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Modulational instability and associated ion-acoustic modulated envelope solitons in a quantum plasma having ion beams

  • Fazal Wahed , Ata-ur-Rahman , S. Neelam Naeem , R. A. Alharbey , Maryam Al Huwayz , Lamiaa S. El-Sherif and Samir A. El-Tantawy EMAIL logo
Published/Copyright: July 18, 2025
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Open Physics
From the journal Open Physics Volume 23 Issue 1

Abstract

This work employs the one-dimensional quantum hydrodynamic model to investigate the nonlinear propagation of modulated ion-acoustic waves (IAWs) in unmagnetized quantum plasma with ion beams. A reductive perturbation technique (RPT) is carried out to reduce the set of fluid equations to a cubic nonlinear Schrödinger equation (NLSE), which governs the propagation of the modulational instability (MI) and its associated modulated structures (envelope solitons). It is demonstrated that plasma configurational parameters, such as ion beam density, quantum diffraction parameter, and ion beam temperature, significantly affect MI and the related nonlinear structures. We also examined the impact of these pertinent physical parameters on the critical wavenumber and the growth rates related to MI. The critical wavenumber and MI growth rate were found to decrease with growing values of quantum diffraction parameters and ion beam temperature while falling with ion beam density. Furthermore, the modulated nonlinear localized structures that appear as bright and dark envelope solitons are discussed in detail. Our results are expected to reveal the mystery of the behavior of the modulated nonlinear phenomena that may arise and propagate in such a type of quantum plasma with ion beams. Moreover, the results can be used to understand the behavior of many modulated nonlinear phenomena and then devote them to various applications.

Keywords: quantum plasma; ion beams; nonlinear Schrödinger equation; modulational instability; ion-acoustic modulated solitons

1 Introduction

Quantum plasmas have been vigorously studied over the past few decades due to their importance in ultracold plasmas [1], strong laser plasma interaction experiments [2], microelectronic devices [3], microplasmas [4], and astrophysical conditions like neutron stars, white dwarfs [5,6]. Quantum plasma consists of ions, electrons, and positrons at high number densities and small temperatures, whereas classical plasma is characterized by small particle number densities and high temperatures. The plasma particles’ de Broglie wavelengths in a classical plasma are substantially less than the system’s size. Nonetheless, the de Broglie thermal wavelengths of the plasma particles in quantum plasma get closer to the system’s spatial scale [7,8]. In the latter case, the Heisenberg and Pauli exclusion principles govern plasma species, and quantum effects can be analyzed through the quantum Bohm potential force and quantum statistics [9]. In quantum plasmas, the charged particles’ length, time, and thermal velocities differ greatly from those in traditional plasmas. Therefore, while dealing with quantum plasmas, one must appropriately modify mathematical formulations used in classical situations. The statistical and hydrodynamic behaviors of plasma particles at the quantum scale are described using well-known mathematical techniques such as Schrödinger–Poisson, Winger–Poisson, and Dirac–Maxwell. The behavior of plasma particles and collective phenomena including waves, nonlinear structures, and instabilities at the quantum scale, however, can be studied effectively using the quantum hydrodynamic (QHD) model [10–12]. The QHD model includes a set of fundamental equations describing the transport of momentum, energy, and charge associated with plasma particles interacting via self-consistent electrostatic potential. The QHD model generalizes the fluid model for plasma by incorporating the quantum correction term, or Borm potential (due to quantum tunneling effects), and the quantum statistical effect through an equation of state. The QHD model has attained significant importance compared to other quantum plasma models due to its analytical tractability and simple numerical modeling, even though it has been failed to explain kinetic effects like the Landau damping of waves [13]. Due to straightforward approach, numerical efficiency, and simplicity, the QHD model has been extensively used by researchers. For example, Haas and Garcia investigated the significance of quantum diffraction in both linear and nonlinear regimes using the QHD model [14]. Ion acoustic solitary waves (IASWs) in a quantum electron–positron–ion (e–p–i) plasma with weakly transverse disturbances were examined by Mushtaq and Khan using the QHD model [15]. Using the same model, Rajabi and Muhammadneja [16] investigated IASWs in dense quantum plasma. Futhermore, Chandra et al. [17] have also statistically and theoretically examined the linear and nonlinear propagation of electron plasma waves in a two-component unmagnetized dense quantum plasma with ion streaming using the QHD model.

Nonlinear wave propagation in ion beam plasma has garnered a lot of attention because of its important uses in heavy ion inertial fusion [18–20], semiconductor lasers [21–24], electron cooling of ion beams [25–29], and intense laser-produced proton beams [30–32]. The research on the latter also contributes to the fields of astrophysics and magnetospheric physics [33–37]. The presence of ion beams significantly affects nonlinear structures in plasmas [38], and the ion beam–plasma interactions process in various plasma environments has been actively studied [39,40]. For example, in an ion beam plasma system with cold ion beams and nonisothermal electrons, Abrol and Tagare derived a modified KdV equation for IASWs [41]. Gell and Roth [42] examined the effect of an ion beam on soliton motion in the ion beam–plasma system. Many authors have since examined solitary waves (SWs) in ion beam plasma, like Das and Deka [43], Misra and Adhikary [44], Huibin and Kelin [45], Zank and McKenzie [46], Karmakar and Das [47], etc. Recently, Kaur et al. [48] studied the nonlinear propagation of IASWs in an unmagnetized plasma that included two temperature electrons, a positive ion beam, and a positive warm ion fluid. Additionally, Kaur et al. [49] investigated dust acoustic solitary and rogue wave (RW) propagation characteristics in an unmagnetized ion beam plasma.

Due to the self-interaction of the carrier wave or intrinsic nonlinearity of the medium, amplitude modulation is frequently observed during nonlinear wave propagation in ion beam plasma or dispersive medium [50]. While investigating amplitude modulation, the multiple space and time-scale technique [51] is commonly used, which results in a Kortwege-de Vires (KdV) equation and nonlinear Schrödinger equation (NLSE). The KdV equation and all its family (e.g., modified KdV, extended/Gardner KdV, Schamel-KdV equations, etc.) describe the dynamics of non-modulated wave packets, which are bare pulses without rapid oscillations within the packets [52–58]. In contrast, the NLSE describes the behavior of modulated wave packets so that wave group dispersion balances nonlinearities [59–61]. The NLSE has stationary envelope solutions, i.e., envelope solitons. These solitons are the localized structures that take the form of localized envelope excitations, which confine, or modulate, a fast internal carrier wave oscillation in space [62]. The detailed analysis of the soliton solution of NLSE can be seen in previous studies [63–68]. Moreover, through analytical and numerical investigations, it has been discovered that modulated wavepackets are related to modulational instability (MI) [69]. MI is a significant nonlinear phenomenon that occurs during propagation of wavepackets and has been studied in many different physical contexts, including solid-state physics [70], hydrodynamics [71], plasma physics [72], and Bose–Einstein condensation [73,74]. The latter has many applications in charge transport in molecular systems [75], signal transmission lines [76], and fiber telecommunications [77]. Watanabe reported the first experimental observation of the MI of a monochromatic ion-acoustic wave (IAW) in 1977 [78]. The MI and associated envelope structures were studied by Irshad et al. [79] in a non-Maxwellian plasma consisting of inertialess κ -deformed Kaniadakis distributed electrons, inertial cold fluid electrons, and stationary positive ions. Wahed et al. [80] investigated the MI and associated low-frequency dust-acoustic waves in a degenerate Thomas-Fermi plasmas. Similarly, the MI of IASWs and associated RWs in ultracold plasmas in the presence of an ion kinematic viscosity were examined by El-Tantawy [81]. Furthermore, Kourakis and Shukla [82] investigated the MI of electron acoustic waves in space plasmas characterized by inertialess Boltzmann-distributed hot electrons, inertial cold electrons, and static ions.

