc.), laboratory experiment (e.g. Q-machine, plasma materials processing reactors^[1]), etc. The wave dynamics, concerning the dust acoustic (DA) waves, dust ion acoustic (DIA) waves, dust acoustic rogue (DAR) waves, etc., facilitate the understanding of the electrostatic density perturbations and potential structures,^[6–8]e.g., soliton, shock, rogue profile, vortices, etc., as observed in dusty plasma systems. The DA waves are produced^[9] due to imbalance between the driving (provided by the inertia of the dust mass) and restoring forces (provided by the thermal pressure of ions and electrons). The linear and nonlinear structures of DA waves in unmagnetized weakly coupled dusty plasmas were investigated theoretically.^[10–15]

Mamun et al.^[16] have investigated the properties of DA solitary and shock waves with strongly coupled electrons and ions by taking into account the Boltzmann (vortex-like) distribution. The positive and negative charged dusts are also observed in space,^[4,17–19] such as in upper mesosphere, cometary tails, Jupiter’s magneto-sphere, etc. The dust grains can be charged positively through three mechanisms:^[20] photoemission due to ultraviolet photon flux, thermionic emission induced by radiative heating, and secondary electron emission from the surface of the dust grains. The existence of non-Maxwellian particles is observed from the satellite missions in astrophysical and space plasmas,^[21–24] such as in the mercury, uranus, earth, and magnetosphere of Saturn. In these plasmas, the electrons’ distribution follows the power-law instead of Maxwellian distribution. On the other hand, the presence of nonthermal particles at high altitudes, e.g., solar wind and space plasmas, is found from the spacecraft observation.^[25–29] Besides, the superthermal plasmas are observed in various experimental^[30] plasmas, such as laser–matter interactions and plasma turbulence, and their properties are studied by considering the kappa distribution function.^[31–33] Kappa distribution was first reported by Vasyliunas^[31] to model the space plasmas and latter it was applied to different plasmas by several researchers.

Recently, several authors^[34–37] have investigated DIA waves with kappa distributed electrons. Alinejad^[37] has shown that the rarefactive DIA solitary waves and double layers may be produced, depending on the population of superthermal electrons. Ghosh et al.^[38] have investigated the nonlinear properties of DIA solitary waves with small amplitude in dusty plasma comprised of electrons, ions, and dust grains, and found that the wave amplitude exponentially decreases with time due to the collisions of dust ions. Alinejad and Mamun^[39] have studied the properties of obliquely propagating electrostatic waves in magnetized plasmas consisting of electrons, ions, and positrons, and noted that the basic features of solitary waves are modified significantly due to oblique collision, densities of superthermal electrons and positrons, and temperature ratio of electron to positron. The productions of solitons and their interactions among themselves are one of the most important nonlinear phenomena in dusty plasmas. Consequently, this has received special attention from plasma researches.

Ghorui et al.^[40] have studied the characteristics of head-on collisions of counter propagating DIA solitary waves in magnetized quantum dusty plasmas by taking into account positively and negatively charged dust grains and found that the quantum diffraction parameter, ion cyclotron frequency, and density ratio of electron to ion play a significant role in modifying the phase shifts. The head-on collisions between the DA solitary waves have studied by Su and Mirie,^[41] Jeffery and Kawahawa,^[42] and Huang and Velarde^[43] through employing extended Poincaré–Lighthill–Kuo (ePLK) method and found that the characteristic collective motion and collisional properties are affected by the dust charge variation. Xue^[44] has also studied the effects of temperature ratio, density ratio of ion to electron, and dust charge variation on phase shifts due to collision of DA solitary waves and noted that the phase shifts are significantly modified by the dust charge variation. Li et al.^[45] have investigated the interaction among Korteweg–de Vries (KdV) solitons in two-dimensional (2D) dusty plasmas. Besides, Han et al.^[46] have studied the production of DA solitary waves and their interactions in weakly relativistic 2D thermal plasmas and determined the phase shifts and trajectories for an arbitrary angle of interactions. Chopra^[47] has investigated the evolution of DAWs and characterized them in dusty plasmas. On the other hand, Boruah et al.^[48] have studied the oblique collisional effects in strongly coupled dusty plasma and noted that dust acoustic solitons are formed at a certain collision angle. Das^[49] has studied the propagation characteristics of the dust ion acoustic solitary waves in dusty plasma with Boltzmann electrons and found the streaming of dust for both compressive and rarefactive KdV solitons. Zahed et al.^[50] have found in their study that the amplitude and width of the pseudo-potential have increased with increasing ion. Ferdousi et al.^[51] have theoretically investigated the characteristic properties of obliquely propagating IASWs in the presence of ambient magnetic field in nonthermal plasma. They have observed that the characteristics of the solitary waves are significantly affected by the nonextensive electrons and positrons as well as external magnetic field. The collisional effects on instabilities of dust acoustic and dust ion acoustic waves were investigated^[52] and it was also observed that the dispersion of the dust acoustic and dust ion acoustic waves are reduced due to the superthermal charge particles.

