Oblique collision of low-frequency dust-acoustic solitons and rogue waves in a multi-dimensional anisotropic self-gravitating electron-depleted complex plasmas

Abstract

This investigation aims to study the effect of dust pressure anisotropy, self-gravitational forces, and magnetic forces on the gravitational instabilities and associated low-frequency rogue wave (RW) profile as well as oblique interaction characteristics of low-frequency dust-acoustic (DA) solitary waves (DASWs) in an electron-depleted dusty plasma. The current plasma model comprises inertial warm, positively and negatively charged massive dust grains and non-extensive ions. Following the quasi-linear theory, the extended Poincaré–Lighthill–Kuo (PLK) perturbation technique, and Jean’s criterion, the two-sided Korteweg-de Vries (KdV) equations and corresponding analytical phase shifts due to collision are derived and analyzed. The critical configuration of the current plasma model that renders the KdV compressive or rarefactive DASWs invalid and leads to the formation of the modified KdV (mKdV)-type solitons is determined. The findings from computational simulations unequivocally demonstrate that parallel and perpendicular dust pressures do not exhibit the same behavior rigorously. Furthermore, the presence of the magnetic field and dust pressure anisotropy has been found to inhibit gravitational instabilities and significantly modify DA RW triplets, which can evolve into super rogue waves through the superposition of triplets. The collision scenarios involving solitons of both similar and opposite polarities traveling toward each other, as well as the influential roles played by the crossing angle, magnetic field, gravitational field, ion entropic index, and dust pressure along and across magnetic field lines on phase shifts and DA RWs, are extensively discussed in detail. The possible applications of the present work for various space plasma scenarios, including dense molecular clouds and interstellar medium, are anticipated. This study will contribute to understanding the dynamics of the generation and propagation of both RWs and the collision of solitons. Once the mechanics of their generation and propagation are understood, these waves can be devoted to many industrial and medical applications.

Introduction

Since the intriguing observations made by Pioneer, Voyager, Giotto, and Vega space missions^1,2 in the interstellar medium, revealing for the first time the presence of submicron-sized metallic (graphite, magnetite, amorphous carbons) or dielectric particles (ices, silicates), knowledge about dust and dusty plasmas, as well as their pivotal roles in laboratory processes and their implications in the formation of various astrophysical objects such as Jupiter’s, Saturn’s, and Neptune’s ring systems^3,4 has undergone a profound expansion. These particles exhibit dynamics never observed before and can absorb and scatter light, resulting in dark spokes observed within gas clouds and planetary rings. By competitive mechanisms like secondary electrons or ions capturing, secondary emission of electrons, ⁵ photoemission under ultraviolet (UV) radiation^6,7 and thermionic emission induced by radiative heating, ⁸ dust grain can attain quickly or not ⁹ exceedingly high positive or negative charges and strongly interact with the adjacent plasma, either via the mediation of long-range electromagnetic fields only (dust size lower than 1 μm) or through the conjoined effects of long-range gravitational and electromagnetic fields (dust size between 1 μm to 15 μm), depending to their mass-to-charge ratio.^10,11 An exceptional case is that of equally important gravitational and electric forces, or self-gravitational plasma $(G{(m_{n,p})}^2/{(Z_{n,p}e)}^2\simeq O(1))$ . m_n,p stand for the negative or positive dust grain mass, Z_n,p is the negative or positive dust grain charge and G denote the universal gravitational constant. Such charging mechanisms can significantly alter the plasma by severely depleting its electron content, as was the case for an electron-depleted dusty plasma (EDDP).^12,13 On the other hand, they can also enhance the plasma’s characteristics^14,15 by promoting the emergence of new instabilities, and novel waves modes.^16,17 In plasmas containing heavy dust, the electric field can be screened while the gravitational field remains attractive under all conditions. Thus, a small disturbance in the uniformly distributed mass can disrupt the local balance between electric and gravitational forces. ⁴ This disruption results in the collapse and fragmentation of the original cloud, ultimately leading to the formation of protostars and planets. This phenomenon, known as gravitational instability, was quantified by James Jeans, ¹⁸ who determined the critical wavelength above which gas clouds become unstable. Dusty plasmas are recognized to support low-frequency wave modes. Among these, we can cite the famous dust-acoustic (DA) mode, predicted by Rao et al. ¹⁹ and repeatedly verified by experiments.^20,21 After Jeans’ work, several researchers have explored the propagation of low-frequency waves and Jeans’ stability criteria for various dusty plasma models, considering magnetohydrodynamics (MHD) approach. Pandey et al. ⁴ have established the conditions for electrostatic levitation, condensation, and dispersion of grains in a dusty plasma. Prerana Sharma ²² studied dust Alfvén waves and found that the Jeans criterion is unaffected by the effects of resistivity and viscosity. Mamun et al. ²³ examined instabilities of self-gravitating dusty clouds in magnetized plasmas, taking into account the external magnetic field and drag forces. Their findings indicated that the external magnetic field plays a stabilizing role in electromagnetic wave propagation. Recently Prajapati et al. ²⁴ demonstrated that the polarization force counteracts gravitational instability. It is instructive to note that in many astrophysical and cosmological systems, the conventional Maxwell–Boltzmann distribution falls in predicting the motion of charged particles due to long-range interactions and memory effects. To address these limitations, Reyni ²⁵ introduced the nonextensive distribution distinguished by the entropic index q. This formalism was further extended and refined by Tsallis. ²⁶

A rogue wave (RW) is a high-amplitude excitation that appears out of nowhere and disappears without a trace. These waves were first observed in the oceans ²⁷ and later in various fields such as optics, Bose-Einstein condensates,^28,29 and subsequently by Bailung ³⁰ in plasmas. The interest in these waves stems from their ability to locally concentrate a very large amount of energy, which can be either an advantage or a disadvantage for their applications. Rogue waves (RWs) can be modeled using the nonlinear Schrödinger equation (NLSE),^31–38 which provides a hierarchy of solutions. The first-order rational solution of the NLSE is called the Peregrine soliton ⁵² or the first-order RWs. The nonlinear interaction of first-order RWs can produce second-order RWs, with the RW triplet ⁵³ being an exceptional case. This triplet consists of three Peregrine solitons at the corners of a triangle, which can superimpose under appropriate conditions to form super RWs. Many different and diverse studies were conducted on different plasma models to study the distinctive properties of first and super RWs and breathers that described by different forms of NLSE. During these studies, the researchers reduced their plasma models’ fluid governing equations to different NLSE forms, including planar NLSE,^31–38 nonplanar NLSE,^39–46 and damped NLSE.^47–51 Accordingly, the researchers studied the properties of the planar RWs and breathers, ⁵³ nonplanar RWs and breathers when the effect of the curvature is considered, as well as the damped RWs and breathers when the impact of the collisions between plasma charges is considered.

Plasmas exhibit extreme dilution in various plasma situations where significant gravitational effects lead to infrequent particle collisions, rendering the fluid approximation inadequate and standard (MHD) equations inapplicable. However, strong magnetic fields generated by magnetic stars or other astrophysical sources induce the Lorentz force on charged particles, causing them to move around the magnetic field lines in circular or helical paths. This confinement effectively aligns particle motion perpendicular to the direction of the magnetic field. Consequently, the plasma demonstrates anisotropic behavior with distinct pressures parallel and perpendicular to the magnetic field lines. Under these conditions, the fluid-like description remains applicable, and the Chew–Goldberger–Low (CGL) equations ⁵⁴ obtained from the moment equations of the collisionless Vlasov equation must be incorporated into the standard MHD equations (as in the present paper) to get a more comprehensive and accurate representation of plasma’s behavior. The effects of pressure anisotropy on wave propagation have been discussed by several researchers using CGL. Bhatia ⁵⁵ demonstrates that the critical wavelength of gravitational instability is strongly influenced by pressure anisotropy, magnetic field, and plasma rotation speed. Prajapati et al ⁵⁶ have discussed self-gravitational instability of a magnetized rotating, anisotropic plasma medium in the presence of heat flux corrections and found that increasing the pressure anisotropy decreases Jean’s instability. Gliddon et al. ⁵⁷ analyzed the gravitational instability of anisotropic plasma and revealed that because of anisotropy, the spiral arm galaxy plasmas can exhibit hose instability as well as gravitational instability. Moreover, he suggested that gravitational instability occurs at smaller wavenumbers in transverse propagation than longitudinal ones for a given frequency. Ariel ⁵⁸ extended this work by adding Hall’s effect. Khalid et al. ⁵⁹ show that the dynamical behavior of ion-acoustic (IA) cnoidal waves (IACWs) is more sensitive to parallel ion pressure rather than perpendicular ones. Chatterjee et al. ⁶⁰ demonstrated that the presence of solitary waves (SWs) in obliquely propagating dust-ion acoustic waves (DIAWs) is greatly influenced by perpendicular and parallel pressures. The study conducted by Bashir et al. ⁶¹ examined the impact of anisotropic dust pressure and superthermal electrons on the propagation of dust-acoustic SWs (DASWs). The findings revealed that dust anisotropy influences both the amplitude and polarity dust-acoustic waves (DAWs). The impact of parallel and perpendicular ion pressures on the propagation of nonlinear electrostatic shock wave structures in anisotropic magnetoplasmas were examined by Albalawi et al. ⁶²

After Zabusky and Kruskal ⁶³ made the remarkable observation that during the interaction of two solitons, both structures remain unperturbed retaining their original amplitudes, widths, and velocities, yet undergo a phase shift between pre-collision and post-collision trajectories, extensive research efforts have been dedicated to exploring wave collisions to gain a comprehensive understanding of energy and transport mechanisms in plasmas, along with resonance phenomena.^64,65 Thus, by adopting a one-dimensional (1D) approach, which leads to two distinct types of interactions, namely, overtaking collisions (angle between wave vector θ = 0) studied using methods like inverse scattering transformation, ⁶⁶ and head-on collisions (θ = π) analyzed using the PLK method, ⁶⁷ Ikezi et al. ⁶⁸ experimentally demonstrated that waves do not add up during the overtaking collision process, whereas the opposite is observed during a head-on collision. Lonngren ⁶⁹ experimentally identified the critical collision angle that leads to the phenomenon of resonance. Kuldeep Singh ⁷⁰ demonstrated that dust concentration significantly affects phase shift between trajectories of dust acoustic ⁷¹ waves. Tiofack et al. also emphasized the impact of self-gravitational field on head on collision. ⁷¹ However, the three-dimensional (3D) approach offers a more realistic and comprehensive framework, enabling the exploration of oblique interactions (0 < θ ≤ π). Using the 3D geometry, El-Shamy et al. ⁷² revealed that the magnitude of the phase shift between the trajectories of two ion acoustic SWs is directly dependent on the electron nonextensive parameter. However, it is inversely proportional to their number density and the strength of the applied magnetic field. Xu et al. ⁷³ addressed the interaction of two SWs in quantum electron-positron-ion plasma and reported that, for a given value of the quantum parameter, the magnitude of the phase shift decreases as the collision angle increases. Irfan et al. ⁷⁴ presented the influence of pressure anisotropy on 3D IA rogons emerging in magnetoplasmas with trapped and untrapped electrons. Liang et al. ⁷⁵ elaborated on the phase shift after the collision of two solitons with an arbitrary angle in a magnetized complex plasma. Ju-Kui ⁷⁶ provided insights into the interaction of DASWs, considering dust charge fluctuations. The oblique collision of two dust-ion acoustic (DIA) solitons in a collisionless magnetized plasma consisting of dynamic ions, static charged dust particles (positive/negative), and inertialess kappa-distributed electrons was investigated by Parveen et al. ⁷⁷ El-Labany et al. ⁷⁸ investigated the phenomenon of oblique collision of dust-acoustic (DA) solitons within a plasma composed of micron-sized dust particles with a negative charge, Boltzmann distributed electrons, and two temperature ions. In their investigation, they disregarded dust pressure anisotropies and gravitational effects. The results indicated that increasing the magnetic field (colliding angle) strength led to a decrease (increase) in phase shifts. Later, El-Labany et al. ⁷⁹ extended the study and incorporated the influence of dust pressure anisotropy. They demonstrated that the dust pressure ratio affects the phase shifts. Kuldeep et al. ⁸⁰ demonstrated that the polarization force enhances the amplitude of dust-acoustic RW (DARW) triplets, whereas a decrease in ion superthermality results in a reduction in the amplitude of DA super RWs. Shimin Guo et al. ⁸¹ analyzed dust ion-acoustic waves and identified the conditions necessary for the formation of RW triplets in a collisionless, unmagnetized plasma. Based on the literature review, it is revealed that the essential physics underlying the oblique collision between two counter-propagating DASWs in a self-gravitating anisotropic plasma remains unexplored. As a result, the main goal of this work is to shed light on impacts of dust pressures anisotropy, self-gravitational and magnetized forces on gravitational instabilities, and oblique interaction features of DASWs in a self-gravitating anisotropic electron depleted dusty plasma consisting of inertial warm positively and negatively charged massive dust grains in addition to non-extensive ions.