To our knowledge, there are no relevant studies in the literature on the significance of the MI and the creation of (un)stable envelope structures in the context of (un)stable electrostatic wavepacket propagation in ion-beam plasma systems. Therefore, in the presence of an ion beam, we studied the impact of various factors on the dynamics of amplitude-modulated IAWs and envelope structures in unmagnetized quantum degenerate plasma. Our findings show that plasma configurational parameters, such as ion beam temperature, diffraction parameter, and density, have a great effect on MI. These parameters can change both the associated critical values and the growth rates of MI.

The structure of the article proceeds as follows: In Section 2, we present the fundamental equations governing the dynamics of IAWs in unmagnetized quantum plasma, considering the influence of an ion beam. Section 3 is devoted to the use of reductive perturbation technique (RPT) to derive the NLSE. The MI and its growth rate are discussed in Section 4. In Section 5, we provide a comprehensive study of both bright and dark envelope solitons, including a parametric investigation of key variables such as the ion beam density ratio n b o , the diffraction parameter H , and the ion beam temperature δ b . Finally, we summarize our findings in Section 6.

2 Basic equations

An unmagnetized, collisionless quantum plasma model made up of ion beams, positive ions, and inertialess electrons is examined in this work. All species in the plasma are assumed to adhere to the Fermi–Dirac statistics, and the positive ions and positive ion beams are regarded as singly ionized. The low-frequency IAWs are sustained via two competing mechanisms: the mass of ion is responsible for the inertia while the restoring force is supplied by the massless electrons to keep IAWs to propagate. The IAWs’ phase velocity is substantially higher than the Fermi velocity of positive ion beams and substantially lower than the electron’s Fermi velocity. At zero temperature, it is assumed that the plasma particles follow the following pressure law and act like a one-dimensional Fermi gas [14]

(1) P j = M j V F j 2 3 N j o 2 × N j 3 ,

where j = i , b , and e stands for ions, ion beams, and electrons, respectively. The Fermi speed is V F j = ( 2 k B T F j ) ⁄ M j , where k B and T F j are the Boltzmann constant and Fermi temperature. The number density and equilibrium number density are represented by N j and N j o , respectively. The electron plasma frequency is ω p e = ( 4 π N o e 2 ) ⁄ M e . The normalized one-dimensional equations governing plasma dynamics [83] are as follows:

(2) ∂ x ϕ − N e ∂ x N e + H 2 2 ∂ x 1 N e ∂ x 2 N e = 0 ,

(3) ∂ t N i + ∂ x ( N i V i ) = 0 ,

(4) ∂ t V i + V i ∂ x V i = − δ i N i ∂ x N i + μ ∂ x ϕ ,

(5) ∂ t N b + ∂ x ( N b V b ) = 0 ,

(6) ∂ t V b + V b ∂ x V b + μ δ b ∂ x ϕ = − δ b N b ∂ x N b ,

(7) ∂ x 2 ϕ = χ N e − N i − N b ( 1 − χ ) .

To simplify the analysis, the normalization of the spacial ( x ) and temporal ( t ) variables is, respectively, carried out via x ω p i ⁄ V F i and t ω p i . The number densities of the current model species N j , electric potential ϕ , and speed u j are normalized by N j ⁄ N o , e φ ⁄ ( 2 k B T F j ) , and V j ⁄ V F e , respectively. The ratio between the masses of positive ion and ion beams, positive ion (beam) Fermi temperature to electron Fermi temperature, and unperturbed electron number density and ion number density are, respectively, given by μ = m i ⁄ m b (mass ratio), δ i , b = T F i , b ⁄ T F e (ion temperature ratio/ion beam temperature ratio), and χ = N e o ⁄ N i o (electron concentration). The ratio of plasma energy ℏ ω p e to Fermi energy k B T F e is equal to the dimensionless quantum parameter H , which is proportional to quantum diffraction.

The QHD model for IAWs in unmagnetized quantum plasma with ion beams is represented by Eqs. (2)–(7). The quantum correction is incorporated through the third term on the left side in Eq. (2), which arises from the quantum correlation of density variations and is referred to as the Bohm potential or quantum pressure. Other contributions to the quantum effects in this model stem from the first terms on the right sides in Eqs (4) and (6), respectively. The latter terms are incorporated through Eq. (1). The electric potential ϕ can be obtained by integrating Eq. (2) with boundary conditions N e = 1 and ϕ = 0 at ± ∞ as

ϕ = − 1 2 + N e 2 2 − H 2 2 1 N e ∂ x 2 N e .

3 Perturbative analysis and derivation of NLSE

Nonlinear partial differential equations (PDEs) can be solved using various techniques, including the RPT, Adomian decomposition technique, variational iteration technique, differential transform technique, reduced differential transform technique, and the simplest equation technique [84,85]. The RPT is particularly effective for studying nonlinear waves with small amplitudes. The RPT was established by Taniuti and Wei [51] and was first used to investigate the dynamics of electron plasma [86] and electron–cyclotron waves [87,88]. This technique has been referred to as an RPT since it reduces the behavior of the system’s PDEs to the solution of nonlinear equations [89]. In the RPT, both spatial and temporal variables are rescaled, and new variables are introduced into the fundamental equations that describe long-wavelength phenomena. This method is widely used in quantum plasma because of its systematic approach, improved computational efficiency, and ability to effectively manage weak nonlinearity and dispersion. According to this technique, the smallness parameter ε ≪ 1 , which represents a weak perturbation impact, is used to expand the state variables related to random perturbations around the equilibrium states [90]. Let S be any of the system variables ( N e , N i , N b , V i , V b , ϕ ) that represent the state of the system at position x and time t . The expansion for S around the equilibrium state S ( 0 ) = ( χ , 1,1 − χ , 0 , 0 , 0 ) reads

S = S ( 0 ) + ∑ n = 1 ∞ S ( n ) ,

with

S ( n ) = ∑ l = − ∞ ∞ S l ( n ) exp [ i l ( k x − ω t ) ] .

All state variables satisfy the reality condition S − l ( n ) = S l ( n ) ∗ . According to Gardner and Morikawa’s [91] suggestion, the space and time coordinates are stretched as ξ = ( x − t v g ) ε and τ = t ε 2 , where v g will be determined later and is the group velocity of the wavepacket. The spatial and time operators then take the following form:

(8) ∂ x ∂ t = ∂ x ∂ t + ε ∂ ξ − v g ∂ ξ + ε 2 0 ∂ τ .

All the dependent variables N e , N i , N b , V i , V b , ϕ can be expanded in terms of ε as

(9) N e ( x , t ) N i ( x , t ) N b ( x , t ) V i ( x , t ) V b ( x , t ) ϕ ( x , t ) = N e o N i o N b o 0 0 0 + ∑ n = 1 ∞ ε n ∑ l = − ∞ ∞ N e l ( n ) ( ξ , τ ) N i l ( n ) ( ξ , τ ) N b l ( n ) ( ξ , τ ) V i l ( n ) ( ξ , τ ) V b l ( n ) ( ξ , τ ) ϕ l ( n ) ( ξ , τ ) e i ( k x − ω t ) l .

Substituting the above Eq. (9) into the system (2)–(7) and using Eq. (8) along with stretched coordinates, the following reduced equations in the lowest order of ε for first-order n = 1 and l = 1 are obtained:

(10) 4 χ ϕ 1 ( 1 ) − N e 1 ( 1 ) ( 4 χ 2 + k 2 H 2 ) = 0 , − ω N i 1 ( 1 ) + k u i 1 ( 1 ) = 0 , − ω V i 1 ( 1 ) + k ϕ 1 ( 1 ) + δ i k N i 1 ( 1 ) = 0 , − ω N b 1 ( 1 ) + k ( 1 − χ ) V b 1 ( 1 ) = 0 , − ω V b 1 ( 1 ) + μ k ϕ 1 ( 1 ) + μ δ b k ( 1 − χ ) N b 1 ( 1 ) = 0 , χ N e 1 ( 1 ) − N i 1 ( 1 ) − ( 1 − χ ) N b 1 ( 1 ) + k 2 ϕ 1 ( 1 ) = 0 .