Owing to their importance and significance the oblique interaction phenomena and phase shifts in dusty plasmas are investigated. The plasma, considered here in this work, consists of inertial positively and negatively charged massive dusts, superthermal kappa distributed electrons and isothermal ions. The rest of this paper is organized as follows. In Section 2, the fluid model equations along with the assumptions are presented. In Section 3 the evolution equations and their analytical solutions for determine the phase shifts are derived. The results and discussion are displayed in Section 4. Finally, the results of this work are summarized in Section 5.

Here, we consider the plasma that consists of inertial warm negatively charged massive dust with mass m_1d and charge q₁ = –Z_1de, positively charged dust with mass m_2d and charge q₂ = +Z_2de, superthermal kappa distributed electrons with mass m_e and charge –e, and isothermal positive ions with mass m_i and charge +e, where Z_1d (Z_2d) is the charge number of negative (positive) dust. The charge neutrality condition can be defined as Z_1dn_1d0 + n_e0 = Z_2dn_2d0 + n_i0, where n_1d0, n_e0, n_2d0, and n_i0 are the equilibrium number density of warm negatively charged massive dust, kappa distributed electrons, positively charged dust, and isothermal ions, respectively. The normalized fluid equations of DA waves are considered as^[53]

3. Derivation of evolution equations and analytical solution with phase shifts

The two-sided Korteweg–de Vries (KdV) equations (KdVEs) are derived by employing ePLK perturbation method to investigate the oblique interaction phenomena of DA waves along with their phase shifts after oblique collision. Accordingly, the expansion of stretched coordinates and perturbed quantities can be considered^[46] as

with τ = ε³t, and

where ε is the small quantity (0 < ε < 1) for measuring the strength of nonlinearity, ξ and η represent the trajectories of two DA waves propagating in r₁ = (l₁, m₁) and r₂ = (l₂, m₂) directions, respectively, r₁ ⋅ r₂ = l₁ l₂ + m₁ m₂ and the collision angle can be obtained from

where

The phase velocities v_phξ, and v_phη of DA waves, and the variables P₀(η,τ) and Q₀(ξ,τ) will be determined latter. Equations (11a) and (11b) yield the following operators

Inserting Eqs. (12a)–(12g) and (13a)–(13c) into Eqs. (1)–(6) and (10), one can obtain a set of partial differential equations (PDEs) in powers of ε. Collecting the terms having equal powers of ε, the leading (smallest) order of ε yields a system of PDEs (see expressions (A1) in Appendix A.

Solving expressions (A1) in Appendix A, one can obtain the following relations:

The phase velocities can be obtained to be

where , , and , , , , , , , , , and are presented in Appendix (A2).

The next higher power of ε provides a set of PDEs and then after some algebraic manipulations and using Eqs. (14a)–(14g), one can find

where C_1N and C_2N (C_1D, C_2D) are the coefficients of nonlinearity (dispersion) of the DA waves (Appendix (A3)), .

Integrating Eq. (17) with respect to ξ and η, respectively, one can obtain

The first and second terms in the integrands on the right-hand side of Eq. (18) are proportional to η and ξ, respectively, for these are independent of η and ξ. Each term of the first and second expressions on the right-hand side of Eq. (18) become secular and these can be eliminated in order to avoid spurious resonances. Thus, the well-known two-sided KdVEs can be obtained from Eq. (18) to be

The third and fourth terms in the right-hand side of Eq. (18) are not secular, thus, the next higher order of ε would be secular and yields

The principal phase functions P₀ and Q₀ can be estimated by solving Eq. (21) with the help of analytical solutions of KdVEs (Eqs. (19) and (20)). It is noted that Eqs. (19) and (20) characterize the two-sided travelling waves in the reference frames ξ and η, respectively. The solutions of KdVEs (Eqs. (19) and (20) can be obtained to be