The paper is structured in the following manner: Section II presents the fundamental equations of the model, whereas Section III examines the dispersion relation. In Section IV, the dynamics of DASWs and their related phase shifts following a collision are modeled using the two-sided KdV equation and the two-sided mKdV equation. These equations are calculated for the generic case and the critical case, respectively, using the PLK approach. Section V is dedicated to the analysis of RW by reducing the mKdV equation to the NLSE using a suitable transformation. Section VI provides a concise summary of the study’s primary findings.

Governing equations

This study examines a three-dimensional self-gravitating EDDP plasma that exhibits dust pressure anisotropy. This plasma model is composed of inertial warmed positively and negatively charged large dust grains, with a mass of (m _i /m _j ≪ 1), moving at a slow velocity (T _j /T _i ≪ 1) in an electron-depleted background that contains nonextensive ions. The symbol T_i,j(m_i,j) represents the temperature (mass) of the ion and the dust. The subscript i denotes ions, while j = n, p represents negative and positive dust grains. The system exhibits charge neutrality, n_i0 + Z _p n_p0 = Z _n n_n0. Additionally, it is subjected to a strong and uniform external magnetic field that is directed along the x-axis: (B ₀ = B₀x). The variables n_i0 and n_j0 represent the unperturbed ions and dust number densities, respectively. It is postulated that because of the magnetic field and the infrequent interparticle collisions, the negative (positive) dust pressure $({\tilde{P}}_j)$ has a tensorial character and takes the form

${\tilde{P}}_j=\stackrel{`}{P_{\perp j}}\widehat{\mathbf{I}}+(\stackrel{`}{P_{\Vert j}}-\stackrel{`}{P_{\perp j}})\widehat{\mathbf{b}}\widehat{\mathbf{b}},$

(1)

where $\widehat{\mathbf{I}}$ is the identity matrix. $\stackrel{`}{P_{\Vert j}}$ and $\stackrel{`}{P_{\perp j}}$ are the parallel and the perpendicular pressure tensor components relative to the magnetic field direction $\widehat{\mathbf{b}}$ ( $\widehat{\mathbf{b}}$ is a unit vector). These scalar components can be equated according to the CGL theory ⁵⁴ as

$\stackrel{`}{P_{\perp j}}=P_{\perp j0}{n}_j,\stackrel{`}{P_{\Vert j}}=P_{\Vert j0}{n_j}^3,$

(2)

where P_⊥j0 = n_j0K _B T_⊥j (P_‖j0 = n_j0K _B T_‖j) and T_⊥j (T_‖j) stand for perpendicular (parallel) dust pressure and temperature at equilibrium. Under the assumptions mentioned above, the following normalized set of equations that govern the motion of positive and negative dust grains in a 3D self-gravitating anisotropic plasma is constructed

${\partial }_t{n}_j+\nabla .(n_j{\mathbf{u}}_{\mathbf{j}})=0$

(3)

${\partial }_t{u}_{j_x}+({\mathbf{u}}_{\mathbf{j}}.\nabla )u_{j_x}={\sigma }_j{\nabla }_x\varphi -{\nabla }_x\psi -{\kappa }_j{P}_{\Vert j}{n}_j{\nabla }_x{n}_j$

(4)

${\partial }_t{u}_{j_y}+({\mathbf{u}}_{\mathbf{j}}.\nabla )u_{j_y}={\sigma }_j{\nabla }_y\varphi -{\nabla }_y\psi -{\sigma }_j{\omega }_{cn}{u}_{j_z}-{\kappa }_j{P}_{\perp j}\frac{{\nabla }_y{n}_j}{n_j}$

(5)

${\partial }_t{u}_{j_z}+({\mathbf{u}}_{\mathbf{j}}.\nabla )u_{j_z}={\sigma }_j{\nabla }_z\varphi -{\nabla }_z\psi +{\sigma }_j{\omega }_{cn}{u}_{j_y}-{\kappa }_j{P}_{\perp j}\frac{{\nabla }_z{n}_j}{n_j}$

(6)

${\nabla }^2\psi =\gamma \left(n_n+\frac{\mu \alpha }{\beta }{n}_p\right).$

(7)

${\nabla }^2\varphi =n_n-\mu n_p+(\mu -1)n_i,$

(8)

with (−1 + μ)n _i = L₀ + L₁ϕ + L₂ϕ² + L₃ϕ³; L₀ = μ − 1, $L_1=-1/2\left(-1+\mu \right)\left(3q-1\right)$ , $L_2=1/8\left(q+1\right)\left(3q-1\right)\left(-1+\mu \right)$ and $L_3=1/48\left(3q-1\right)\left(q-3\right)\left(q+1\right)\left(-1+\mu \right)$ . The notations ∂/∂x = ∂ _x and ∇ = (∂ _x , ∂ _y , ∂ _z ) are used. Equations (3)–(8) are normalized adopting the following rescalings: n _i → n _i /n_n0, n _j → n _j /n_n0, $u_{j_{x_s}}\to u_{j_{x_s}}/C_{sn}$ , ϕ → ϕ|e|/K_BT_i, $\psi \to \psi |e|/{C_{sn}}^2$ , x _s → x _s /λ _Dn , $t\to t/{{\omega }_{pn}}^{-1}$ . Where $C_{sn}={\left(Z_n{K}_{\text{B}}{T}_{\text{i}}/m_n\right)}^{1/2}$ , ${\lambda }_{Dn}={(K_{\text{B}}{T}_{\text{i}}/4\pi Z_n{e}^2{n}_{n0})}^{1/2}$ , and ${{\omega }_{pn}}^{-1}={(m_n/4\pi e^2{n}_{n0})}^{1/2}$ refer to the negative dust acoustic speed, the negative dust Debye length and the inverse of negative dust plasma frequency. $u_{j_{x_s}}$ , ϕ, ψ, K_B, e, x _s = (x, y, z), and t stand, respectively, for dust velocity, electrostatic potential, gravitational potential, Boltzmann constant, electron charge, space, and time coordinates. Here, P_‖j = 3T_‖j/T_i, P_⊥j = T_⊥j/T_i, α = Z _n /Z _p , β = m _n /m _p , μ = Z _p n_p0/Z _n n_n0, $\gamma ={\omega }_{jn}^2/{\omega }_{pn}^2=G{(m_n/Z_{n}e)}^2$ , ${\omega }_{jn}={(4\pi G{m}_n{n}_{n0})}^{1/2}$ , ω _cn = Ω _cn /ω _pn , ${\mathrm{\Omega }}_{cn}=\left|Z_{\text{n}}e\right|B_0/m_n$ , κ _j = (1, β), and σ _j = (1, − β/α). Notice that ω _jn refers to Jean’s frequency, while Ω _cn represents the cyclotron frequency.

Linear dispersion relation

Consider that the plasma is slightly disturbed from its equilibrium position, generating a low-frequency disturbance and propagating in the form of a plane wave. All the dependent quantities can, therefore, be expressed in terms of their equilibrium and their perturbed parts as

${[n_j,u_j,\varphi ,\psi ]}^T={[\mathrm{1,0,0,0}]}^T+{[\delta n_j,\delta u_j,\delta \varphi ,\delta \psi ]}^T,$

(9)

where all the perturbed quantities are proportional to $e^{i(k_{x}x+k_{y}y+k_{z}z-\omega t)}$ . k and ω stand, respectively, for the wave number and the frequency. By substituting equation (9) into equations (3)–(8), and applying space and time transformations ∂ _t → −iω and ${\partial }_{x_s}\to i{k}_{x_s}$ ; x _s = x, y, z, the following dispersion relation is obtained, providing information on the plasma’s ability to support (transmit) specific wave modes

${\omega }_{f,s}=\frac{N_{\varsigma }\pm \sqrt{{N_{\varsigma }}^2-4{\mathrm{\Gamma }}_{\varsigma }{D}_{\varsigma }}}{2{\mathrm{\Gamma }}_{\varsigma }};(\varsigma =\mathrm{1,2}).$

(10)