In terms of ϕ 1 ( 1 ) , the first-order quantities are given by

(11) N e 1 ( 1 ) = 4 χ 4 χ 2 + k 2 H 2 ϕ 1 ( 1 ) , N i 1 ( 1 ) = − k 2 k 2 δ i − ω 2 ϕ 1 ( 1 ) , V i 1 ( 1 ) = − k ω k 2 δ i − ω 2 ϕ 1 ( 1 ) , N b 1 ( 1 ) = k 2 ( 1 − χ ) μ ω 2 − k 2 μ δ b ( 1 − χ ) 2 ϕ 1 ( 1 ) , V b 1 ( 1 ) = − k ω μ − ω 2 + k 2 μ δ b − 2 k 2 χ μ δ b + k 2 χ 2 μ δ b ϕ 1 ( 1 ) .

The following dispersion relation is the result of the above first-order quantities:

(12) ω 2 = 1 2 H 2 k 4 + 8 ( 1 + k 2 ) χ 2 ( k 2 ( H 2 k 2 + 4 χ 2 ) × [ 1 + ( χ − 1 ) 2 μ ] + [ k 4 ( ( H 2 k 2 + 4 χ 2 ) 2 ( 1 + ( χ − 1 ) 2 μ ) 2 + ( H 2 k 4 + 4 ( 1 + k 2 ) χ 2 ) ( 2 ( H 2 k 2 + 4 χ 2 ) ( − 1 + ( − 1 + χ ) 2 μ ) + ( H 2 k 4 + 4 ( 1 + k 2 ) χ 2 ) ( ( χ − 1 ) 2 μ δ b − δ i ) ) × ( ( χ − 1 ) 2 μ δ b − δ i ) ) ] 1 2 + k 2 ( H 2 k 4 + 4 ( 1 + k 2 ) χ 2 ) ( ( χ − 1 ) 2 μ δ b + δ i ) ) .

In the presence of ion beams, the dispersion relation for the IAWs in an unmagnetized quantum (degenerate) plasma is represented by Eq. (12). It is noted that the dispersion relation depends on ion beam density n b o , quantum parameter H , and ion beam temperature δ b . Figure 1(a) illustrates that ω increases as the values of k increase for a fixed value of n b o . It is also noted that when n b o is increased, ω is shifted toward larger values. Figure 1(b) and (c) illustrates that for smaller values of k , the frequency ω has the same value for given values of H and δ b ; however, for larger values of k , the frequency ω shifts toward larger values for increasing values of H and δ b .

Figure 1 The wave frequency ω \omega is analyzed against (a) ion beam density n b o {n}_{bo} for fixed values of H = 0.1 H=0.1 and δ b = 0.1 {\delta }_{b}=0.1 , (b) quantum diffraction parameter H H for fixed values of n b o = 0.5 {n}_{bo}=0.5 and δ b = 0.1 {\delta }_{b}=0.1 , and (c) ion beam temperature δ b {\delta }_{b} for fixed values of n b o = 0.5 {n}_{bo}=0.5 and H = 0.1 H=0.1 .
Figure 1

The wave frequency ω is analyzed against (a) ion beam density n b o for fixed values of H = 0.1 and δ b = 0.1 , (b) quantum diffraction parameter H for fixed values of n b o = 0.5 and δ b = 0.1 , and (c) ion beam temperature δ b for fixed values of n b o = 0.5 and H = 0.1 .

The reduced equations for ( n , l ) = ( 2 , 1 ) are

(13) i k ϕ 1 ( 2 ) − i k χ + k 2 H 2 4 χ N e 1 ( 2 ) = − ∂ ξ ϕ 1 ( 1 ) + χ + k 2 H 2 2 χ ∂ ξ N e 1 ( 1 ) , − i ω N i 1 ( 2 ) + i k V i 1 ( 2 ) = v g ∂ ξ N i 1 ( 1 ) − ∂ ξ V i 1 ( 1 ) , − i ω V i 1 ( 2 ) + i k ϕ 1 ( 2 ) + i δ i k N i 1 ( 2 ) = v g ∂ ξ V i 1 ( 1 ) − δ i ∂ ξ N i 1 ( 1 ) − ∂ ξ ϕ 1 ( 1 ) , − i ω N b 1 ( 2 ) + i k ( 1 − χ ) V b 1 ( 2 ) = v g ∂ ξ N b 1 ( 1 ) − ( 1 − χ ) ∂ ξ V i 1 ( 1 ) , − i ω V b 1 ( 2 ) + i k μ ϕ 1 ( 2 ) + i μ δ b k ( 1 − χ ) N b 1 ( 2 ) = v g ∂ ξ V b 1 ( 1 ) − μ δ b ( 1 − χ ) ∂ ξ N b 1 ( 1 ) − μ ∂ ξ ϕ 1 ( 1 ) , χ N e 1 ( 2 ) − N i 1 ( 2 ) − ( 1 − χ ) N b 1 ( 2 ) + k 2 ϕ 1 ( 2 ) = 2 i k ∂ ξ ϕ 1 ( 1 ) .

The compatibility condition obtained from the above-reduced equations is

(14) v g = − 1 + 2 H 2 χ 2 ( H 2 k 2 + 4 χ 2 ) + ω 2 ( χ − 1 ) 2 μ ( ω 2 − k 2 ( χ − 1 ) 2 μ δ b ) 2 + 1 ( ω 2 − k 2 δ i ) 2 k ω ( χ − 1 ) 2 μ ( ω 2 − k 2 ( χ − 1 ) 2 μ δ b ) 2 + 1 ( ω 2 − k 2 δ i ) 2 = ∂ ω ∂ k .

In Eq. (14), v g represents the group velocity of the wavepackets. The variation in v g with k for different values of plasma parameters n b o , H , and δ b is illustrated in Figure 2. In every case, the group velocity begins large at small k values and decreases as k increases. It is observed from Figure 2(a) that v g rises with an increase in n b o ; however, for large values of k , it reaches zero slowly. Figure 2(b) shows that, for smaller k values, v g increases with an increase in H ; however, after some values of k , the trend changes and v g decreases with an increase in H . Furthermore, for small k values, v g is not influenced by δ b , and for large k values, v g rises as δ b increases, as shown in Figure 2(c).

Figure 2 The wave packets group velocity v g {v}_{g} is analyzed against (a) ion beam density n b o {n}_{bo} for fixed values of H = 0.1 H=0.1 and δ b = 0.1 {\delta }_{b}=0.1 , (b) quantum diffraction parameter H H for fixed values of n b o = 0.5 {n}_{bo}=0.5 and δ b = 0.1 {\delta }_{b}=0.1 , and (c) ion beam temperature δ b {\delta }_{b} for fixed values of n b o = 0.5 {n}_{bo}=0.5 and H = 0.1 H=0.1 .
Figure 2

The wave packets group velocity v g is analyzed against (a) ion beam density n b o for fixed values of H = 0.1 and δ b = 0.1 , (b) quantum diffraction parameter H for fixed values of n b o = 0.5 and δ b = 0.1 , and (c) ion beam temperature δ b for fixed values of n b o = 0.5 and H = 0.1 .