Here, Χ = ξ – U₀τ, Χ₁ = η + U₀ τ, and φ_{ξ 0} = 3U₀/C_1N are the amplitudes (width) of the right traveling solitons and is the amplitude (width) of the left traveling solitons at their initial positions, U₀ being the constant velocity of the solitons. To determine the phase shifts due to head-on collision between two solitons say S_ξ (S_η) traveling toward right (left) direction, it is assumed that initially (t → –∞)S_ξ at ξ = 0, η → –∞, and Sη at η = 0, ξ → + ∞ are asymptotically away from each other. After some time they will collide and then depart. That is, after collision S_ξ will be at ξ = 0, η → + ∞ and S_η will be at η = 0, ξ → –∞. The phase shifts can be determined from the solutions (Eqs. (22) and (23)) of KdVEs along with Eq. (21) to be

The un-magnetized dusty plasma consisting of inertial warm negatively charged massive dusts, positively charged dusts, superthermal kappa distributed electrons and isothermal ions as observed in space mentioned earlier, is considered to investigate the oblique interaction phenomena of DA waves. In this regard, the ePLK method is used to derive the two-sided KdVEs. The KdV-like soliton solutions presented in Eqs. (22) and (23) are existent due to the balance between nonlinearity (C_1N and C_2N) and dispersion (C_1D and C_2D). So, if C_1NC_1D (C_2NC_2D) > 0 and both φ_{ξ 0} and φ_{η 0} are positive, the KdV solitons are obtained from Eqs. (19) and (20) and known as compressive DA wave solitons. Besides, the KdV soliton obtained is called rarefactive DA wave soliton, while C_1NC_1D (C_2NC_2D) < 0 and both φ_{ξ 0} and φ_η0 are negative. As it is assumed that the solitons S_ξ and S_η travel toward right and left directions, respectively, therefore, one can see from Eq. (24) that each soliton has a phase shift due to collision in their respective direction of propagation. The plasma parameters are considered here in this work as follows: M_{1d (2d)} = m_1d/m_1d m_2d. m_2d = 150, n_2d(1d) = n_2d0 / n_2d0 n_1d0. n_1d0 = 0.1, n_i(1d) = n₁₀ / n₁₀Z_1dn_1d0. Z_1dn_1d0 = 0.1–0.6, T_i(e) = T_iT_i T_e. T_e = 0.1–0.3, T_{1d (i)} = T_1d T_1d Z_1dT_i. Z_1dT_i = 10⁻⁵, T_2d(i) = T_1dm_1dT_1dm_1d m_2dZ_1dT_i. m_2dZ_1dT_i = 10⁻⁴, Z_2d / Z_2d Z_1d = 0.01–0.09. Z_1d = 0.01–0.09, κ_e = 2.0–2.6, and the remaining parameters are all constant. The head-on collisional dynamics of DA waves for KdV solitons are displayed in Fig. 1 with velocities v_phξ = 0.045 and v_{ph η} = 0.45, but they travel in opposite directions with the obliqueness θ. In Fig. 1, panels 1(b) and 1(d) represent the contour of the panels 1(a) and 1(c), while panels 1(a) and 1(c) show the colisional effects at θ = 30° and θ = 45°, respectively. These figures reveal the formation of new nonlinear wave structures in the colliding region, which is in good agreement with the finding of El-Shamy et al.^[54] Besides, the pronounced soliton structures are produced for small obliqueness. Figure 2(a) (for θ = 45°) is the 2D plot of Fig. 1(b), while figure 2(b) is the 2D plot of Fig. 1(a).

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. Oblique colisional effects of dust acoustic waves on electrostatic potential for θ = 30° (a) and θ = 45° (c) with n2d(1d) = 0.1, ni(1d) = 0.4, Z = 0.01, T1d(i) = 10–5, T2d(i) = 10–4, κe = 3, Ti(e) = 0.1, vphξ = 0.045, vphη = 0.045, M1d(2d) = 150, and U0 = 0.09. Panels (b) and (d) are contour plots of panels (a) and (c).

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Oblique colisional effects of dust acoustic waves on electrostatic potential for θ = 45° (a) and θ = 30° (b) with n2d(1d) = 0.1, ni(1d) = 0.4, Z = 0.01, T1d(i) = 10–5, T2d(i) = 10–4, κe = 3, Ti (e) = 0.1, vph ξ = 0.045, vphη = 0.045, M1d (2d) = 150, and U0 = 0.09.