Equation (10) indicates that the fast acoustic mode (denoted by the symbol f and corresponding to the positive sign) and the slow acoustic mode (denoted by the symbol s and corresponding to the negative sign) are possible in the plasma. Index ς = 1 is associated with propagation along the magnetic field (k _x = k_‖; k _y = 0; k _z = 0) and does not depend on magnetic field strength while ς = 2 refers to propagation perpendicular to the magnetic field $(k_{\Vert }=0;k_y^2+k_z^2=k_{\perp }^2)$ and depends on magnetic field. Noted that $k_{\Vert }^2+k_{\perp }^2=k^2$ . N _ς , Γ _ς , and D _ς are given in the appendix. The perpendicular dispersion relation is similar to one obtained in Ref. 71 for the same assumptions. The effects of non-uniform pressures, gravitation, and the magnetic field on the excitation dispersion will now be highlighted by considering typical physical parameters representative of astrophysical mediums, such as the photoassociation regions separating H II regions from dense molecular clouds ⁸² : m_(p,n) = 10⁻¹² − 10⁻⁷g, Z _n = 3 × 10⁴, Z _p = 2 × 10⁴, n_p0 = 4 × 10⁻⁷ cm⁻³, n_n0 = 5 × 10⁻⁷ cm⁻³, T _i = 10 K, T_(p,n) = 1 − 3K, and B₀ = 10⁻³ − 10⁻²G. Assuming m _p = 2 × 10⁻⁷g, m _n = 4 × 10⁻⁷g, and B₀ = 7 × 10⁻²G, the following quantity is obtained: n_i0 = 7 × 10⁻³ cm⁻³, Ω _cn = 8.71 × 10⁻¹¹ Hz, ω _pn = 6.23 × 10⁻¹⁰ Hz, ω _Jd = 4.08 × 10⁻¹⁰ Hz, ω _cn = 0.1, μ = 0.53, β = 2, and α = 1.5. Additionally, γ = 0 − 0.9 and values of q > 3/5 are considered based on Ref 83. Figure 1 illustrates the frequency variations of a longitudinal wave as a function of the wavenumber k for the two modes when P_‖n and γ increase. By successively observing pairs in Figures 1(a, d); (b, e); and (c, f), it can be concluded without any ambiguity that although the fast mode remains insensitive to P_‖n for low values of k, DAWs frequencies increase when the wavenumber and the parallel dust pressure increase. To simplify the discussion, only the pressures associated with the heavier negative dust grains are considered, as observations made for negative dust are also valid for positive dust. Moreover, it is noted that in the presence of a gravitational field, there are ranges of frequencies that the medium cannot transmit (the slow mode exclusively). Indeed, when the criterion of the onset of instability k < k _crit is satisfied, ${\omega }_s^2<0$ means that ω _s is purely imaginary, then the disturbance loses its wave character and grows exponentially with time (Γ = |Img(ω _s )| measures the growth rate of the gravitational instability). The critical point is associated with a state of the plasma for which gravitational and electrostatic forces are in equilibrium. A careful comparative study of Figures 1(b, e) and (c, f) reveals that increasing parallel dust pressure causes a considerable decrease in critical wavenumber. In contrast, increasing gravitation slightly lowers DAWs frequencies but strongly increases the maximum growth rate of instability (Γ_max) and k _crit . For P_‖n = 0.8, we find k _crit = 0.3 and based on the typical parameters of photo-association regions that separate H II regions from dense, dusty molecular clouds, we have determined the critical values λ _crit = 3.41 × 10⁶ km and M _crit = 3.3 × 10¹⁰ kg. These values represent the radius and mass of the dust cloud, respectively, beyond which the cloud collapses and fragments, leading to the formation of planets or stars. Figure 2 depicts impact of P_⊥n, γ and ω _cn over the dispersion of a transversal wave. It is observed that P_⊥n has similar effects to those previously attributed to P_‖n. However, instability occurs at smaller wavenumbers in perpendicular propagation than in parallel one, which means that P_⊥n promotes instability less than P_‖n. Influence of γ remains analogous (see Figures 2(a, d); (b, e); and (c, f)). Figure 2(c, f);(d, g); and (e, h) were obtained by keeping parameters of Figures 2(c) and (f) fixed and changing only values of ω _cn . It is clear that enforcing magnetic field strength leads to enhancement of frequencies, reduction, and suppression of Jean-like instabilities in the perpendicular direction, so the magnetic field played a stabilizing role for waves this result is consistent with those made in Refs. 71, 84. Finally, it can be concluded that, although pressure inhomogeneities, like the magnetic field, cannot eliminate Jeans-type instabilities on their own, they do help increase the stability of the waves.

Figure 1.

Evolution of the longitudinal dispersion relationship associated to the fast mode (a–c) and the corresponding slow mode (d–f) for different values of P_‖n. When (a) γ = 0, (b) γ = 0.3, and (c) γ = 0.6. Along with α = 1.5, β = 2, P_⊥n = 0.3, P_‖p = 0.6, P_⊥p = 0.4, μ _p = 0.53, and q = 0.7.

Figure 2.

Evolution of the transversal dispersion relationship associated to the fast mode (a–e) and the corresponding slow mode (d–h) for different values of P_⊥n. When (a) γ = 0 and ω _cn = 0.1, (b) γ = 0.3 and ω _cn = 0.1, (c) γ = 0.6 and ω _cn = 0.1, (d) γ = 0.6 and ω _cn = 0.4, (e) γ = 0.6 and ω _cn = 0.7. Along with α = 1.5, β = 2, P_‖n = 0.3, P_‖p = 0.6, P_⊥p = 0.4, μ _p = 0.53, and q = 0.7.

Derivation of wave interaction equations

Two small amplitude dust acoustic excitations, namely R and L, generated in two asymptotically distant plasma regions in the initial state and traveling obliquely toward each other, are assumed. After a finite time, they meet and interact quasi-elastically so that the only memory of their past overlap is the measurable phase shift between the pre-and post-collision trajectories. The approximate study of this exchange mechanism can be examined using the PLK method, ⁶⁷ which skillfully combines the strained coordinates technique and standard reductive perturbation method. Accordingly the stretched coordinates ξ, η, and τ are introduced as

$\xi =\epsilon (L_{x}x+L_{y}y+L_{z}z-{\lambda }_{\xi }t)+{\epsilon }^{2}P(\eta ,\tau )+\dots ,$

(11)

$\eta =\epsilon (M_{x}x+M_{y}y+M_{z}z+{\lambda }_{\eta }t)+{\epsilon }^{2}Q(\xi ,\tau )+\dots ,$

(12)

$\tau ={\epsilon }^{3}t.$

(13)

The dependent perturbed quantities around their equilibrium values are expanded as

${[n_j,u_{j_x},u_{j_y},u_{j_z},\varphi ,\psi ]}^T={[\mathrm{1,0,0,0,0,0}]}^T+\sum _{l=1}^{\infty }{\epsilon }^l{[n_{\mathit{j}}^{(l)},u_{j_x}^{(l)},u_{j_y}^{(l)},u_{j_z}^{(l)},{\varphi }^{(l)},{\psi }^{(l)}]}^T$

(14)

where ɛ ≪ 1 is a small dimensionless parameter, ξ, and η, respectively, denote the trajectories for the right and left moving wave. At P(η, τ) = Q(ξ, τ) = 0, solitons, respectively, follow the directions r(= L _x x + L _y y + L _z z) and r′(= M _x x + M _y y + M _z z). The unknown functions P(η, τ) and Q(ξ, τ) are the phase records of the interacting waves induced by collision. λ _ξ and λ _η refer to the wave’s phase velocity moving, respectively, to the right and left. The phase records and velocities are to be determined subsequently. Initial wave trajectories are characterized by vectors r _ξ (L_x, L_y, L_z) and r _η (M_x, M_y, M_z). L_(x,y,z) and M_(x,y,z) stand, respectively, for the direction cosines of the right wave vector and the left one along x-, y-, and z-axes so that $L_x^2$ + $L_y^2$ + $L_z^2=1$ , $M_x^2$ + $M_y^2$ + $M_z^2=1$ and colliding angle θ is defined by cos(θ) = L_xM_x + L_yM_y + L_zM_z. From equations (11)–(13) we find

$\left\{\begin{array}{c}{\partial }_t=\epsilon \widehat{T}+{\epsilon }^3{\widehat{T}}_1+{\epsilon }^3{\partial }_{\tau }+\dots ,\hfill \\ {\partial }_{x_s}=\epsilon {\widehat{\chi }}_{x_s}+{\epsilon }^3{\widehat{\chi }}_{{x_s}_1}+\dots ,\hfill \end{array}\right.$

(15)

where $\widehat{T}={\lambda }_{\eta }{\partial }_{\eta }-{\lambda }_{\xi }{\partial }_{\xi }$ ; ${\widehat{T}}_1={\lambda }_{\eta }{\partial }_{\eta }P{\partial }_{\xi }-{\lambda }_{\xi }{\partial }_{\xi }Q{\partial }_{\eta }$ ; ${\widehat{\chi }}_{x_s}=L_{x_s}{\partial }_{\xi }+M_{x_s}{\partial }_{\eta }$ and ${\widehat{\chi }}_{{x_s}_1}=L_{x_s}{\partial }_{\xi }Q{\partial }_{\eta }+M_{x_s}{\partial }_{\eta }P{\partial }_{\xi }$ . Inserting equations (14) and (15) into equations (3)–(8), and collecting the terms in different power of ɛ, the lowest-order read

$\begin{array}{c}\widehat{T}{n}_j^{(1)}+{\widehat{\chi }}_x{u}_{j_x}^{(1)}=0,\hfill \\ \widehat{T}{u}_{j_x}^{(1)}-{\sigma }_j{\widehat{\chi }}_x{\varphi }^{(1)}+{\widehat{\chi }}_x{\psi }^{(1)}+{\kappa }_j{P}_{\Vert j}{\widehat{\chi }}_x{n}_j^{(1)}=0,\hfill \\ {\sigma }_j{\omega }_{cn}{u}_{j_z}^{(1)}-{\sigma }_j{\widehat{\chi }}_y{\varphi }^{(1)}+{\widehat{\chi }}_y{\psi }^{(1)}+{\kappa }_j{P}_{\perp j}{\widehat{\chi }}_y{n}_j^{(1)}=0,\hfill \\ -{\sigma }_j{\omega }_{cn}{u}_{j_y}^{(1)}-{\sigma }_j{\widehat{\chi }}_z{\varphi }^{(1)}+{\widehat{\chi }}_z{\psi }^{(1)}+{\kappa }_j{P}_{\perp j}{\widehat{\chi }}_z{n}_j^{(1)}=0,\hfill \\ L_1{\varphi }^{(1)}+n_n^{(1)}-\mu n_p^{(1)}=0,\hfill \\ n_n^{(1)}+\frac{\mu \alpha }{\beta }{n}_p^{(1)}=0.\hfill \end{array}$

(16)

By eliminating $n_j^{(1)}$ , ψ⁽¹⁾ and $u_{j_x}^{(1)}$ in equation (16), it is found that

$\frac{4{\lambda }_{\eta }{\lambda }_{\xi }{L}_{1}B}{{\mu }_{p}A}{\partial }_{\xi \eta }^2{\varphi }^{(1)}=0$

(17)

under the condition that the phase velocities satisfied

${{\lambda }_{\xi }}^2=\frac{{L_{\text{x}}}^2\left(\alpha L_{1}C+\mu A^2\right)}{\alpha L_{1}B};{\lambda }_{\eta }=\frac{M_{\text{x}}{\lambda }_{\xi }}{L_{\text{x}}}.$

(18)

To the leading order, equation (17) indicates that two DA waves are expected to propagate towards each other because ϕ⁽¹⁾ can be expressed as the combination of two functions: one (ϕ_1ξ ≡ ϕ_1ξ(ξ, τ)) depends on ξ and τ, and the other (ϕ_1η ≡ ϕ_1η(η, τ)) depends on η and τ. Consequently, solutions of the set of equation (16) consistent with the requirements of equations (17) and (18) are as follows

$\begin{array}{cc}{\varphi }^{(1)}={\varphi }_{1\xi }+{\varphi }_{1\eta },\hfill & {\psi }^{(1)}=E({\varphi }_{1\xi }+{\varphi }_{1\eta }),\hfill \\ n_n^{(1)}=-\frac{\alpha L_1({\varphi }_{1\xi }+{\varphi }_{1\eta })}{A},\hfill & u_{n_y}^{(1)}=F{L}_{\text{z}}{\partial }_{\xi }{\varphi }_{1\xi }+F{M}_{\text{z}}{\partial }_{\eta }{\varphi }_{1\eta },\hfill \\ n_p^{(1)}=\frac{\beta L_1({\varphi }_{1\xi }+{\varphi }_{1\eta })}{\mu A},\hfill & u_{p_y}^{(1)}=G{L}_{\text{z}}{\partial }_{\xi }{\varphi }_{1\xi }+G{M}_{\text{z}}{\partial }_{\eta }{\varphi }_{1\eta },\hfill \\ u_{n_x}^{(1)}=-\frac{\left({\varphi }_{1\xi }{\lambda }_{\xi }{M}_{\text{x}}-{\varphi }_{1\eta }{\lambda }_{\eta }{L}_{\text{x}}\right)\alpha L_1}{L_{\text{x}}{M}_{\text{x}}A},\hfill & u_{n_z}^{(1)}=-F{L}_{\text{y}}{\partial }_{\xi }{\varphi }_{1\xi }-F{M}_{\text{y}}{\partial }_{\eta }{\varphi }_{1\eta },\hfill \\ u_{p_x}^{(1)}=\frac{\left({\varphi }_{1\xi }{\lambda }_{\xi }{M}_{\text{x}}-{\varphi }_{1\eta }{\lambda }_{\eta }{L}_{\text{x}}\right)\beta L_1}{\mu L_{\text{x}}{M}_{\text{x}}A},\hfill & u_{p_z}^{(1)}=-G{L}_{\text{y}}{\partial }_{\xi }{\varphi }_{1\xi }-G{M}_{\text{y}}{\partial }_{\eta }{\varphi }_{1\eta }.\hfill \end{array}$