The reduced equations of second-order, n = 2 , l = 1 , result in

(15) N e 1 ( 2 ) = 4 H 2 k 2 χ + 16 χ 3 ( 4 χ 2 + k 2 H 2 ) 2 ϕ 1 ( 2 ) + i 4 H 2 k χ ( 4 χ 2 + k 2 H 2 ) 2 ∂ ξ ϕ 1 ( 1 ) , N i 1 ( 2 ) = − k 4 δ i − k 2 ω 2 ( ω 2 − k 2 δ i ) 2 ϕ 1 ( 2 ) − i 2 k ω ( ω − k v g ) ( ω 2 − k 2 δ i ) 2 ∂ ξ ϕ 1 ( 1 ) , V i 1 ( 2 ) = − k ω 3 + k 3 ω δ i ( ω 2 − k 2 δ i ) 2 ϕ 1 ( 2 ) + i − ( ω 3 + k 2 ω δ i ) ( ω 2 − k 2 δ i ) 2 + k ω 2 v g + k 3 V g δ i ( ω 2 − k 2 δ i ) 2 ∂ ξ ϕ 1 ( 1 ) , N b 1 ( 2 ) = k 2 ( χ − 1 ) μ ( − ω 2 + k 2 ( χ − 1 ) 2 μ δ b ) ( ω 2 − k 2 μ δ b ( χ − 1 ) 2 ) 2 ϕ 1 ( 2 ) + i A b 1 ∂ ξ ϕ 1 ( 1 ) , V b 1 ( 2 ) = − μ − k ω 3 + k 3 ( χ − 1 ) 2 μ ω δ b ( ω 2 − k 2 μ δ b ( χ − 1 ) 2 ) 2 ϕ 1 ( 2 ) + i A b 2 ∂ ξ ϕ 1 ( 1 ) .

The coefficients A b 1 and A b 2 are given in the Appendix. The second harmonic quantities obtained from the reduced equations for ( n , l ) = ( 2,2 ) read

(16) N e 2 ( 2 ) = ( A 3 3 + A 4 4 A 9 9 ) ∣ ϕ 1 ( 1 ) ∣ 2 , N i 2 ( 2 ) = − ( A 5 5 + A 6 6 A 9 9 ) ∣ ϕ 1 ( 1 ) ∣ 2 , V i 2 ( 2 ) = ( A 10 10 + A 11 11 A 9 9 ) ∣ ϕ 1 ( 1 ) ∣ 2 , N b 2 ( 2 ) = ( − A 7 7 + A 8 8 A 9 9 ) ∣ ϕ 1 ( 1 ) ∣ 2 , V b 2 ( 2 ) = ( − A 12 12 + A 13 13 A 9 9 ) ∣ ϕ 1 ( 1 ) ∣ 2 , ϕ 2 ( 2 ) = A 9 9 ∣ ϕ 1 ( 1 ) ∣ 2 .

The second-order zero harmonic quantities obtained from reduced equations for ( n , l ) = (2, 0) are

(17) N e 0 ( 2 ) = − 16 χ ( H 2 k 2 + 4 χ 2 ) 2 − A 22 22 χ ∣ ϕ 1 ( 1 ) ∣ 2 , N i 0 ( 2 ) = − ( A 14 14 + A 15 15 A 22 22 ) ∣ ϕ 1 ( 1 ) ∣ 2 , V i 0 ( 2 ) = − ( A 16 16 + A 17 17 A 22 22 ) ∣ ϕ 1 ( 1 ) ∣ 2 , N b 0 ( 2 ) = − ( A 18 18 + A 19 19 A 22 22 ) ∣ ϕ 1 ( 1 ) ∣ 2 , V b 0 ( 2 ) = ( A 20 20 + A 21 21 A 22 22 ) ∣ ϕ 1 ( 1 ) ∣ 2 , ϕ 0 ( 2 ) = A 22 22 ∣ ϕ 1 ( 1 ) ∣ 2 .

Finally, we obtain the following cubic NLSE by substituting the above derived expressions into the l = 1 component of the third-order part of the reduced equations:

(18) i ∂ Ψ ∂ τ + P ∂ ξ 2 Ψ + Q ∣ Ψ ∣ 2 Ψ = 0 .

Eq. (18) is referred to as NLSE, with ϕ 1 ( 1 ) = Ψ , where the group dispersion coefficient P and the nonlinear coefficient Q are given by

(19) P = ( − ω 2 + k 2 ( χ − 1 ) 2 μ δ b ) ( ω 2 − k 2 δ i ) 2 ( ω 2 − k 2 δ i ) 2 ( ( 1 − χ ) ω A 1 1 + k ( 1 − χ ) ( χ − 1 ) A 2 2 ) − ( − ω 2 + k 2 ( χ − 1 ) 2 μ δ b ) ( 2 k 2 ω 3 + 2 k 4 ω δ i ) × 1 − 16 ( H 4 k 2 χ 4 + 4 H 2 χ 6 ) ( H 2 k 2 + 4 χ 2 ) 4 + ( k v g − ω ) ( ω 2 ( ω − 3 k v g ) + k 2 ( 3 ω − k v g ) δ i ) ( ω 2 − k 2 δ i ) 3 + ( 1 − χ ) ( ( χ − 1 ) A b 2 ( ω − k v g ) + 2 A b 1 ( ω v g − k ( χ − 1 ) 2 μ δ b ) ) − ω 2 + k 2 ( χ − 1 ) μ δ b ,

and

(20) Q = ( − ω 2 + k 2 ( χ − 1 ) 2 μ δ b ) ( ω 2 − k 2 δ i ) 2 ( ω 2 − k 2 δ i ) 2 ( ( 1 − χ ) ω A 1 1 + k ( 1 − χ ) ( χ − 1 ) A 2 2 ) − ( − ω 2 + k 2 ( χ − 1 ) 2 μ δ b ) ( 2 k 2 ω 3 + 2 k 4 ω δ i ) × − 8 χ 2 ( − 15 H 2 k 2 χ 4 + 32 χ 2 ) ( H 2 k 2 + 4 χ 2 ) 4 − 2 ( 3 H 2 k 2 − 8 χ 2 ) ( χ A 3 3 + χ A 4 4 A 9 9 + A 22 22 ) ( H 2 k 2 + 4 χ 2 ) 2 − k ( 1 − χ ) ( ω A 2 2 ( A 7 7 − A 8 8 A 9 9 ) ) − ω 2 + k 2 ( χ − 1 ) 2 μ δ b − k ( 1 − χ ) ( ω A 1 1 ( − A 12 12 + A 9 9 A 13 13 + A 20 20 + A 21 21 A 22 22 ) − k ( 1 − χ ) A 2 2 ( − A 12 12 + A 9 9 A 13 13 + A 20 20 + A 21 21 A 22 22 ) + k ( χ − 1 ) μ A 1 1 ( A 7 7 − A 8 8 A 9 9 ) δ b ) − ω 2 + k 2 ( χ − 1 ) 2 μ δ b − k 2 ( − ω ( ω A 5 5 + ω A 6 6 A 9 9 + ω ( A 14 14 + A 15 15 A 22 22 ) + 2 k ( − A 10 10 − A 9 9 A 11 11 + A 16 16 + A 17 17 A 22 22 ) ) − k 2 ( A 5 5 + A 6 6 A 9 9 + A 14 14 + A 15 15 A 22 22 ) δ i ) ( ω 2 − k 2 δ i ) 2 − k ( 1 − χ ) ( A 18 18 + A 19 19 A 22 22 ) ( ω A 2 2 + k ( χ − 1 ) μ A 1 1 δ b ) − ω 2 + k 2 ( χ − 1 ) 2 μ δ b

The coefficient P related to wave packets group velocity v g via P = 1 2 ∂ v g ∂ k is examined against k for different values of n b o , H , and δ b as displayed in Figure 3. It is shown that P is always negative in the given range for all parameters ( n b o , H , δ b ). The absolute value of P first rises and reaches its maximum value, then decreases as k increases. It is also observed that a rise in the maximum absolute value of P occurs with an increase in the n b o , as depicted in Figure 3(a). Furthermore, the maximum absolute value of P falls with an increase in H and δ b , as shown in Figure 3(b) and (c).

