Interactions of ion acoustic multi-soliton and rogue wave with Bohm quantum potential in degenerate plasma
1. IntroductionIt is well known that the dense electron–positron–ion (epi) plasmas are prevailed in the astrophysical environments, such as in
white dwarfs, neutron stars, and active galactic nuclei,[1–4] and are produced in the laboratory, such as in the intense laser–solid interaction experiments.[5,6] The relativistic or quantum effect becomes prominent due to the fact that the interelectron distance is comparable to the thermal de Broglie wavelength[1] in such high density (of the order of 1030 cm−3) plasmas, e.g., in white dwarfs. In such extremely high density plasmas, the electron thermal energy is much less the electron Fermi energy. According to Pauli’s exclusion principle, the electron thermal pressure can thus be neglected[7] compared to the Fermi degeneracy pressure. The astrophysical objects sustain against the enormous gravitational forces due to the extremely dense degenerate electron pressure. In such case, the tunneling of the plasma species associated with the Bohm potential and statistical Fermi–Dirac pressure plays a significant role[8] on the structures and dynamics of the ion acoustic (IA) waves as observed in different astrophysical and space plasmas,[9,10] such as in white dwarf[1] and magnetars.[11]
References [4] and [12]–[21] have investigated the structures and nonlinear propagation of IA waves considering either degenerate or non-degenerate plasmas. Mamun and Shukla[12] investigated the structures of the shock waves (SWs) in degenerate dense plasma in order to explain the phenomenon observed in white dwarfs. Rahman et al.[17] showed that the positron concentration and the relativistic plasma parameters significantly modify the amplitude, width, and phase velocity of the IA waves in epi plasmas. Haas et al.[18] found that the interaction of plasma particles with the Bohm potential significantly affects the properties of the nonlinear IA waves. Bhowmik et al.[19] studied the effects of quantum diffraction parameter (H) and equilibrium plasma species density ratio on the propagation of electron acoustic waves in the quantum plasmas. Recently, Hossen et al.[8] considered a degenerate epi plasma system consisting of inertial non-relativistic light ions, degenerate electrons, and positrons and noted the effects of tunneling of the plasma particles with the Bohm potential on the propagation of IA waves by deriving the KdV, mKdV, and mixed mKdV (mmKdV) equations. On the other hand, many authors[22–29] have studied the propagation phenomena of the rogue waves by transforming the KdV, mKdV, and mmKdV equations to the corresponding NLS equations. Besides, Mandal et al.[30] have investigated only the overtaking collisions and phase shifts of dust acoustic multi-solitions in a four-component dusty plasma. However, the interactions between single- and multi-solitons and hence the production of rogue waves along with their structures and dynamics are still unrevealed in the plasmas[8] for better understanding the physical issues observed in space plasmas.[25–29,31–38] Being motivated, for the significance of the problems related to the astrophysical and laboratory plasmas, we study the interaction processes of the IA single-, and multi-solitons, and their phase shifts as well as rogue waves in an unmagnetized plasmas composing degenerate electrons and positrons including Bohm quantum potential and an inertial non-relativistic light ions. The effects of helium ion mass and density, degenerate electrons, and positrons densities and temperatures on the phase shift, interactions among the IA single- and multi-solitons, and the production of rogue waves are investigated.
The paper is organized as follows. The theoretical model and derivations of two-sided KdV equations are presented in Section 2. The single and multi-solitons solutions of the KdV equations and their corresponding phase shifts are illustrated in Section 3. Derivation of nonlinear Schrödinger equation (NLSE) along with the solution of its rational function is displayed in Section 4. The results and discussion are described in Section 5. Finally, the conclusion is drawn in Section 6.