(19)

To the next higher-order, the governing equations (3)–(8) generate the new system

$\begin{array}{c}\widehat{T}{n}_j^{(2)}+{\widehat{\chi }}_z{u}_{j_z}^{(1)}+{\widehat{\chi }}_x{u}_{j_x}^{(2)}+{\widehat{\chi }}_y{u}_{j_y}^{(1)}+{\widehat{\chi }}_x(u_{j_x}^{(1)}{n}_j^{(1)})=0,\hfill \\ \widehat{T}{u}_{j_x}^{(2)}+{\widehat{\chi }}_x{\psi }^{(2)}-{\sigma }_j{\widehat{\chi }}_x{\varphi }^{(2)}+{\kappa }_j{P}_{\Vert j}{\widehat{\chi }}_x{n}_j^{(2)}+{\kappa }_j{P}_{\Vert j}{n}_j^{(1)}{\widehat{\chi }}_x{n}_j^{(1)}+u_{j_x}^{(1)}{\widehat{\chi }}_x{u}_{j_x}^{(1)}=0,\hfill \\ \widehat{T}{u}_{j_y}^{(1)}+{\widehat{\chi }}_y{\psi }^{(2)}-{\sigma }_j{\widehat{\chi }}_y{\varphi }^{(2)}+{\kappa }_j{P}_{\perp j}{\widehat{\chi }}_y{n}_j^{(2)}-{\kappa }_j{P}_{\perp j}{n}_j^{(1)}{\widehat{\chi }}_y{n}_j^{(1)}+{\sigma }_j{\omega }_{cn}{u}_{j_z}^{(2)}=0,\hfill \\ \widehat{T}{u}_{j_z}^{(1)}+{\widehat{\chi }}_z{\psi }^{(2)}-{\sigma }_j{\widehat{\chi }}_z{\varphi }^{(2)}+{\kappa }_j{P}_{\perp j}{\widehat{\chi }}_z{n}_j^{(2)}-{\kappa }_j{P}_{\perp j}{n}_j^{(1)}{\widehat{\chi }}_z{n}_j^{(1)}-{\sigma }_j{\omega }_{cn}{u}_{j_y}^{(2)}=0,\hfill \\ n_n^{(2)}+\frac{\mu \alpha }{\beta }{n}_p^{(2)}=0,\hfill \\ -L_1{\varphi }_2-L_2{\varphi }_1^2+\mu n_p^{(2)}-n_n^{(2)}=0.\hfill \end{array}$

(20)

By eliminating $n_j^{(2)}$ , $u_{j_x}^{(2)}$ , and ψ⁽²⁾ in equation (20), the following result is obtained

$\begin{array}{c}-\frac{\left(\alpha \left(\alpha P_{\Vert n}\mu \left(4\alpha \mu -3\beta \right)+{\beta }^2{P}_{\Vert p}\left(3\alpha \mu -4\beta \right)\right)L_1+\mu A^2\left(\alpha \mu -\beta \right)\right)L_1}{A^3{\mu }^2}{\partial }_{\xi }{\varphi }_{1\xi }{\partial }_{\eta }{\varphi }_{1\eta }\hfill \\ +\frac{\left(\alpha \mu -\beta \right)L_1}{\mu A{M}_{\text{x}}{L}_{\text{x}}}(L_{\text{x}}^2{\varphi }_{1\eta }{\partial }_{\xi \xi }^2{\varphi }_{1\xi }+M_{\text{x}}^2{\varphi }_{1\xi }{\partial }_{\eta \eta }^2{\varphi }_{1\eta })+{\partial }_{\xi \eta }^2{\varphi }^{(2)}=0,\hfill \end{array}$

(21)

and

$\begin{array}{c}(\delta -L_2){\varphi }_{1\xi ,1\eta }=0.\hfill \end{array}$

(22)

Equation (21) suggests that ϕ⁽²⁾ has a term $({\tilde{\varphi }}_2(\xi ,\eta ,\tau ))$ that depends on τ, ξ and η in a way that is not possible to disentangled. Coefficients A, B, E, F, G, and δ are provided in the Appendix. Using equations (18), (19), and (21) the disturbed quantities of the second order are expressed as follows

$\begin{array}{cc}{\varphi }^{(2)}={\varphi }_{2\xi }+{\varphi }_{2\eta }+{\tilde{\varphi }}_2,\hfill & {\psi }^{(2)}=H_1({\varphi }_{1\xi }^2+{\varphi }_{1\eta }^2)+H_2({\varphi }_{2\xi }+{\varphi }_{2\eta })+{\tilde{\psi }}_2,\hfill \\ n_n^{(2)}=H_3({\varphi }_{1\xi }^2+{\varphi }_{1\eta }^2)+\frac{{L_{\text{x}}}^2\mu A({\varphi }_{2\xi }+{\varphi }_{2\eta })}{-B{{\lambda }_{\xi }}^2+C{L_{\text{x}}}^2}+{\tilde{n}}_{2n},\hfill & n_p^{(2)}=\frac{-\beta }{\alpha \mu }{H}_3({\varphi }_{1\xi }^2+{\varphi }_{1\eta }^2)-\frac{\beta {L_{\text{x}}}^{2}A({\varphi }_{2\xi }+{\varphi }_{2\eta })}{\left(-B{{\lambda }_{\xi }}^2+C{L_{\text{\hspace{0.17em}x}}}^2\right)\alpha }+{\tilde{n}}_{2p},\hfill \\ u_{n_x}^{(2)}=H_6({\varphi }_{1\eta }^2-{\varphi }_{1\xi }^2)-\frac{L_{\text{x}}{\lambda }_{\xi }\mu A({\varphi }_{2\eta }-{\varphi }_{2\xi })}{-B{{\lambda }_{\xi }}^2+C{L_{\text{x\hspace{0.17em}}}}^2}+{\tilde{u}}_{2n_x},\hfill & u_{p_x}^{(2)}=H_7({\varphi }_{1\eta }^2-{\varphi }_{1\xi }^2)+\frac{L_{\text{x}}{\lambda }_{\xi }\beta A({\varphi }_{2\eta }-{\varphi }_{2\xi })}{\left(-B{{\lambda }_{\xi }}^2+C{L_{\text{x}}}^2\right)\alpha }+{\tilde{u}}_{2p_x},\hfill \end{array}$

(23)

$\begin{array}{cc}{u}_{n_y}^{(2)}=\hfill & H_4(L_z{\varphi }_{1\xi }{\partial }_{\xi }{\varphi }_{1\xi }+M_z{\varphi }_{1\eta }{\partial }_{\eta }{\varphi }_{1\eta })+\frac{{\lambda }_{\xi }{I}_3}{L_{\text{x}}{{\omega }_{cn}}^{2}AB}(L_y{L}_x{\partial }_{\xi }^2{\varphi }_{1\xi }-M_x{M}_y{\partial }_{\eta }^2{\varphi }_{1\eta })\hfill \\ \hfill & -\frac{\mu A\left({L_{\text{x}}}^2\left(P_{\Vert n}-P_{\perp n}\right)-{{\lambda }_{\xi }}^2\right)}{\left(-B{{\lambda }_{\xi }}^2+C{L_{\text{x}}}^2\right){\omega }_{cn}}(L_z{\partial }_{\xi }{\varphi }_{2\xi }+M_z{\partial }_{\eta }{\varphi }_{2\eta })+{\tilde{u}}_{2n_y},\hfill \\ u_{p_y}^{(2)}=\hfill & H_5(L_z{\varphi }_{1\xi }{\partial }_{\xi }{\varphi }_{1\xi }+M_z{\varphi }_{1\eta }{\partial }_{\eta }{\varphi }_{1\eta })+\frac{{\lambda }_{\xi }{I}_6\alpha }{L_{\text{x}}\mu {{\omega }_{cn}}^{2}AB\beta }(L_y{L}_x{\partial }_{\xi }^2{\varphi }_{1\xi }-M_x{M}_y{\partial }_{\eta }^2{\varphi }_{1\eta })\hfill \\ \hfill & -\frac{A\left({L_{\text{x}}}^2\beta \left(P_{\Vert p}-P_{\perp p}\right)-{{\lambda }_{\xi }}^2\right)}{\left(-B{{\lambda }_{\xi }}^2+C{L_{\text{x}}}^2\right){\omega }_{cn}}(L_z{\partial }_{\xi }{\varphi }_{2\xi }+M_z{\partial }_{\eta }{\varphi }_{2\eta })+{\tilde{u}}_{2p_y},\hfill \\ u_{n_z}^{(2)}=\hfill & -H_4(L_y{\varphi }_{1\xi }{\partial }_{\xi }{\varphi }_{1\xi }+M_y{\varphi }_{1\eta }{\partial }_{\eta }{\varphi }_{1\eta })+\frac{{\lambda }_{\xi }{I}_3}{L_{\text{x}}{{\omega }_{cn}}^{2}AB}(L_z{L}_x{\partial }_{\xi }^2{\varphi }_{1\xi }-M_x{M}_z{\partial }_{\eta }^2{\varphi }_{1\eta })\hfill \\ \hfill & +\frac{\mu A\left({L_{\text{x}}}^2\beta \left(P_{\Vert n}-P_{\perp n}\right)-{{\lambda }_{\xi }}^2\right)}{\left(-B{{\lambda }_{\xi }}^2+C{L_{\text{x}}}^2\right){\omega }_{cn}}(L_y{\partial }_{\xi }{\varphi }_{2\xi }+M_y{\partial }_{\eta }{\varphi }_{2\eta })+{\tilde{u}}_{2n_z},\hfill \\ u_{p_z}^{(2)}=\hfill & -H_5(L_y{\varphi }_{1\xi }{\partial }_{\xi }{\varphi }_{1\xi }+M_y{\varphi }_{1\eta }{\partial }_{\eta }{\varphi }_{1\eta })+\frac{{\lambda }_{\xi }{I}_6\alpha }{L_{\text{x}}{\mu }_p{{\omega }_{cn}}^{2}AB\beta }(L_z{L}_x{\partial }_{\xi }^2{\varphi }_{1\xi }-M_x{M}_z{\partial }_{\eta }^2{\varphi }_{1\eta })\hfill \\ \hfill & +\frac{A\left({L_{\text{x}}}^2\beta \left(P_{\Vert p}-P_{\perp p}\right)-{{\lambda }_{\xi }}^2\right)}{\left(-B{{\lambda }_{\xi }}^2+C{L_{\text{x}}}^2\right){\omega }_{cn}}(L_y{\partial }_{\xi }{\varphi }_{2\xi }+M_y{\partial }_{\eta }{\varphi }_{2\eta })+{\tilde{u}}_{2p_z}.\hfill \end{array}$

Equation (22) generates two independent situations; the first is where ϕ_1ξ = ϕ_1η = 0 and leads to the KdV equation, the second invalidates the KdV equation and results from an exceptional composition of the plasma (L₂ = δ), this state gives rise to the mKdV equation. Coefficients H₁₋₇ and I₁₋₆ are available in the appendix. The following subsections are devoted to the in-depth study of these eventualities.