Figure 3 The coefficient P P is analyzed against (a) ion beam density n b o {n}_{bo} for fixed values of H = 0.1 H=0.1 and δ b = 0.1 {\delta }_{b}=0.1 , (b) quantum diffraction parameter H H for fixed values of n b o = 0.5 {n}_{bo}=0.5 and δ b = 0.1 {\delta }_{b}=0.1 , and (c) ion beam temperature δ b {\delta }_{b} for fixed values of n b o = 0.5 {n}_{bo}=0.5 and H = 0.1 H=0.1 .
Figure 3

The coefficient P is analyzed against (a) ion beam density n b o for fixed values of H = 0.1 and δ b = 0.1 , (b) quantum diffraction parameter H for fixed values of n b o = 0.5 and δ b = 0.1 , and (c) ion beam temperature δ b for fixed values of n b o = 0.5 and H = 0.1 .

The nonlinearity coefficient Q due to carrier wave self-interaction turns negative for larger wave number k , as shown in Figure 4. The modulational stable region corresponds to the positive sign of Q (i.e., Q > 0 or k < k C ). In contrast, the negative sign of Q indicates a modulational unstable region (i.e., Q < 0 or k C < k ).

Figure 4 The coefficient Q Q is analyzed against (a) ion beam density n b o {n}_{bo} for fixed values of H = 0.1 H=0.1 and δ b = 0.1 {\delta }_{b}=0.1 , (b) quantum diffraction parameter H H for fixed values of n b o = 0.5 {n}_{bo}=0.5 and δ b = 0.1 {\delta }_{b}=0.1 , and (c) ion beam temperature δ b {\delta }_{b} for fixed values of n b o = 0.5 {n}_{bo}=0.5 and H = 0.1 H=0.1 .
Figure 4

The coefficient Q is analyzed against (a) ion beam density n b o for fixed values of H = 0.1 and δ b = 0.1 , (b) quantum diffraction parameter H for fixed values of n b o = 0.5 and δ b = 0.1 , and (c) ion beam temperature δ b for fixed values of n b o = 0.5 and H = 0.1 .

4 MI analysis

We now provide a quick overview of the NLSE (18) stability analysis by linearizing around the monochromatic (Stokes’) wave solution [92,93] of the form Ψ = ψ exp [ i Q ∣ ψ ∣ 2 τ ] , where ψ is set to equal ψ o + ε ψ 1 . The perturbation ψ 1 is represented in this case as ψ 1 = ψ 1,0 exp [ i ( k ^ ξ − ω ^ τ ) ] + c.c. , where c.c for the complex conjugate, ω ^ , and k ^ denote perturbed frequency and perturbed wave number. It is important to distinguish ω ^ and k ^ from k and ω . The following nonlinear dispersion relation is obtained for amplitude modulation by substituting it into Eq. (18):

(21) ω 2 ^ = P 2 k 2 ^ k 2 ^ − 2 Q P ∣ ψ 1,0 ∣ 2 .

The stability/instability of plane-wave solutions under external perturbations can be inferred from the signs of P and Q [94]. The carrier wave’s amplitude is modulationally stable if Q ⁄ P is negative, as demonstrated by Eq. (21). In this case, the angular frequency ω ^ remains real for all values of k ^ , and no instability occurs. However, MI begins when ω ^ becomes imaginary for the positive sign of Q ⁄ P . The MI is attained at perturbation wave number k ^ = k C ^ = 2 Q P ∣ ψ ∣ , where ψ denotes the amplitude of the carrier waves. The expression for the MI growth rate obtained from Eq. (21) is

(22) Γ = P k ^ 2 k C 2 k ^ 2 − 1 = Im [ ω ^ ] .

Before we proceed to envelope solitons formation, we should first identify parametric stable and unstable regions based on the sign of Q ⁄ P . The sign of Q ⁄ P depends upon the wave number k and plasma parameters ( n b o , H , δ b ), as shown in Figure 5. For small wave numbers k (i.e., for larger wavelengths), the ratio Q ⁄ P is negative, which suggests modulational stability of the wave packets as expected physically. However, the ratio Q ⁄ P becomes positive for the wave numbers greater than certain critical wave number k C (where Q = 0 or Q ⁄ P = 0 ), indicating that MI begins above k C . For n b o = 0.5 , the value of k C is approximately 1.5, and it approaches greater values as n b o increases, as displayed in Figure 5(a). This depicts that the modulationally unstable region (or growth rate) grows with higher values of n b o . The value of k C is approximately 1.42 for H = 0.1 and reduces as H increases. This shows that the modulationally unstable region diminishes with rising values of H . Our findings are consistent with those of Mishra and Ghosh [95]. The quantitative impact of H and δ b on the k C is quite comparable. The impact of diverse plasma parameters ( n b o , H , δ b ) on the ratio Q ⁄ P is shown in Figure 5. We noted from Figure 5(a) that the absolute value of the ratio Q ⁄ P both in the modulationally stable ( k < k C ) and unstable ( k c < k ) regions increases with rising values of n b o , suggesting the formation of narrower black and bright envelope solitons. The impact of the diffraction parameter H on the ratio Q ⁄ P as a function of k (keeping n b o and δ b constant) is illustrated in Figure 5(b). It is demonstrated that in modulationally stable (i.e., k < k C ) and unstable (i.e., k C < k ) regions, the absolute value of Q ⁄ P increases with higher H values for a given value of k . An analogous quantitative pattern for increasing values of δ b is also seen in Figure 5(c).

Figure 5 The ratio Q ⁄ P Q/P is analyzed against (a) ion beam density n b o {n}_{bo} for fixed values of H = 0.1 H=0.1 and δ b = 0.1 {\delta }_{b}=0.1 , (b) quantum diffraction parameter H H for fixed values of n b o = 0.5 {n}_{bo}=0.5 and δ b = 0.1 {\delta }_{b}=0.1 , and (c) ion beam temperature δ b {\delta }_{b} for fixed values of n b o = 0.5 {n}_{bo}=0.5 and H = 0.1 H=0.1 .
Figure 5

The ratio Q ⁄ P is analyzed against (a) ion beam density n b o for fixed values of H = 0.1 and δ b = 0.1 , (b) quantum diffraction parameter H for fixed values of n b o = 0.5 and δ b = 0.1 , and (c) ion beam temperature δ b for fixed values of n b o = 0.5 and H = 0.1 .

5 Envelope soliton formation

In this study, we examined IAWs in quantum plasma with an ion beam. Model quantum plasma is considered for the numerical analysis by choosing the values for ion beam density n b o and ion beam temperature δ b based on the work of Indrani Paul et al. [83], and quantum diffraction parameter H values from Hass et al. [14], which corresponds to the laser-produced quantum plasmas.

The nonlinear excitations in the form of bright and dark envelope solitons are described by the localized solution of the NLSE (18). By using Ψ ( τ , ξ ) = ( ρ ) 1 2 exp [ i Θ ] . in Eq. (18) and equating real and imaginary parts, the exact expressions for these envelope structures can be determined. For a detailed derivation, the reader is advised to see previous studies [64,93].

The bright soliton solution of NLSE (18) is associated with a positive sign of Q ⁄ P . In the region of larger wave numbers or shorter wavelengths, the carrier wave is modulationally unstable, and it can either grow to bright-type solitons or collapse as a result of random external perturbations. Bright solitons solution represents localized envelope pulses that restrict the fast carrier wave. The exact expressions for bright-type solitons. [93] are given by

(23) Ψ = Ψ m sec h ξ − V τ L , Θ = 1 2 P [ V ξ − ( V 2 ⁄ 2 + Λ ) τ ] ,

where V , L , and Λ denote the envelope velocity, pulse width, and oscillation frequency at rest. It is observed that, in contrast to the KdV equation where L 2 Ψ m = cons tan t , Ψ m and L satisfy L Ψ m = 2 P ⁄ Q = cons tan t (available in complete detail in the study of Kourakis and Shukla [64]). The pulse amplitude is independent of V , and the phase profile of the bright envelope solitons remains the same during their propagation. The slight deformation in the internal structure of the envelope is due to the slow dependency of its phase on space and time. The bright envelope soliton solution (23) is numerically examined against the relevant plasma parameters ( n b o , H , δ b ), as shown in Figure 6. We noted that relevant plasma parameters influence the structure of bright solitons. The amplitude and phase of the bright solitons are shifted with relevant plasma parameters ( n b o , H , δ b ).