2. Governing equations2.1. Model equationsThe electrons (positrons) remain non-relativistic if their Fermi energy is less than their rest mass energy[39] in high density plasmas. On the other hand, the electrons have Fermi energy that is comparable to or greater than their rest mass energy, and hence the electron Fermi speed turns out to be comparable to the speed of light in vacuum for the number densities in the range of 1029–1034 cm−3. Chandrasekhar[4,40] explained the equation of state for degenerate plasmas, as observed in an astrophysical compact object, namely, white dwarf, considering degenerate pressure exerted by plasma fluid as 
, where α = 5/3, Ks ≈ (3/5)Λcℏc, (Λc = πℏ/msc = 1.2 × 10−10 cm, ℏ = h/2π) for the non-relativistic and γ = 4/3, Ks ≈ (3/4)ℏc for the ultra-relativistic cases, ms and ns are the plasma species mass and density, and h and c are the Planck constant and velocity of light in free space, respectively. In this report, an unmagnetized dense epi plasma system is considered composing both non-relativistic and relativistic degenerate electrons and positrons, and inertial non-relativistic light ions. ne0 = np0+ ni0 is considered as the quasi-neutrality condition, where ne0, np0, and ni0 are the densities of the unperturbed electrons, positrons, and ions, respectively. To study the structures and dynamics of the IA waves including their electrostatic resonance phenomena, corresponding phase shifts, and production of rogue waves in the considered plasmas, the normalized fluid equations[8] can be written as
 | |
 | |
 | |
 | |
 | |
Here,
ns (
s = i, e, p) is normalized by their equilibrium counterparts
ns0,
ui is the ion fluid speed normalized by ion acoustic speed
Ci = (
kBTFe/
mi)
1/2,
ϕ is the electrostatic potential normalized by (
kBTFe/
e),
η1 = (
TFp/
TFe),
μ = (
ne0/
ni0),
σ = (
np0/
ni0),
,
,
j = e, p,
kB is the Boltzmann constant,
mi is the ion rest mass, and
TFp (
TFe) is the Fermi temperature of positron (electron). The time and space variables are normalized by
and
λDi = (
kBTFe/4
πni0e2)
1/2, respectively. The remaining dimensionless quantum parameters are
and
λ (
), where
He =
ℏωpe/
kBTFe and
Hp =
ℏωpp/
kBTFp introduced in Eqs. (
3) and (
4) due to the tunneling effect associated with the Bohm potential.
[41] 2.2. Derivation of two-sided KdV equationsThe two-sided KdV equations are derived by employing the extended Poincaré–Lighthill–Kuo (PLK) method[38]. According to the extended PLK method, the scaling variables x and t are defined as
and the perturbed quantities can be written as
where
ξ and
η indicate the trajectories between the IA waves that are propagating toward each other,
Vp is the unknown phase velocity of the IA waves normalized by
Ci, and
ε is a small parameter. The unknown phase functions
X0(
η,
τ) and
Y0(
ξ,
τ) will be evaluated later. Substituting Eqs. (
6) and (
7) into Eqs. (
1)–(
5) and equating the quantities with the same orders of
ε, one can obtain coupled equations. Taking the lowest order of
ε gives the following equations:
where
K1′ =
K1α and
K2′ =
K2γ. One may define the relations along with different physical quantities
and
taking Eqs. (
8) to (
11) into account as
Equation (
13) indicates the two-sided electrostatic waves, one of which
is propagating to the right direction from
ξ = 0,
η → −∞ to
ξ = 0,
η → +∞ and the other
is propagating to the left direction from
η = 0,
ξ → +∞ to
η = 0,
ξ → +∞. By inserting Eq. (
13) into Eq. (
12), the phase velocity is obtained as
Again, the next order of
ε provides a set of equations in terms of the second order perturbed quantities, similar to Eqs. (
8)–(
12). One may also define the physical quantities
and
similar to those of Eq. (
13). Considering the next higher order of
ε yields a set of nonlinear evolution equations that are presented in Appendix A (Eqs. (
A1)–(
A5)). Simplifying Eqs. (
A1)–(
A5) with the help of Eq. (
13) and then integrating with regards to
ξ and
η yield
where
It is clearly seen that the first and the second terms in the left hand side of Eq. (
15) are proportional to
η and
ξ, respectively. Therefore, all the terms involved in the first two expressions of the left hand side of Eq. (
15) become secular, which may be eliminated in order to stay away from resonances.