The effect on nonlinear coefficient (C_1N) with κ_e for T_i(e) is displayed in Fig. 3(a) and revels the enhancement of C_1N with the increase of T_i(e) of the DA wave solitons. Further, the decreasing rate of C_1N becomes larger in a lower range of κ_e (κ_e < 3.5) than that for a higher range of κ_e (κ_e≥ 3.5). The effect of n_i(1d) on C_1N with Z is described in Fig. 3(b) and it is shown that C_1N is increasing with Z of the DA wave solitons increasing, but no effect on n_i(1d) is observed except in the critical region, while C_1N is decreasing due to the increase of n_i(1d) around the critical region. The driving force (provided by the inertia of n_1d) is decreasing due to the depopulation of n_1d around the critical region (n_i(1d) = 0.46), consequently the nonlinearity of DA waves decreases with n_i(1d) increasing.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. Oblique colisional effects of dust acoustic waves on nonlinear coefficients C1N of rightward traveling KdV solitons for (a) Ti(e) taking with n2d(1d) = 0.1, ni(1d) = 0.4, Z = 0.04, T1d(i) = 10–5, T2d(i) = 10–4, vphξ = 0.045, vphη = 0.045, M1d(2d) = 150, θ = 45° and U0 = 0.09; (b) Z taking with κe = 2, Ti(e) = 0.1, and the remaining parameters are the same as those in Fig. 1(a).

It is mentioned earlier that the solitons S_ξ and S_η travel toward the right and left direction, respectively. Therefore, one can see from Eq. (24) that each soliton produces a phase shift due to collision in its direction of propagation. The phase shifts of the solitons are produced due to gaining energy from the medium preserving their original shapes. The phase shifts are produced due to head-on collisions between the two counter-propagating solitons. They may be either positive or negative depending on the sign of L / M. or L₁ / L₁M₁. M₁ (Eq. (24)). Figures 4(a) (θ=30°) and 4(b) (θ = 45°) successively show the effect of Z with n_i(1d) on the phase shift. These figures show that the phase shifts decrease with both n_i(1d) and Z increasing. It is noted that the phase shifts become pronounced with both n_i(1d) and Z decreasing. Thus, obliqueness plays a crucial role in changing phase shift. Figures 4(c) (θ=30°) and 4(d) (θ = 45°) show the effect of κ_e with T_i(e) on phase shift. It is observed that the phase shift decreasing with T_i(e) increasing, while it increases almost linearly with κ_e increasing. It is also noted that the decreasing rate of phase shift becomes more pronounced for small values of T_i(e). The effects of obliqueness on the phase shift are insignificant which can also be observed from Figs. 4(c) and 4(d). Furthermore, it is seen from Fig. 4 that the phase shift is positive in each case, which means that the trajectories of the travelling solitons are slower than those of solitons without collision.

Fig. 4.

Figure Option ViewDownloadNew Window

Fig. 4. Effects on phase shift Δδη for (a) ni(1d) taking with θ = 30°, n2d(1d) = 0.1, ni(1d) = 0.4, T1d(i) = 10–5, T2d(i) = 10–4, Ti(e) = 0.1, vphξ = 0.045, vphη = 0.045, M1d(2d) = 150, ε = 0.1, and U0 = 0.09; (b) ni(1d), θ = 45° taking with the remaining parameters being the same as those in Fig. 4(a) with Z; (c) Ti(1d), θ = 30° and (d) Ti(1d), θ = 45° taking with the remaining parameters being the same as those in Fig. 4(a) with κe.

Figures 5(a) and 5(b) show the effects of obliqueness and temperature ratio of isothermal ion to electron (T_i(e)) on amplitudes of the DA wave solitons, respectively. Figure 5(a) displays that the amplitude increases with both θ and κ_e increasing. This can be explained as the fact that the coefficient of nonlinearity drops down while the dispersive coefficient goes up due to the enhancement of κ_e, consequently the amplitude of soliton increases. Besides, the amplitude decreases with T_i(e) increasing, but asymptotically increases with n_i(1d) increasing as observed from Fig. 5(b). Due to the enhancement of T_i the loss rate of energy becomes pronounced, as a result the amplitudes decrease. Besides, the restoring force provided by the isothermal ions increases due to the increase in n_i (depopulation of n_1d), consequently the amplitude increases with n_i(1d) increasing. It is obvious that the amplitudes of the coupled KdV solitons and their corresponding phase shifts obtained from the Eqs. (19) and (20) become infinite while C_1N → 0 and C_2N → 0 for the critical values of κ_e. One can obtain (for rightward traveling soliton) the critical value of κ_e: κ_ec ≅ 1.507 for C_1N = 0 with the plasma parameters taken as n_2d(1d) = 0.1, n_i(1d) = 0.4, Z = 0.04, T_1d(i) = 10^–5, T_2d(i) = 10^–4, T_i(e) = 0.1, v_phξ = 0.045, v_phη = 0.45, M_1d(2d) = 150, θ = 45° and U₀ = 0.09. Balance between the nonlinearity and dispersion is the necessary condition for producing the KdV solitons. Thus, for κ_e≅ κ_ec, the nonlinearity terms of the coupled KdV equations become zero, consequently it is impossible for KdV solitons to exist. Since the dispersive coefficients (C_1D and C_2D) are always positive, therefore it is dependent on the signs of nonlinear coefficients (C_1N and C_2N) that the KdV solitons are either compressive or rarefactive. In the present study for the rightward traveling soliton C_1N ≶ 0 holds, thus the rarefactive and compressive DA soliton structure are formed.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. Variations of amplitude of rightward traveling solitons with (a) κe for θ = 30°, θ = 45° for n2d(1d) = 0.1, ni(1d) = 0.4, T1d(i) = 10–5, T2d(i) = 10–4, Ti(e) = 0.1, M1d(2d) = 150, ε = 0.1, and U0 = 0.09; (b) ni(1d) for θ = 45°, Ti(e) = 0.20, 0.25, 0.30 taking with the remaining parameters being the same as those in Fig. 5(a).