Figure 6 Bright envelope soliton solution (23) is plotted in unstable region (i.e., Q ⁄ P > 0 Q/P\gt 0 ) against (a) ion beam density n b o {n}_{bo} for fixed values of H = 0.1 H=0.1 and δ b = 0.1 {\delta }_{b}=0.1 , (b) quantum diffraction parameter H H for fixed values of n b o = 0.5 {n}_{bo}=0.5 and δ b = 0.1 {\delta }_{b}=0.1 , and (c) ion beam temperature δ b {\delta }_{b} for fixed values of n b o = 0.5 {n}_{bo}=0.5 and H = 0.1 H=0.1 .
Figure 6

Bright envelope soliton solution (23) is plotted in unstable region (i.e., Q ⁄ P > 0 ) against (a) ion beam density n b o for fixed values of H = 0.1 and δ b = 0.1 , (b) quantum diffraction parameter H for fixed values of n b o = 0.5 and δ b = 0.1 , and (c) ion beam temperature δ b for fixed values of n b o = 0.5 and H = 0.1 .

Dark envelope soliton solution of NLSE (18) is associated with a negative sign of Q ⁄ P . In the region represented by small k values or longer wavelenghts, the carrier wave is modulationally stable and it may take the form of dark-type solitons, which are propagating localized envelope holes (voids). The exact expressions for dark solitons [93] are given by

(24) Ψ = Ψ m 1 − d 2 sec h 2 ξ − u τ L 1 , Θ = 1 2 P ( u 0 ξ − ( 1 2 u o 2 − 2 P Q Ψ m ) τ ) .

These excitations can appear as black solitons (vanishing potential at ξ = 0 ) or gray solitons (finite potential at ξ = 0 ), the asymptotic values in both cases at infinite are constant and finite. The value of constant d is equal to one for black solitons and less than one for gray solitons. The numerical examination of the dark-type envelope soliton solution (24) against plasma parameters ( n b o , H , δ b ) is displayed in Figures 7 and 8. These figures show that plasma parameters ( n b o , H , δ b ) influence the structure of both dark and gray envelope solitons that slowly shift the amplitude and phase of the modulated dark solitons.

Figure 7 Dark (black) envelope soliton solution (24) is plotted in stable region (i.e., Q ⁄ P < 0 Q/P\lt 0 and d = 1 d=1 ) against (a) ion beam density n b o {n}_{bo} for fixed values of H = 0.1 H=0.1 and δ b = 0.1 {\delta }_{b}=0.1 , (b) quantum diffraction parameter H H for fixed values of n b o = 0.5 {n}_{bo}=0.5 and δ b = 0.1 {\delta }_{b}=0.1 , and (c) ion beam temperature δ b {\delta }_{b} for fixed values of n b o = 0.5 {n}_{bo}=0.5 and H = 0.1 H=0.1 .
Figure 7

Dark (black) envelope soliton solution (24) is plotted in stable region (i.e., Q ⁄ P < 0 and d = 1 ) against (a) ion beam density n b o for fixed values of H = 0.1 and δ b = 0.1 , (b) quantum diffraction parameter H for fixed values of n b o = 0.5 and δ b = 0.1 , and (c) ion beam temperature δ b for fixed values of n b o = 0.5 and H = 0.1 .

Figure 8 Dark (gray) envelope soliton solution (24) is plotted in stable region (i.e., Q ⁄ P < 0 Q/P\lt 0 and d = 0.95 d=0.95 ) against (a) n b o {n}_{bo} while keeping H = 0.1 H=0.1 and δ b = 0.1 {\delta }_{b}=0.1 , (b) H H whiel keeping n b o = 0.5 {n}_{bo}=0.5 and δ b = 0.1 {\delta }_{b}=0.1 , and (c) δ b {\delta }_{b} whiel keeping n b o = 0.5 {n}_{bo}=0.5 and H = 0.1 H=0.1 .
Figure 8

Dark (gray) envelope soliton solution (24) is plotted in stable region (i.e., Q ⁄ P < 0 and d = 0.95 ) against (a) n b o while keeping H = 0.1 and δ b = 0.1 , (b) H whiel keeping n b o = 0.5 and δ b = 0.1 , and (c) δ b whiel keeping n b o = 0.5 and H = 0.1 .

6 Summary

In this article, we have studied the low-frequency IAWs in unmagnetized quantum plasma including the effects of ion beams. One of the reductive perturbation methods, i.e., DEM has been used for reducing the basic plasma model equations to the cubic NLSE. The nonlinearity and dispersion coefficients of the NLSE have been explicitly expressed, and their parametric dependency on the relevant plasma parameters has been investigated. The MI and growth rate of envelope excitations have been estimated based on the coefficients of the NLSE. We noted that the critical wavenumber k C grows as the ion beam density n b o values increase. Conversely, k C decreases when the quantum parameter H and the ion beam temperature ratio δ b increase, suggesting that MI occurs at lower k . It was shown that the instability growth rate decreases when H and δ b values increase. However, it was found that increasing the values of n b o results in an increase in the instability growth rate. Furthermore, the rising values of all relevant plasma parameters ( n b o , H , δ b ) lead to narrower dark and bright solitons in modulationally stable and unstable regions.

For clarity and precision, it is important to highlight that we have utilized a QHD model, which incorporates two key quantum effects, i.e., Bohm potential and and quantum statistical terms. Our model is valid for small values of H . The existence of quantum plasma has been confirmed in astrophysical environments, (viz., neutron stars, white dwarfs, and supernovae remnants) and in laser-driven plasma compression experiments [5]. The number densities in these systems can reach values as high as 1 0 23 and 1 0 32 m − 3 at T Fe = 1 0 5 K , resulting in numerical values of H ∼ 1 and 1 0 − 2 . For technological and laboratory plasmas, the values of H are typically less than one, while they exceed unity for superdense astrophysical plasmas [96]. One of the most significant anticipated applications of this study is the utilization of quantum plasma for the self-cooling of quantum computer components. This entails regulating the instability regions by manipulating plasma parameters that influence their thickness, potentially mitigating the enigmatic phenomena that can raise the temperatures of electronic chips. The study may also be beneficial for analyzing nonlinear effects in sensing signals, signal propagation in optical communications, and the design of optical fibers [97]. Furthermore, our findings may contribute to a better understanding of beam–plasma interactions and the dynamics of nonlinear structures within dense quantum plasmas.

7 Future work

The phenomena of RWs and breathers are among the most enigmatic occurrences, producing many consequences that can significantly influence the features of the system under investigation. Consequently, future research may investigate the characteristics of nonplanar RWs and breathers [98–101], dissipative RWs and breathers [102,103], as well as the methods for regulating their manifestation in the system under examination, depending on the necessity for their presence or avoidance.

Acknowledgments

The authors aknowledge the financial support provided by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R439), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The support of Prince sattam bin Abdulaziz University project number (PSAU/2025/R/1446) is also acknowledged.

  1. Funding information: The authors are thankful for the financial support provided by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R439), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The support of Prince sattam bin Abdulaziz University project number (PSAU/2025/R/1446) is also acknowledged.

  2. Author contributions: Fazal Wahed: formal analysis, investigation, writing – original draft. Ata-ur-Rahman: methodology, writing – review and editing. S. Neelam Naeem: investigation, writing – original draft. R. A. Alharbey: investigation, methodology. Maryam Al Huwayz: formal analysis, writing – original draft. Lamiaa S. El-Sherif: investigation, writing – review and editing. Samir A. El-Tantawy: formal analysis, investigation, methodology, supervision, writing – review and editing. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: All data generated or analysed during this study are included in this published article.