Two-sided KdV equations and phase shifts

Let’s first look at the generic case (ϕ⁽¹⁾ = 0) thus the first-order state variables as well as the second-order mixed terms cancel; the calculations are therefore shortened, and the third-order gives

$\begin{array}{c}\widehat{T}{n}_j^{(3)}+{\widehat{\chi }}_x{u}_{j_x}^{(3)}=0,\hfill \\ \widehat{T}{u}_{j_x}^{(2)}+{\widehat{\chi }}_x{\psi }^{(3)}-{\sigma }_j{\widehat{\chi }}_x{\varphi }^{(3)}+{\kappa }_j{P}_{\Vert j}{\widehat{\chi }}_x{n}_j^{(3)}=0,\hfill \\ \widehat{T}{u}_{j_y}^{(2)}+{\widehat{\chi }}_y{\psi }^{(3)}-{\sigma }_j{\widehat{\chi }}_y{\varphi }^{(3)}+{\kappa }_j{P}_{\perp j}{\widehat{\chi }}_y{n}_j^{(3)}+{\sigma }_j{\omega }_{cn}{u}_{j_z}^{(3)}=0,\hfill \\ \widehat{T}{u}_{j_z}^{(2)}+{\widehat{\chi }}_z{\psi }^{(3)}-{\sigma }_j{\widehat{\chi }}_z{\varphi }^{(3)}+{\kappa }_j{P}_{\perp j}{\widehat{\chi }}_z{n}_j^{(3)}-{\sigma }_j{\omega }_{cn}{u}_{j_y}^{(3)}=0,\hfill \\ n_n^{(3)}+\frac{\mu \alpha }{\beta }{n}_p^{(3)}=0,\hfill \\ -L_1{\varphi }_3+\mu n_p^{(3)}-n_n^{(3)}=0.\hfill \end{array}$

(24)

A hasty comparison between equations (16) and (24) shows that the third-order presents, just like, the first-order equations of linear form. Moreover, the state variables of this order do not occur at the next order; one can choose $n_j^{(3)}=u_{j_y}^{(3)}={\psi }^{(3)}={\varphi }^{(3)}=0$ . Hence, the fourth-order gives

$\begin{array}{c}\widehat{T}{n}_j^{(4)}+{\widehat{T}}_1{n}_j^{(2)}+{\widehat{\chi }}_{x1}{u}_{j_x}^{(2)}+{\widehat{\chi }}_x{u}_{j_x}^{(4)}+{\widehat{\chi }}_x(u_{j_x}^{(2)}{n}_j^{(2)})+{\partial }_{\tau }{n}_j^{(2)}=0,\hfill \\ \widehat{T}{u}_{j_x}^{(4)}+{\widehat{\chi }}_x{\psi }^{(4)}+{\widehat{\chi }}_{x1}{\psi }^{(2)}-{\sigma }_j({\widehat{\chi }}_x{\varphi }^{(4)}+{\widehat{\chi }}_{x1}{\varphi }^{(2)})+{\kappa }_j{P}_{\Vert j}({\widehat{\chi }}_x{n}_j^{(4)}+{\widehat{\chi }}_{x1}{n}_j^{(2)}+n_j^{(2)}{\widehat{\chi }}_x{n}_j^{(2)})\hfill \\ +{\widehat{T}}_1{u}_{j_x}^{(2)}+u_{j_x}^{(2)}{\widehat{\chi }}_x{u}_{j_x}^{(2)}+{\partial }_{\tau }{u}_{j_x}^{(2)}=0,\hfill \\ -\gamma n_n^{(4)}-\frac{\gamma \mu \alpha }{\beta }{n}_p^{(4)}+2\cos (\theta ){\partial }_{\xi \eta }^2{\psi }^{(2)}+{\partial }_{\xi \xi }^2{\psi }^{(2)}+{\partial }_{\eta \eta }^2{\psi }^{(2)}=0,\hfill \\ -n_n^{(4)}+\mu n_p^{(4)}+2\cos (\theta ){\partial }_{\xi \eta }^2{\varphi }^{(2)}+{\partial }_{\xi \xi }^2{\varphi }^{(2)}+{\partial }_{\eta \eta }^2{\varphi }^{(2)}-L_1{\varphi }^{(4)}-L_2{[{\varphi }^{(2)}]}^2=0.\hfill \end{array}$

(25)

Using equation (25) with the suitably simplified solutions of equation (23), one obtains after some straightforward calculations the two-sided KdV equations which, respectively, control the oblique dynamics of the right- and left-going DAWs in addition with phase functions equations

$\left\{\begin{array}{c}{\partial }_{\tau }{\varphi }_{2\xi }+S_1{\varphi }_{2\xi }{\partial }_{\xi }{\varphi }_{2\xi }+T_1{\partial }_{\xi }^3{\varphi }_{2\xi }=0,\hfill \\ {\partial }_{\tau }{\varphi }_{2\eta }-S_1^{\prime }{\varphi }_{2\eta }{\partial }_{\eta }{\varphi }_{2\eta }-T_1^{\prime }{\partial }_{\eta }^3{\varphi }_{2\eta }=0,\hfill \end{array}\right.$

(26)

and

$\left\{\begin{array}{c}{\partial }_{\xi }Q(\xi ,\tau )=U_1{\varphi }_{2\xi },\hfill \\ {\partial }_{\eta }P(\eta ,\tau )=V_1{\varphi }_{2\eta }.\hfill \end{array}\right.$

(27)

Coefficients S₁, $S_1^{\prime }$ , T₁, $T_1^{\prime }$ , U₁ and V₁ are detailed in the appendix. The one-soliton solutions of equation (26) can be written as

$\left\{\begin{array}{c}{\varphi }_{2\xi }={\mathrm{\Phi }}_{2\xi }sec{h}^2\left(\frac{\xi -{\lambda }_{{\xi }_0}\tau }{W_{\xi }}\right),\hfill \\ {\varphi }_{2\eta }={\mathrm{\Phi }}_{2\eta }sec{h}^2\left(\frac{\eta +{\lambda }_{{\eta }_0}\tau }{W_{\eta }}\right),\hfill \end{array}\right.$

(28)

where ${\mathrm{\Phi }}_{2\xi }=3{\lambda }_{{\xi }_0}/S_1$ , ${\mathrm{\Phi }}_{2\eta }=3{\lambda }_{{\eta }_0}/S_1^{\prime }$ , $W_{\xi }=\sqrt{4T_1/{\lambda }_{{\xi }_0}}$ , $W_{\eta }=\sqrt{4T_1^{\prime }/{\lambda }_{{\eta }_0}}$ , ${\lambda }_{{\xi }_0}$ , and ${\lambda }_{{\eta }_0}$ stand for the amplitudes, widths, and the constant velocity of right and left running DA solitons at their initial positions, respectively. Figure 3 facilitates the polarity exploration of soliton propagating in the medium for a specific constitution. It is clear from this figure that the plasma supports rarefactive and compressive solitons. Figure 3 shows that dispersion increases (decreases) as gravitation (parallel dust pressure) increases. Moreover, nonlinearity grows with the enhancement of negative dust grain mass. For γ < 0.1 (γ > 0.1), T₁ is negative (positive), indicating that ${\lambda }_{{\eta }_0,{\xi }_0}<0$ $({\lambda }_{{\eta }_0,{\xi }_0}>0)$ as shown in Figure 3(a). Thus, examining Figure 3(b) for q = 1.6, it is evident that the nonlinear coefficient S₁ is negative for β < β _crit . Consequently, for such a configuration, only compressive (rarefactive) DA solitons exist in the medium. While for β > β _crit , S₁ is positive and only rarefactive (compressive) structure is expecting in the plasma. Choosing q = 1.6, β = β _crit = 0.45 corresponds to S₁ = 0 and Φ_2ξ,2η → ∞, leading to the breakdown of KdV solitons and their failure to describe DAWs. Curve observations also indicate that β_crit decreases with decreasing ion nonextensivity. Additionally, for low values of q, the polarity is determined exclusively by the sign of the dispersion. Section B is entirely devoted to clarifying the phenomena occurring at the critical point. By inserting equation (28) into equation (27), the phase changes due to the collision given

$\left\{\begin{array}{c}Q(\xi ,\tau )=U_1{W}_{\xi }{\mathrm{\Phi }}_{2\xi }\left[\tanh \left(\frac{\xi -{\lambda }_{{\xi }_0}\tau }{W_{\xi }}\right)-1\right],\hfill \\ P(\eta ,\tau )=V_1{W}_{\eta }{\mathrm{\Phi }}_{2\eta }\left[\tanh \left(\frac{\eta +{\lambda }_{{\eta }_0}\tau }{W_{\eta }}\right)+1\right].\hfill \end{array}\right.$

(29)

Assuming two solitons asymptotically distant from each other at the beginning and the end of the interaction, that is, when τ → −∞, Π _R is at ξ = 0, η = −∞ and Π _L is at ξ = +∞, η = 0, however, when τ → +∞, Π _R is at ξ = 0, η = +∞ and Π _L is at ξ = −∞, η = 0, The following phase shifts are obtained

$\left\{\begin{array}{c}\mathrm{\Delta }Q\left(\xi ,\tau \right)=2U_1{W}_{\xi }{\mathrm{\Phi }}_{2\xi }{\epsilon }^2,\hfill \\ \mathrm{\Delta }P\left(\eta ,\tau \right)=-2V_1{W}_{\eta }{\mathrm{\Phi }}_{2\eta }{\epsilon }^2.\hfill \end{array}\right.$

(30)

Figure 3.

(a) Coefficient of dispersion, T₁ against γ for different values of P_‖n, with β = 2 and q = 1.6 (b) Coefficient of nonlinearity, S₁ against β for different values of q with γ = 0.6 and P_‖n = 0.5. Along with α = 1.5, P_⊥n = 0.3, P_‖p = 0.6, P_⊥p = 0.4, and μ _p = 0.53.

ΔP and ΔQ are witnesses of the energy exchanged during the colliding process, for this purpose, they can also be expressed in terms of the right or the left soliton energy (E _R , E _L ) by evaluating integral $E_R={\int }_{-\infty }^{+\infty }{\varphi }_{2\xi }^2\mathrm{d}\mathrm{\xi }$ and $E_L={\int }_{-\infty }^{+\infty }{\varphi }_{2\eta }^2\mathrm{d}\mathrm{\eta }$ . It is found that $\mathrm{\Delta }Q\left(\xi ,\tau \right)=3U_1{\epsilon }^2/2{\mathrm{\Phi }}_{2\xi }{E}_R$ and $\mathrm{\Delta }P\left(\eta ,\tau \right)=-3V_1{\epsilon }^2/2{\mathrm{\Phi }}_{2\eta }{E}_L$ with $E_R=4/3W_{\xi }{\mathrm{\Phi }}_{2\xi }^2$ and $E_L=4/3W_{\eta }{\mathrm{\Phi }}_{2\eta }^2$ . Thus, the conserving law for solitons having the same amplitude can be deduced

$\mathrm{\Delta }P\left(\eta ,\tau \right)+\mathrm{\Delta }Q\left(\xi ,\tau \right)=E_L-E_R=0.$

(31)

Equation (31) instructs us that overlapping solitons of the same energies undergo deviations of the same magnitude concerning their initial trajectories. Recalling that a positive (negative) phase shift means a loss (gain) of soliton velocity so that the postcollisions parts of the soliton lags behind (move ahead) the initial trajectory. Figures 4(a) and (b) elaborate the impact of dimensionless parameters γ, P_‖n, and β on the phase shift. From Figure 4(a), it is found that the phase shift overgrows to reach a maximum for weak gravitational fields; this peak remains unchanged for any further growth in gravity intensity. The figure also indicates that the phase shift decreases in magnitude as the parallel dust pressure increases. Moreover, inspecting Figure 4(b) reveals that on either side of the critical point previously identified in Figure 3(b), the monotony of the phase shift is reversed. Specifically, when β < β_crit, the phase shift increases sharply as the negative dust grain mass increases, while it drops suddenly and is canceled when β > β_crit. As expected, the model loses its relevance near the critical point (ΔP → ∞). Figure 5 highlights the sensitivity of ΔP to parameters q and θ. Figure 5(a) shows that excess of nonextensivity thins the phase shift. Furthermore, as the angle between wave vectors involved in the collision increases within 0 < θ < π/2 (acute oblique interaction), the interaction generates a positive and ever-decreasing phase shift, conversely as the crossing angle increases between the bounds π/2 < θ ≤ π (obtuse oblique interaction) a negative phase shift, whose magnitude increases and reaches its peak during a frontal collision (θ = π), is observed. A zero phase shift is noted at θ = π/2 (see Figure 5(b)). These findings are consistent with both experimental and theoretical studies on soliton collisions.^64,85 It should be noted that the figure exhibits an inherent symmetry as supplementary crossing angles result in the same absolute phase shift. Altogether, the exchanges of energy during impact are minimal in the vicinity of the right angle. For the sake of completeness, Figure 5(c) has been drawn; it gives a more enlightened overview by exposing three distinct angles (7π/12, 3π/2, and π) the evolution of the difference between the trajectories very long time before the collision and very long time after the collision.