Figures 6(a)–6(c) show the effects of θ for κ_e < κ_ec on the rightward travelling solitons due to oblique collisions. Figures 6(a)–6(c) explain the effects of θ = 30°, θ = 45°, and θ = 75°, respectively, for the rightward travelling solitons. One can observe from these figures that the dip shape (rarefactive) structures of the potential profiles are formed in each case of obliqueness. Thus, the structures of potential profiles become pronounced for the small values of θ.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. Effects on structure of potential profile of rightward traveling solitons for (a) θ = 30°, (b) θ = 45°, and (c) θ = 75°, with parameters taking as κe = 1.504, n2d(1d) = 0.1, ni(1d) = 0.2, Z = 0.01, T1d(i) = 10–5, T2d(i) = 10–4, Ti(e) = 0.3, vphξ = 0.045, vphη = 0.045, M1 d(2d) = 150, ε = 0.1, and U0 = 0.09.

Figures 7(a)–7(c) display the effects of obliqueness (θ = 30°, θ = 45°, and θ = 75°) for κ_e > κ_ec on the structures of the potential profiles for the rightward travelling solitons due to oblique collision. These figures reveal that the humpshaped (compressive) soliton is produced in each case due to the effect of obliqueness θ and that the largest amplitude of soliton is produced for small θ. Besides, figures 6 and 7 expose that the smallest value of obliqueness is prominent and the soliton propagates from left to right direction with time, which is in good agreement with analytical result.^[55,56] Finally, the rarefactive (compressive) soliton structure is formed for κ_e < κ_ec (κ_e > κ_ec).

Fig. 7.

	Figure Option ViewDownloadNew Window
	Fig. 7. Effects on structure of potential profile of rightward traveling solitons for (a) θ = 30°, (b) θ = 45°, and (c) θ = 75°, with parameters taking as κe = 1.52, n2d(1d) = 0.1, ni(1d) = 0.2, Z = 0.01, T1d (i) = 10–5, T2d (i) = 10–4, Ti (e) = 0.3, vph ξ = 0.045, vph η = 0.045, M1d (2d) = 150, ε = 0.1, and U0 = 0.09.

The fully ionized un-magnetized dusty plasma is considered to investigate the oblique interaction phenomena of the DA waves. The plasma system consists of inertial warm negatively charged massive dusts, positively charged dusts, superthermal kappa distributed electrons, and isothermal ions. The KdVEs are derived by employing the ePLK method. The important results obtained in this paper are summed up as follows. New nonlinear wave structures are formed in the colliding region due to the head-on collision of the two counter propagating DA waves that depend on the degree of obliqueness. Near the critical value of n_i(1d) (n_i(1d) = 0.46), the nonlinearity decreases with n_i(1d) increasing. The phase shifts decrease (increase) with T_i(e) (κ_e) increasing. The humpshaped (compressive) and dipshaped (rarefactive) solitons are produced depending on θ for κ_e < κ_ec and κ_e > κ_ec, respectively. The proposed model may be useful for explaining the results for both space (such as, lower and upper mesospheres, cometary tails, planetary rings, planetary magnetospheres, inter-planetary spaces, interstellar plasma media, etc.) and laboratory experiments (e.g., Q-machine plasma, RF plasma, microwave plasma, fusion plasma, etc.), where the inertial warm negatively charged massive dusts, positively charged dusts, superthermal kappa distributed electrons, and isothermal ions are the combined attribution.