Appendix

The expressions for the coefficients of Eqs (15), (16), and (17) are:

A b 1 = 2 k μ ω ( χ − 1 ) ( ω − k v g ) ( ω 2 − k 2 μ δ b ( χ − 1 ) 2 ) 2 , A b 2 = μ ( − ω 3 − k 2 ( χ − 1 ) ) 2 μ ω δ b + k v g ( ω 2 + k 2 ( χ − 1 ) 2 μ δ b ) ( ω 2 − k 2 μ δ b ( χ − 1 ) 2 ) , A 1 1 = k 2 ( 1 − χ ) μ ω 2 − k 2 ( 1 − χ ) 2 μ δ b , A 2 2 = k μ ω − ω 2 + k 2 μ δ b − 2 k 2 χ μ δ b + k 2 χ 2 μ δ b , A 3 3 = χ ( 6 H 2 k 3 − 8 k χ 2 ) k ( H 2 k 2 + χ 2 ) ( H 2 k 2 + 4 χ 2 ) 2 , A 4 4 = χ ( H 4 k 5 + 8 H 2 k 3 χ 2 + 16 k χ 4 ) k ( H 2 k 2 + χ 2 ) ( H 2 k 2 + 4 χ 2 ) 2 , A 5 5 = − 3 k 4 ω 2 − k 6 δ i 2 ( ω 2 − k 2 δ i ) 3 ,

A 6 6 = − 2 k 2 ω 4 + 4 k 4 ω 2 δ i − 2 k 6 δ i 2 2 ( ω 2 − k 2 δ i ) 3 , A 7 7 = 2 k ω A 1 1 A 2 2 − k 2 ( 1 − χ ) ( A 2 2 ) 2 − k 2 ( 1 − χ ) μ ( A 1 1 ) 2 δ b 2 ω 2 − 2 k 2 ( 1 − χ ) μ δ b , A 8 8 = k 2 ( 1 − χ ) μ ω 2 − k 2 ( 1 − χ ) 2 μ δ b , A 9 9 = − χ A 3 3 − A 5 5 − A 7 7 + χ A 7 7 4 k 2 + χ A 4 4 + A 6 6 − A 8 8 + χ A 8 8 , A 10 10 = ω ( k 3 ω 2 + 3 k 5 δ i ) 2 ( ω 2 − k 2 δ i ) 3 , A 11 11 = ω ( 2 k ω 4 − 4 k 3 ω 2 δ i + 2 k 5 δ i 2 ) 2 ( ω 2 − k 2 δ , A 12 12 = − k ω ( A 2 2 ) 2 − k μ ω ( A 1 1 ) 2 δ b + 2 k 2 μ A 1 1 A 2 2 δ b − 2 k 2 χ μ A 1 1 A 2 2 δ b 2 ( ω 2 − k 2 μ δ b + 2 k 2 χ μ δ b − k 2 χ 2 μ δ b ) ,

A 13 13 = k μ ω ω 2 − k 2 μ δ b + 2 k 2 χ μ δ b − k 2 χ 2 μ δ b A 14 14 = − k 2 ω 2 − 2 k 3 ω v g − k 4 δ i ( v g 2 − δ i ) ( k 2 δ i − ω 2 ) 2 , A 15 15 = − ω 4 + 2 k 2 ω 2 δ i − k 4 δ i 2 ( v g 2 − δ i ) ( k 2 δ i − ω 2 ) 2 , A 16 16 = k 2 ω 2 v g + 2 k 3 ω δ i + k 4 v g δ i ( δ i − v g 2 ) ( ω 2 − k 2 δ i ) 2 , A 17 17 = ω 4 v g − 2 k 2 ω 2 v g δ i + k 4 v g δ i 2 ( δ i − v g 2 ) ( ω 2 − k 2 δ i ) 2 ,

A 18 18 = μ ( χ − 1 ) ( k 2 μ ω 2 + k 3 μ ( 2 ω v g + k ( χ − 1 ) 2 μ δ b ) ( v g 2 − ( χ − 1 ) 2 μ δ b ) ( ω 2 − k 2 ( χ − 1 ) 2 μ δ b ) 2 , A 19 19 = μ ( χ − 1 ) v g 2 − ( χ − 1 ) 2 μ δ b , A 20 20 = μ ( 2 k 3 ( χ − 1 ) 2 μ 2 ω δ b + k 2 μ v g ( ω 2 + k 2 ( χ − 1 ) 2 μ δ b ) ( v g 2 − ( χ − 1 ) 2 μ δ b ) ( ω 2 − k 2 ( χ − 1 ) 2 μ δ b ) 2 , A 21 21 = μ v g v g 2 − ( χ − 1 ) 2 μ δ b , A 22 22 = 16 χ 2 ( H 2 k 2 + 4 χ 2 ) 2 − A 14 14 + ( χ − 1 ) A 18 18 1 + A 15 15 − ( χ − 1 ) A 19 19 .

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Received: 2025-03-08
Revised: 2025-05-11
Accepted: 2025-05-17
Published Online: 2025-07-18