Figure 4.

Variation of colliding phase shift (Δ _p ) (a) against γ with β = 2, (b) against β with γ = 0.6 for different values of P_‖n. Along with α = 1.5, P_⊥n = 0.3, P_‖p = 0.6, P_⊥p = 0.4, q = 1.6, μ _p = 0.53, ${\lambda }_{{\xi }_0}=0.01$ , ${\lambda }_{{\eta }_0}=0.02$ , ɛ = 0.5, and θ = π/2.55.

Figure 5.

Variation of colliding phase shift (Δ _p ) (a) against q with θ = π/2.55, (b) against θ with q = 1.6 for different values of P_‖n. (c) Evolution of the shift between incoming and outgoing trajectories of the right-running soliton against the crossing angle. Along with γ = 0.6 α = 1.5, P_⊥n = 0.3, P_‖p = 0.6, P_⊥p = 0.4, μ _p = 0.53, ${\lambda }_{{\xi }_0}=0.01$ , ${\lambda }_{{\eta }_0}=0.02$ , and ɛ = 0.5.

Using the parameters conducive to the propagation of hump or dip type solitons as identified in Figure 3, Figure 6 presents the oblique interaction scenario of compressive-compressive (rarefactive-rarefactive) solitons moving with different amplitudes when the collision angle is π/2.55. It is observed from Figure 6(a) that far from the interaction zone, the two solitons propagate towards each other with a constant profile (amplitude and width), and velocities $({\lambda }_{{\xi }_0}=0.01,{\lambda }_{{\eta }_0}=0.02)$ . As the solitons approach, the kinetic energy they carry undergoes conversion (velocity redistribution) into potential energy, giving rise to a distorted potential that gradually grows and refines to take the contours of a familiar nonlinear structure when all the kinetic energy has been converted into potential energy. This newly formed spontaneous structure exhibiting an amplitude and width that surpass those of incoming solitons (reminiscent of RWs) marks the end of the brief first stage of collision, characterized by a loss of velocity in incoming solitons, and the beginning of the second slower stage, characterized by a gain in velocity. Shortly after its formation, this ephemeral dissipates, leading again to an uncouth and composite potential that gradually splits into two distinct solitons. At the same time, the intensity and width decrease progressively as potential energy is converted back into kinetic energy due to the soliton’s separation. Ultimately, the post-collision solitons, having exchanged their positions, move away from each other while preserving their original shapes and velocities but following trajectories different from their initial paths. Numerical analysis reveals a non-zero cumulative phase shift (ΔP + ΔQ = 0.0071). Specifically, the slower soliton experiences a more significant phase shift (ΔP = 0.0242) than the faster soliton (ΔQ = −0.0171). This discrepancy arises from the longer interaction time experienced by the slower soliton relative to the faster one. Generally speaking, the phase shift experienced by solitons is commonly attributed to the deceleration and subsequent acceleration experienced during the overlap. ⁶⁴ Figure 6(c) provides a more comprehensive insight into the collision process detailed in Figure 6(a). The previous description for positive pulses holds true for negative pulses as well, with the only difference being that ΔP = −0.1511, ΔQ = 0.1068, and consequently ΔP + ΔQ = −0.0443 (see Figures 6(b) and (d)).

Figure 6.

Time evolution and corresponding colliding profiles of two compressive DASWs (a, c) for β = 2 and two rarefactive DASWs (b, d) for β = 0.1. The parameters are set as α = 1.5, P_‖n = 0.5, P_⊥n = 0.3, P_‖p = 0.6, P_⊥p = 0.4, q = 1.6, μ _p = 0.53, γ = 0.6, ${\lambda }_{{\xi }_0}=0.01$ , ${\lambda }_{{\eta }_0}=0.02$ , ɛ = 0.5, and θ = π/2.55. ΔP + ΔQ ≠ 0.

Two-sided mKdV equations and phase shifts at the critical composition

The formation of K-dV solitons is not valid for the parametric regime corresponding to L₂ = δ, because for this eventuality, $S_1=S_1^{\prime }=0$ , and nonlinearity cannot balances dispersion. Therefore, it is necessary to consider higher-order nonlinearity to determine the evolution laws of waves specific to this configuration. Without distorting the reasoning, solutions of the linear operator (21) can be canceled (ϕ_2ξ = ϕ_2η = 0) while retaining all the mixed terms ( ${\tilde{\varphi }}_2\ne 0$ , that is, ${\tilde{\psi }}_2,{\tilde{n}}_{2j}$ and ${\tilde{u}}_{2j_{x_s}}$ differ from zero). Consequently, equations (3)–(8) reduce to

$\begin{array}{c}\widehat{T}{n}_j^{(3)}+{\widehat{\chi }}_x{u}_{j_x}^{(3)}+{\widehat{\chi }}_z{u}_{j_z}^{(2)}+{\widehat{\chi }}_y{u}_{j_y}^{(2)}+{\widehat{\chi }}_{x1}{u}_{j_x}^{(1)}+{\widehat{T}}_1{n}_j^{(1)}+{\widehat{\chi }}_x(u_{j_x}^{(1)}{n}_j^{(2)})+{\widehat{\chi }}_x(u_{j_x}^{(2)}{n}_j^{(1)})\hfill \\ +{\widehat{\chi }}_y(u_{j_y}^{(1)}{n}_j^{(1)})+{\widehat{\chi }}_z(u_{j_z}^{(1)}{n}_j^{(1)})+{\partial }_{\tau }{n}_j^{(1)}=0,\hfill \\ \widehat{T}{u}_{j_x}^{(3)}+{\widehat{T}}_1{u}_{j_x}^{(1)}-{\sigma }_j({\widehat{\chi }}_x{\varphi }^{(3)}+{\widehat{\chi }}_{x1}{\varphi }^{(1)})+{\widehat{\chi }}_x{\psi }^{(3)}+{\kappa }_j{P}_{\Vert j}\left({\widehat{\chi }}_x{n}_j^{(3)}+{\widehat{\chi }}_{x1}{n}_j^{(1)}+{\widehat{\chi }}_x(n_j^{(2)}{n}_j^{(1)})\right.\hfill \\ +u_{j_x}^{(1)}{\widehat{\chi }}_x{u}_{j_x}^{(2)}+u_{j_y}^{(1)}{\widehat{\chi }}_y{u}_{j_x}^{(1)}+u_{j_x}^{(2)}{\widehat{\chi }}_x{u}_{j_x}^{(1)}+u_{j_z}^{(1)}{\widehat{\chi }}_z{u}_{j_x}^{(1)}+{\widehat{\chi }}_{x1}{\psi }^{(1)}+{\partial }_{\tau }{u}_{j_x}^{(1)}=0,\hfill \\ -\gamma n_n^{(3)}-\frac{\gamma \mu \alpha }{\beta }{n}_p^{(3)}+2\cos (\theta ){\partial }_{\xi \eta }^2{\psi }^{(1)}+{\partial }_{\xi \xi }^2{\psi }^{(1)}+{\partial }_{\eta \eta }^2{\psi }^{(1)}=0,\hfill \\ -n_n^{(3)}+\mu n_p^{(3)}+2\cos (\theta ){\partial }_{\xi \eta }^2{\varphi }^{(1)}+{\partial }_{\xi \xi }^2{\varphi }^{(1)}+{\partial }_{\eta \eta }^2{\varphi }^{(1)}-L_1{\varphi }^{(3)}-2L_2{\varphi }^{(1)}{\varphi }^{(2)}-L_3{[{\varphi }^{(1)}]}^3=0.\hfill \end{array}$

(32)

By applying the constraints mentioned above $({\varphi }_{2\xi }={\varphi }_{2\eta }=0,{\tilde{\varphi }}_2\ne 0)$ to solutions (23) and equation (32), and after some cumbersome mathematical calculations, the two-sided mKdV equations of the right- and the left-running DAWs are obtained, along with the corresponding phase functions equations

$\left\{\begin{array}{c}{\partial }_{\tau }{\varphi }_{1\xi }+S_2{\varphi }_{1\xi }^2{\partial }_{\xi }{\varphi }_{1\xi }+T_2{\partial }_{\xi \xi \xi }^3{\varphi }_{1\xi }=0,\hfill \\ {\partial }_{\tau }{\varphi }_{1\eta }-S_2^{\prime }{\varphi }_{1\eta }^2{\partial }_{\eta }{\varphi }_{1\eta }-T_2^{\prime }{\partial }_{\eta \eta \eta }^3{\varphi }_{1\eta }=0,\hfill \end{array}\right.$

(33)

$\left\{\begin{array}{c}{\partial }_{\xi }Q(\xi ,\tau )=U_2{\varphi }_{1\xi },\hfill \\ {\partial }_{\eta }P(\eta ,\tau )=V_2{\varphi }_{1\eta },\hfill \end{array}\right.$

(34)

where S₂, $S_2^{\prime }$ , T₂, $T_2^{\prime }$ , U₂ and V₂ are given in the Appendix. As is well known, the mKdV equations given in equation (33) have the following one-soliton solutions

$\left\{\begin{array}{c}{\varphi }_{1\xi }=\pm {\mathrm{\Phi }}_{1\xi }sech\left(\frac{\xi -{\lambda }_{{\xi }_0}\tau }{W_{\xi }^{\prime }}\right),\hfill \\ {\varphi }_{1\eta }=\pm {\mathrm{\Phi }}_{1\eta }sech\left(\frac{\eta +{\lambda }_{{\eta }_0}\tau }{W_{\eta }^{\prime }}\right).\hfill \end{array}\right.$

(35)

Here, the following definition is used: ${\mathrm{\Phi }}_{1\xi }=\sqrt{6{\lambda }_{{\xi }_0}/S_2}$ , ${\mathrm{\Phi }}_{1\eta }=\sqrt{6{\lambda }_{{\eta }_0}/S_2^{\prime }}$ , $W_{\xi }^{\prime }=\sqrt{T_2/{\lambda }_{{\xi }_0}}$ , and $W_{\eta }^{\prime }=\sqrt{T_2^{\prime }/{\lambda }_{{\eta }_0}}$ . Here, signs are not coupled ⁸⁵ because equation (33) is invariant for sign permutation of ϕ_1ξ and ϕ_1η. From equations (11), (12), (34), and (35), the leading phase functions and the phase shifts associated with two asymptotically distant mKdV solitons can be deduced as follows

$\left\{\begin{array}{c}{Q}^{\prime }(\xi ,\tau )=U_2{W}_{\xi }^{\prime }{\mathrm{\Phi }}_{1\xi }^2\left[\tanh \left(\frac{\xi -{\lambda }_{{\xi }_0}\tau }{W_{\xi }^{\prime }}\right)-1\right],\hfill \\ P^{\prime }(\eta ,\tau )=V_2{W}_{\eta }^{\prime }{\mathrm{\Phi }}_{1\eta }^2\left[\tanh \left(\frac{\eta +{\lambda }_{{\eta }_0}\tau }{W_{\eta }^{\prime }}\right)+1\right].\hfill \end{array}\right.$

(36)

$\left\{\begin{array}{c}\mathrm{\Delta }{Q}^{\prime }\left(\xi ,\tau \right)=2U_2{W}_{\xi }^{\prime }{\mathrm{\Phi }}_{1\xi }^2{\epsilon }^2,\hfill \\ \mathrm{\Delta }{P}^{\prime }\left(\eta ,\tau \right)=-2V_2{W}_{\eta }^{\prime }{\mathrm{\Phi }}_{1\eta }^2{\epsilon }^2.\hfill \end{array}\right.$

(37)

In contrast to the phase shift experienced by KdV-type solitons, the phase shift of mKdV-type solitons depends on the perpendicular dust pressure and magnetic field. Hence we have constructed Figure 7 to access the effects of parameters γ, P_‖n, P_⊥n, ω _cn , q, and θ on ΔP′ under critical plasma conditions. The selected parameters β _crit = 0.45, q = 1.6, align with the scenario depicted in Figure 3(b)). As already mentioned, when gravity increases, the phase shift rapidly escalates for weak field intensities, reaching a maximum value that remains unaltered regardless of further increases. Conversely, as parallel or perpendicular dust pressure intensifies, the phase shift between trajectories grows. Yet, parallel pressure induces a more pronounced phase shift compared to perpendicular pressure (see Figures 7(a) and (b)). The magnetic field’s effects are not to be underestimated, as it engenders a reduction of the path separation, as indicated by Figure 7(c). This result aligns with that of El-Labany et al. ⁷⁸ Furthermore, Figure 7(d) unveils an increase in the phase shift with an increase of ion nonextensivity. Moreover, as previously observed, two supplementary collision angles result in phase shifts of opposite signs but with identical magnitudes. Crossing angles in close proximity to the right angle, whether slightly greater or smaller, tend to result in phase shifts approaching nullification. It increases as one deviates from these angles, reaching its maximum at non-zero but small angles or angles near π. Undoubtedly, critical composition unveils a wealth of phenomena, allowing for not only the conventional collisions between solitons of the same polarity but also the collisions between solitons of opposite polarities, as indicated by the non-coupling of signs elucidated in equation (35). Figure 8 depicts such a situation, where a compressive and a rarefactive soliton of different velocities $({\lambda }_{{\xi }_0}=0.01,{\lambda }_{{\eta }_0}=0.02)$ are engaged in a frontal collision. The process closely resembles that portrayed in Figure 6 with the noteworthy distinction that within the overlap region, the evanescent potential (which does not exist when amplitudes are identical) assumes the polarity of the more intensive soliton, despite having an amplitude lower than that of the pre-collision solitons. Furthermore, numerical analysis reveals that the solitons undergo distinct deviation (ΔP′ + ΔQ′ = −0.1105 with ΔP′ = −0.3770 and ΔQ′ = 0.2665). Ultimately, collisions under critical composition exhibit substantial differences from those under normal composition.

Figure 7.

Variation of colliding phase shift $({\mathrm{\Delta }}_p^{\prime })$ (a) against γ with P_⊥n = 0.3, ω _cn = 0.8 and q = 1.6 for different values of P_‖n (b) against γ with P_‖n = 0.3, ω _cn = 0.8 and q = 1.6 for different values of P_⊥n (c) against P_‖n with γ = 0.6, P_⊥n = 0.3, and q = 1.6 for different values of ω _cn (d) against θ with γ = 0.6, P_‖n = 0.5, P_⊥n = 0.3, ω _cn = 0.8, and q = 1.6. Along with β = 0.45, α = 1.5, P_‖p = 0.6, P_⊥p = 0.4, μ _p = 0.53, ${\lambda }_{{\xi }_0}=0.01$ , ${\lambda }_{{\eta }_0}=0.02$ , and ɛ = 0.5.

Figure 8.

Time evolution and corresponding colliding profile of a compressive and a rarefactive DASWs. The parameters are set as β = 0.45, α = 1.5, P_‖n = 0.5, P_⊥n = 0.3, P_‖p = 0.6, P_⊥p = 0.4, q = 1.6, μ _p = 0.53, γ = 0.6, ω _cn = 0.6, ${\lambda }_{{\xi }_0}=0.01$ , ${\lambda }_{{\eta }_0}=0.02$ , ɛ = 0.5, and θ = π/2.55. ΔP′ + ΔQ′ ≠ 0.

Dust-acoustic RWs at the critical composition

The critical composition allows exploring nonlinear modulated acoustic excitations, focusing on the RWs phenomenon. By virtue of equation (26) and (33), one can derive the nonlinear Schrödinger equation (NLSE) through a suitable variable transformation. Notably, the NLSE derived from the mKdV equation fulfills the essential conditions required for the RWs formation. ⁸⁶ In this context, the following expansions are adopted

${\varphi }_{1\xi }(\xi ,\tau )=\sum _{l=-\infty }^{\infty }\left(\sum _{m=1}^{\infty }{\epsilon }^m{\varphi }_{{\xi }_l}^{(m)}(\tilde{\xi },\tilde{\tau }){\mathrm{e}}^{il\left(k\xi +\omega \tau \right)}\right);\tilde{\xi }=\epsilon (\xi -v_g\tau );\tilde{\tau }={\epsilon }^2\tau .$

(38)

where k, ω, and v _g stand for the wave number, the frequency, and the group velocity of DAWs. Substituting transformations (38) into equation (33), then collecting the terms in different power of ɛ, the first-order approximation (m = 1) with the first harmonic (l = 1) yields the dispersion relation ω = −T₂k³. For the second-order approximation (m = 2), the results v _g = −3 T₂k² and ${\varphi }_{{\xi }_2}^{(2)}=0$ are obtained. Finally, the third harmonic modes (m = 3, l = 1) yields the following NLSE

$i{\partial }_{\tilde{\tau }}{\varphi }_{{\xi }_1}^{(1)}+\frac{\mathrm{\Upsilon }}{2}{\partial }_{\tilde{\xi }\tilde{\xi }}^2{\varphi }_{{\xi }_1}^{(1)}+\mathrm{\Lambda }{\left|{\varphi }_{{\xi }_1}^{(1)}\right|}^2{\varphi }_{{\xi }_1}^{(1)}=0,$

(39)

where ϒ = −6 T₂k and Λ = −S₂k stand, respectively, for dispersion and nonlinear coefficients. As indicated by the relationship ϒ/Λ = 6 T₂/S₂, DAWs stability is strongly connected to signs of coefficients in equation (33). Indeed, it is well known that for, ϒ/Λ < 0, waves are stable, while they are modulationally unstable against external perturbations when ϒ/Λ > 0. In this later scenario, the NLSE (39) features an incredible localized first-order rational solution known as the Peregrine breather, which appears as suddenly as it disappears and focuses a large amount of energy in a confined spatio-temporal region. The first-order rational solution of equation (39) can be written as ^87,88

${\varphi }_{{\xi }_1}^{(1)}=\sqrt{\frac{\mathrm{\Upsilon }}{\mathrm{\Lambda }}}\left[\frac{4\left(1+2i\mathrm{\Upsilon }\tilde{\tau }\right)}{1+4{\mathrm{\Upsilon }}^2{\tilde{\tau }}^2+4{\tilde{\xi }}^2}-1\right]{\mathrm{e}}^{i\mathrm{\Upsilon }\tilde{\tau }}.$

(40)

It is important to underline that applying transformations (38) to equation (26), yields a NLSE which does not support RWs solution, as $\mathrm{\Upsilon }/\mathrm{\Lambda }=-12T_1^2{k}^2/S_1^2$ is always negative. Figure 9 serves as an illustrative tool to elucidate connexion between dust pressure anisotropy (P_‖n, P_⊥n), magnetic field (ω _cn ), nonextensive index (q), gravitational parameter (γ), and RWs. Figures 9(a) and (b) demonstrate the non-proportional antagonistic effects of pressures along and across the magnetic field on RWs. While parallel dust pressure strongly damps waves, an increase in perpendicular pressure yields a modest amplification. This unequivocally indicates that P_‖n and P_⊥n work differently. Additionally, Figures 9(c) and (d) reveal that for low values of γ, an intensified gravitational field induces a sharp rise in energy of RWs, while the magnetic field and entropic index act as damping factors. By disregarding the effects of anisotropy, these results agree with Ref. 71.

Figure 9.

Variation of $|{\varphi }_{{\xi }_1}|$ for different values of (a) P_‖n, (b) P_⊥n, (c) ω _cn , (d), γ, and q. According to the figure, the fixed parameters are usually chosen as P_⊥n = 0.3, P_‖n = 0.5, ω _cn = 0.6, P_‖p = 0.6, P_⊥p = 0.4, β = 0.45, α = 1.5, μ _p = 0.53, and q = 2.5.

Figure 10(a) provides a more detailed view of the Peregrine soliton. The nonlinear interaction of a first-order RW can lead to the formation of higher-order RWs, a surprising case being the DARW triplets. This second-order RW, composed of three Peregrine solitons arranged at the corners of a triangle, can also be interpreted as a set of first-order RWs with phase shifts (θ₁ and θ₂) that allow the relative positions of the peaks to be adjusted. The DARW triplet solution of equation (39) ⁵³ is

${\varphi }_{{\xi }_1}^{(1)}=\sqrt{\frac{\mathrm{\Upsilon }}{\mathrm{\Lambda }}}\left[1+\frac{R_1+i{R}_2}{R_3}\right]{\mathrm{e}}^{i\mathrm{\Upsilon }\tilde{\tau }},$

(41)

where

$\begin{array}{cc}\hfill R_1=& 36-48{\xi }_1^4-144{\mathrm{\Upsilon }}^2\left(4{(\mathrm{\Upsilon }\tilde{\tau })}^2+1\right)-24\sqrt{2}{\theta }_1{\xi }_1-960{(\mathrm{\Upsilon }\tilde{\tau })}^4-864{(\mathrm{\Upsilon }\tilde{\tau })}^2+48{\theta }_2\mathrm{\Upsilon }\tilde{\tau };\hfill \\ R_2=\hfill & 24\mathrm{\Upsilon }\tilde{\tau }\left(15-4{\xi }_1^4+12{\xi }_1^2-2\sqrt{2}{\theta }_1{\xi }_1\right)-384{(\mathrm{\Upsilon }\tilde{\tau })}^5-192{(\mathrm{\Upsilon }\tilde{\tau })}^3\left(2{\xi }_1^2+1\right)+24{\theta }_2\left(2{(\mathrm{\Upsilon }\tilde{\tau })}^2-{\xi }_1^2-\frac{1}{2}\right);\hfill \\ R_3=\hfill & 8{\xi }_1^6+12{\xi }_1^4\left(4{(\mathrm{\Upsilon }\tilde{\tau })}^2+1\right)+6{\xi }_1^2{\left(-4{(\mathrm{\Upsilon }\tilde{\tau })}^2+3\right)}^2+{\theta }_1\left({\theta }_1+2\sqrt{2}{\xi }_1\left(12{(\mathrm{\Upsilon }\tilde{\tau })}^2-2{\xi }_1^2+3\right)\right);\hfill \\ \hfill & +{\theta }_2\left({\theta }_2+4\mathrm{\Upsilon }\tilde{\tau }\left(-4{(\mathrm{\Upsilon }\tilde{\tau })}^2+6{\xi }_1^2-9\right)\right)+396{(\mathrm{\Upsilon }\tilde{\tau })}^2+9+64{(\mathrm{\Upsilon }\tilde{\tau })}^6.\hfill \end{array}$

(42)

Figure 10.