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2025-0180
Keywords for this article
quantum plasma; ion beams; nonlinear Schrödinger equation; modulational instability; ion-acoustic modulated solitons
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  28. Solitonic wave solutions of a Hamiltonian nonlinear atom chain model through the Hirota bilinear transformation method
  29. Bilinear form and soltion solutions for (3+1)-dimensional negative-order KdV-CBS equation
  30. Solitary chirp pulses and soliton control for variable coefficients cubic–quintic nonlinear Schrödinger equation in nonuniform management system
  31. Influence of decaying heat source and temperature-dependent thermal conductivity on photo-hydro-elasto semiconductor media
  32. Dissipative disorder optimization in the radiative thin film flow of partially ionized non-Newtonian hybrid nanofluid with second-order slip condition
  33. Bifurcation, chaotic behavior, and traveling wave solutions for the fractional (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili model
  34. New investigation on soliton solutions of two nonlinear PDEs in mathematical physics with a dynamical property: Bifurcation analysis
  35. Mathematical analysis of nanoparticle type and volume fraction on heat transfer efficiency of nanofluids
  36. Creation of single-wing Lorenz-like attractors via a ten-ninths-degree term
  37. Optical soliton solutions, bifurcation analysis, chaotic behaviors of nonlinear Schrödinger equation and modulation instability in optical fiber
  38. Chaotic dynamics and some solutions for the (n + 1)-dimensional modified Zakharov–Kuznetsov equation in plasma physics
  39. Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation
  40. Single-step fabrication of Mn(iv) oxide-Mn(ii) sulfide/poly-2-mercaptoaniline porous network nanocomposite for pseudo-supercapacitors and charge storage
  41. Novel constructed dynamical analytical solutions and conserved quantities of the new (2+1)-dimensional KdV model describing acoustic wave propagation
  42. Tavis–Cummings model in the presence of a deformed field and time-dependent coupling
  43. Spinning dynamics of stress-dependent viscosity of generalized Cross-nonlinear materials affected by gravitationally swirling disk
  44. Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies
  45. Robust control and preservation of quantum steering, nonlocality, and coherence in open atomic systems
  46. Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow
  47. Dynamic analysis, circuit realization, and its synchronization of a new chaotic hyperjerk system
  48. Decoherence of steerability and coherence dynamics induced by nonlinear qubit–cavity interactions
  49. Finite element analysis of turbulent thermal enhancement in grooved channels with flat- and plus-shaped fins
  50. Modulational instability and associated ion-acoustic modulated envelope solitons in a quantum plasma having ion beams
  51. Statistical inference of constant-stress partially accelerated life tests under type II generalized hybrid censored data from Burr III distribution
  52. On solutions of the Dirac equation for 1D hydrogenic atoms or ions
  53. Entropy optimization for chemically reactive magnetized unsteady thin film hybrid nanofluid flow on inclined surface subject to nonlinear mixed convection and variable temperature
  54. Stability analysis, circuit simulation, and color image encryption of a novel four-dimensional hyperchaotic model with hidden and self-excited attractors
  55. A high-accuracy exponential time integration scheme for the Darcy–Forchheimer Williamson fluid flow with temperature-dependent conductivity
  56. Novel analysis of fractional regularized long-wave equation in plasma dynamics
  57. Development of a photoelectrode based on a bismuth(iii) oxyiodide/intercalated iodide-poly(1H-pyrrole) rough spherical nanocomposite for green hydrogen generation
  58. Investigation of solar radiation effects on the energy performance of the (Al2O3–CuO–Cu)/H2O ternary nanofluidic system through a convectively heated cylinder
  59. Quantum resources for a system of two atoms interacting with a deformed field in the presence of intensity-dependent coupling
  60. Studying bifurcations and chaotic dynamics in the generalized hyperelastic-rod wave equation through Hamiltonian mechanics
  61. A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions
  62. Interaction solutions of high-order breathers and lumps for a (3+1)-dimensional conformable fractional potential-YTSF-like model
  63. Hydraulic fracturing radioactive source tracing technology based on hydraulic fracturing tracing mechanics model
  64. Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet
  65. Numerical investigation of mixed convection and viscous dissipation in couple stress nanofluid flow: A merged Adomian decomposition method and Mohand transform
  66. Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves
  67. Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy
  68. Solitons and travelling waves structure for M-fractional Kairat-II equation using three explicit methods
  69. Impact of nanoparticle shapes on the heat transfer properties of Cu and CuO nanofluids flowing over a stretching surface with slip effects: A computational study
  70. Computational simulation of heat transfer and nanofluid flow for two-sided lid-driven square cavity under the influence of magnetic field
  71. Irreversibility analysis of a bioconvective two-phase nanofluid in a Maxwell (non-Newtonian) flow induced by a rotating disk with thermal radiation
  72. Hydrodynamic and sensitivity analysis of a polymeric calendering process for non-Newtonian fluids with temperature-dependent viscosity
  73. Exploring the peakon solitons molecules and solitary wave structure to the nonlinear damped Kortewege–de Vries equation through efficient technique
  74. Modeling and heat transfer analysis of magnetized hybrid micropolar blood-based nanofluid flow in Darcy–Forchheimer porous stenosis narrow arteries
  75. Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating
  76. Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints
  77. Influence of a magnetic field on double-porosity photo-thermoelastic materials under Lord–Shulman theory
  78. Soliton-like solutions for a nonlinear doubly dispersive equation in an elastic Murnaghan's rod via Hirota's bilinear method
  79. Analytical and numerical investigation of exact wave patterns and chaotic dynamics in the extended improved Boussinesq equation
  80. Nonclassical correlation dynamics of Heisenberg XYZ states with (x, y)-spin--orbit interaction, x-magnetic field, and intrinsic decoherence effects
  81. Exact traveling wave and soliton solutions for chemotaxis model and (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation
  82. Unveiling the transformative role of samarium in ZnO: Exploring structural and optical modifications for advanced functional applications
  83. On the derivation of solitary wave solutions for the time-fractional Rosenau equation through two analytical techniques
  84. Analyzing the role of length and radius of MWCNTs in a nanofluid flow influenced by variable thermal conductivity and viscosity considering Marangoni convection
  85. Advanced mathematical analysis of heat and mass transfer in oscillatory micropolar bio-nanofluid flows via peristaltic waves and electroosmotic effects
  86. Exact bound state solutions of the radial Schrödinger equation for the Coulomb potential by conformable Nikiforov–Uvarov approach
  87. Some anisotropic and perfect fluid plane symmetric solutions of Einstein's field equations using killing symmetries
  88. Nonlinear dynamics of the dissipative ion-acoustic solitary waves in anisotropic rotating magnetoplasmas
  89. Curves in multiplicative equiaffine plane
  90. Exact solution of the three-dimensional (3D) Z2 lattice gauge theory
  91. Propagation properties of Airyprime pulses in relaxing nonlinear media
  92. Symbolic computation: Analytical solutions and dynamics of a shallow water wave equation in coastal engineering
  93. Wave propagation in nonlocal piezo-photo-hygrothermoelastic semiconductors subjected to heat and moisture flux
  94. Comparative reaction dynamics in rotating nanofluid systems: Quartic and cubic kinetics under MHD influence
  95. Laplace transform technique and probabilistic analysis-based hypothesis testing in medical and engineering applications
  96. Physical properties of ternary chloro-perovskites KTCl3 (T = Ge, Al) for optoelectronic applications
  97. Gravitational length stretching: Curvature-induced modulation of quantum probability densities
  98. The search for the cosmological cold dark matter axion – A new refined narrow mass window and detection scheme
  99. A comparative study of quantum resources in bipartite Lipkin–Meshkov–Glick model under DM interaction and Zeeman splitting
  100. PbO-doped K2O–BaO–Al2O3–B2O3–TeO2-glasses: Mechanical and shielding efficacy
  101. Nanospherical arsenic(iii) oxoiodide/iodide-intercalated poly(N-methylpyrrole) composite synthesis for broad-spectrum optical detection
  102. Sine power Burr X distribution with estimation and applications in physics and other fields
  103. Numerical modeling of enhanced reactive oxygen plasma in pulsed laser deposition of metal oxide thin films
  104. Dynamical analyses and dispersive soliton solutions to the nonlinear fractional model in stratified fluids
  105. Computation of exact analytical soliton solutions and their dynamics in advanced optical system
  106. An innovative approximation concerning the diffusion and electrical conductivity tensor at critical altitudes within the F-region of ionospheric plasma at low latitudes
  107. An analytical investigation to the (3+1)-dimensional Yu–Toda–Sassa–Fukuyama equation with dynamical analysis: Bifurcation
  108. Swirling-annular-flow-induced instability of a micro shell considering Knudsen number and viscosity effects
  109. Numerical analysis of non-similar convection flows of a two-phase nanofluid past a semi-infinite vertical plate with thermal radiation
  110. MgO NPs reinforced PCL/PVC nanocomposite films with enhanced UV shielding and thermal stability for packaging applications
  111. Optimal conditions for indoor air purification using non-thermal Corona discharge electrostatic precipitator
  112. Investigation of thermal conductivity and Raman spectra for HfAlB, TaAlB, and WAlB based on first-principles calculations
  113. Tunable double plasmon-induced transparency based on monolayer patterned graphene metamaterial
  114. DSC: depth data quality optimization framework for RGBD camouflaged object detection
  115. A new family of Poisson-exponential distributions with applications to cancer data and glass fiber reliability
  116. Numerical investigation of couple stress under slip conditions via modified Adomian decomposition method
  117. Monitoring plateau lake area changes in Yunnan province, southwestern China using medium-resolution remote sensing imagery: applicability of water indices and environmental dependencies
  118. Heterodyne interferometric fiber-optic gyroscope
  119. Exact solutions of Einstein’s field equations via homothetic symmetries of non-static plane symmetric spacetime
  120. A widespread study of discrete entropic model and its distribution along with fluctuations of energy
  121. Empirical model integration for accurate charge carrier mobility simulation in silicon MOSFETs
  122. The influence of scattering correction effect based on optical path distribution on CO2 retrieval
  123. Anisotropic dissociation and spectral response of 1-Bromo-4-chlorobenzene under static directional electric fields
  124. Role of tungsten oxide (WO3) on thermal and optical properties of smart polymer composites
  125. Analysis of iterative deblurring: no explicit noise
  126. Review Article
  127. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  128. Erratum
  129. Erratum to “On Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model (Open Physics 2021;19:679–682)”
  130. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  131. Possible explanation for the neutron lifetime puzzle
  132. Special Issue on Nanomaterial utilization and structural optimization - Part III
  133. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  134. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  135. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  136. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  137. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
  138. Novel exact solitons to the fractional modified mixed-Korteweg--de Vries model with a stability analysis
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