(a) The Peregrine soliton profile for the wave potential, (b) DA Rogue wave triplets profile for θ₁ = 1000 and θ₂ = 0, and (c) DA super RW profile get for θ₁ = θ₂ = 0. Along for P_⊥n = 0.3, P_‖n = 0.5, ω _cn = 0.6, P_‖p = 0.6, P_⊥p = 0.4, β = 0.45, α = 1.5, μ _p = 0.53, and q = 2.5.

When θ₁ = 1000 and θ₂ = 0, the three peaks are positioned at $A:({\theta }_1^{\frac{1}{3}}/\mathrm{4,0})$ , $B:(-\sqrt{2}{\theta }_1^{\frac{1}{3}}/4,\sqrt{3}{\theta }_1^{\frac{1}{3}}/4\mathrm{\Upsilon })$ , and $C:(-\sqrt{2}{\theta }_1^{\frac{1}{3}}/4,-\sqrt{3}{\theta }_1^{\frac{1}{3}}/4\mathrm{\Upsilon })$ in the (ξ₁, τ) plane. Figure 10(b) depicts a RW triplets, revealing a structure with three peaks, each separated by 60 radians. Two of the peaks, B and C, are symmetrical and share the same amplitude $(|{\varphi }_{{\xi }_1}^{(1)}|=29.4)$ , though they are slightly more intense than peak A $(|{\varphi }_{{\xi }_1}^{(1)}|=28.90)$ . By fixing θ₂ = 0 and gradually reducing θ₁, the peaks move closer together, and when θ₁ = θ₂ = 0, the first-order DARWs coalesce, forming a single peak called DA super RW (Figure 10(c)). Table 1 provides a summary of the effects of pressure anisotropy on RW triplets and super RWs. As previously demonstrated for the Peregrine soliton, P_‖n markedly reduces the amplitude, whereas P_⊥n slightly increases the intensity of both the triplets and super RWs. This underscores the critical influence of pressure anisotropy on the propagation of these waves.

Table 1.

The amplitude of DA RWs, triplets, and super RWs (×10).

Amplitude of RWs →	Single RW		Triplets RW		Super RW
	P⊥n = 0.3	P⊥n = 0.9	P⊥n = 0.3	P⊥n = 0.9	P⊥n = 0.3	P⊥n = 0.9
P‖n = 0.2	2.930	3.054	2.940	3.059	4.893	5.090
P‖n = 0.3	1.742	1.808	1.743	1.811	2.903	3.014
P‖n = 0.4	1.380	1.430	1.382	1.432	2.300	2.383

Summary

In this paper, the effects of dust pressure anisotropy, gravitational and magnetic fields on linear and nonlinear wave propagation, as well as oblique collisions of dust-acoustic solitary waves (DASWs) in an electron-depleted dusty plasma composed of massive dust grains with opposite polarities and non-extensive ions, were investigated through the linear analysis, and the Poincaré–Lighthill–Kuo (PLK) perturbation technique. Respectively, the two-side Korteweg-de Vries (KdV) and two-side modified KdV (mKdV) equations were successfully derived for the generic and the critical cases, along with their corresponding phase shifts. Numerical analysis was performed using representative data for dust molecular clouds. ⁸² Computations revealed that gravitational forces introduced the possibility of gravitational instabilities characterized by a critical wavenumber and a growth rate. Moreover, magnetic field and dust pressures combine to quench instabilities by attenuating gravitational forces. However, it should be noted that despite the parallel and perpendicular dust pressures showing qualitatively similar impacts on dust-acoustic wave oscillations, their quantitative effects differ. Specifically, it has been shown that perpendicular dust pressures promote less instability. Additionally, we have determined the critical radius and mass beyond which the dust cloud becomes unstable and may fragment, leading to star formation. Ranges of physical parameters conducive to the emergence, propagation, and collision of solitons of identical (opposite) polarities have been identified. The analysis also reveals that parallel pressures induced a more intense phase shift between trajectories compared to perpendicular dust pressures. An intensifying gravitational field has been observed to enhance the phase shift, while the presence of a magnetic field diminishes this effect. It has been demonstrated that during the collision process, the slowest soliton undergoes the most significant phase shift, primarily due to its extended interaction time. Notably, two distinct collision processes, characterized by supplementary crossing angles, induce phase shifts of opposite signs but equal magnitude. On the other hand, as the collision angle gradually becomes acute or obtuse, the phase shift increases and reaches its maximum value at non-zero but small angles or angles near π. Finally, a detailed investigation into the hierarchy of solutions to the nonlinear Schrödinger equation has been undertaken, with particular emphasis on the Peregrine soliton, the DA rogue wave triplets, and its transformation into super rogue waves through the superposition of triplets. Specifically, an increase in parallel pressure drastically reduces DA rogue waves triplets amplitude, whereas an increase in perpendicular pressure leads to a modest amplitude growth. Additionally, the gravitational field amplifies RWs while the magnetic field attenuates them. The findings presented here are valuable in providing relevant prediction concerning wave stability, propagation, and oblique collisions of solitons in self-gravitating magnetized EDDP containing massive dust grains and non-extensive ions found in astrophysical media such as dust molecular clouds ⁸² and interstellar medium ⁸⁹ where, gravitational and magnetized effects, along with pressure anisotropy, are supposed to be significant. ⁵⁷

Future work

In some collisional plasma models, the collisional force between plasma charges with each other or with some neutral particles inside the plasma cannot be neglected because this effect will significantly impact the behavior of nonlinear structures propagating in the studied model. Thus in this case, and after applying the extended PLK method, the fluid equations to a model under consideration will be reduced to the following the two coupled damped KdV equations

$\left\{\begin{array}{c}{\partial }_{\tau }{\varphi }_{2\xi }+S_1{\varphi }_{2\xi }{\partial }_{\xi }{\varphi }_{2\xi }+T_1{\partial }_{\xi }^3{\varphi }_{2\xi }+\mathrm{\Gamma }{\varphi }_{2\xi }=0,\hfill \\ {\partial }_{\tau }{\varphi }_{2\eta }-S_1^{\prime }{\varphi }_{2\eta }{\partial }_{\xi }{\varphi }_{2\eta }-T_1^{\prime }{\partial }_{\eta }^3{\varphi }_{2\eta }+\mathrm{\Gamma }{\varphi }_{2\eta }=0,\hfill \end{array}\right.$

(43)

and for generic/critical case, the following two coupled damped mKdV equations are obtained

$\left\{\begin{array}{c}{\partial }_{\tau }{\varphi }_{1\xi }+S_2{\varphi }_{1\xi }^2{\partial }_{\xi }{\varphi }_{1\xi }+T_2{\partial }_{\xi }^3{\varphi }_{1\xi }+\mathrm{\Gamma }{\varphi }_{1\xi }=0,\hfill \\ {\partial }_{\tau }{\varphi }_{1\eta }-S_2^{\prime }{\varphi }_{1\eta }^2{\partial }_{\xi }{\varphi }_{1\eta }-T_2^{\prime }{\partial }_{\eta }^3{\varphi }_{1\eta }+\mathrm{\Gamma }{\varphi }_{1\eta }=0.\hfill \end{array}\right.$

(44)

Moreover, in many plasma cases, the impact of nonplanar geometry on the behavior of different waves that arise and propagate in different plasma systems must be addressed, which in this case, the following two coupled nonplanar KdV/mKdV equations are obtained

$\left\{\begin{array}{c}{\partial }_{\tau }{\varphi }_{2\xi }+S_1{\varphi }_{2\xi }{\partial }_{\xi }{\varphi }_{2\xi }+T_1{\partial }_{\xi }^3{\varphi }_{2\xi }+f\left(t\right){\varphi }_{2\xi }=0,\hfill \\ {\partial }_{\tau }{\varphi }_{2\eta }-S_1^{\prime }{\varphi }_{2\eta }{\partial }_{\xi }{\varphi }_{2\eta }-T_1^{\prime }{\partial }_{\eta }^3{\varphi }_{2\eta }+f\left(t\right){\varphi }_{2\eta }=0,\hfill \end{array}\right.$

(45)

and

$\left\{\begin{array}{c}{\partial }_{\tau }{\varphi }_{1\xi }+S_2{\varphi }_{1\xi }^2{\partial }_{\xi }{\varphi }_{1\xi }+T_2{\partial }_{\xi }^3{\varphi }_{1\xi }+f\left(t\right){\varphi }_{1\xi }=0,\hfill \\ {\partial }_{\tau }{\varphi }_{1\eta }-S_2^{\prime }{\varphi }_{1\eta }^2{\partial }_{\xi }{\varphi }_{1\eta }-T_2^{\prime }{\partial }_{\eta }^3{\varphi }_{1\eta }+f\left(t\right){\varphi }_{1\eta }=0.\hfill \end{array}\right.$

(46)

Thus, when these effects are considered in specific plasma models, the governing fluid equations of the model will be reduced to non-integrable evolution equations (43)–(46). Consequently, to examine the features of these waves and their interaction, it is necessary to analyze and solve these equations using some numerical or semi-analytical methods (e.g., HPM, ansatz method, and so on) to provide semi-analytical solutions that facilitate the analysis of these wave aspects. The HPM has effectively analyzed various evolution equations,^90–97 making it a highly regarded tool for studying multiple non-integrable wave equations.^98,99 Therefore, the HPM/ansatz method can be used to analyze some derived evolution equations, such as the coupled damped/nonplanar KdV and coupled damped/nonplanar mKdV equations, to study the properties of waves/waves interaction that propagate at phase velocity, such as solitary waves, solitons collision, shocks, and shocks collision.

Moreover, if the fluid equations of the studied model are reduced to one of the non-integrable nonlinear Schrödinger equations (such as the below nonplanar NLSE^39–46 or the damped NLSE^47–51), in this case, the HPM/ansatz method can also be applied to study the properties of modulated waves that propagate at group velocity, such as RWs or dark/bright modulated SWs:the damped NLSE reads

$i{\partial }_{\tilde{\tau }}{\varphi }_{{\xi }_1}^{\left(1\right)}+\frac{\mathrm{\Upsilon }}{2}{\partial }_{\tilde{\xi }}^2{\varphi }_{{\xi }_1}^{\left(1\right)}+\mathrm{\Lambda }|{\varphi }_{{\xi }_1}^{\left(1\right)}{|}^2{\varphi }_{{\xi }_1}^{\left(1\right)}+i\mathrm{\Gamma }{\varphi }_{{\xi }_1}^{\left(1\right)}=0,$

(47)

and the nonplanar NLSE reads

$i{\partial }_{\tilde{\tau }}{\varphi }_{{\xi }_1}^{\left(1\right)}+\frac{\mathrm{\Upsilon }}{2}{\partial }_{\tilde{\xi }}^2{\varphi }_{{\xi }_1}^{\left(1\right)}+\mathrm{\Lambda }|{\varphi }_{{\xi }_1}^{\left(1\right)}{|}^2{\varphi }_{{\xi }_1}^{\left(1\right)}+if\left(\tilde{\tau }\right){\varphi }_{{\xi }_1}^{\left(1\right)}=0.$

(48)