[42] The following two-sided KdV equations are obtained to study the resonance phenomena of the electrostatic potential:
The third and the fourth terms of the left hand side of Eq. (
15) may become secular terms in the next higher order and yield the following equations:
Equations (
18) and (
19) indicate that
X0 and
Y0 are the functions of
η and
ξ, respectively, which may be calculated with the help of analytical soliton solutions of KdV equations (
16) and (
17).
3. Soliton solutions and phase shiftsTo study the nonlinear propagation characteristics of the interactions of the IA solitons and their phase shifts in the plasmas, one has to derive the analytical soliton solutions of the KdV Eqs. (16) and (17) using the well established Hirota bilinear method.[43,44] According to this method, the single soliton solutions of Eqs. (16) and (17) can be obtained as
where
and
. Simplifying Eqs. (
18) and (
19) and taking Eqs. (
20) and (
21) into account, the leading phase changes due to the interaction of IA solitary waves can be defined as
The trajectories of the solitary waves for weak interactions can be reduced to
To evaluate the phase shifts after interactions between the IA solitons, one may assume that the solitons, say
R and
L are asymptotically far away from each other at the initial time, that is, soliton
R is at
ξ = 0,
η → + ∞ and
L is at
η = 0,
ξ → + ∞. After interaction,
R is at
ξ = 0,
η → + ∞ and
L is at
η = 0,
ξ → −∞ to the right of
L. The corresponding phase shifts are obtained as
Again, the double-soliton solutions of the KdV equations can be written as
where
,
and
a12 = (
k2−
k1)
2/(
k2+
k1)
2 with
i = 1, 2. By using Eqs. (
28) and (
29), the solution of Eqs. (
18) and (
19) can written as
and the corresponding phase shifts may be obtained due to the interactions of the double solitons as
Finally, the triple-soliton solutions of the KdV equations can be written as
where
,
,
a12 = (
k1−
k2)
2/(
k1+
k2)
2,
a23 = (
k2−
k3)
2/(
k2 +
k3)
2,
a13 = (
k1−
k3)
2/(
k1+
k3)
2, and
a123 =
a12a23a13 with
i = 1,2,3, and their corresponding phase shifts due to the interaction of triple solitons may be written as
4. Derivation of NLSE with rogue wave solutionTo study the behavior of weakly nonlinear wave packets in the plasmas, one can derive the NLSE considering the KdV equation as mentioned in Eq. (16). For simplicity, one can use the following equation for the transformation of variables:
where
,
k is the wave number,
ω is the angular frequency, and
vg is the group velocity of the nonlinear IA waves. By using the KdV Eq. (
16), the following NLSE is obtained:
where
P = 6B
k and
Q = −(
A2/6Bk). The rational function solution of Eq. (
38) can be written as
It is seen that the ratio of
P and
Q obtained is always negative, that is
P/
Q = −1/
A2. Moreover, the weakly nonlinear theory predicts that quasi-monochromatic wave packets are always modulationally stable and the rogue waves cannot propagate due to the existence of the negative nonlinear coefficient terms in the NLSE. This indicates that the NLSE obtained from the KdV equation may not support the rogue wave solution.
On the other hand, there may arise the cases where the nonlinear coefficient (A) vanishes at the critical value (σc) for certain plasma parameters and equation (16) fails to study the nonlinear evolution of perturbation. For instance, A vanishes at σ = σc ≈ 0.0195 for λ = 0.10, β = 0.10, γ = 4/3, μ = 0.1 and η1 = 0.8. In such a case, one can derive the mKdV equation by employing the stretched coordinates ξ = ε (x − Vpt), τ = ε3t and perturbed quantities as 
into Eqs. (1)–(5), where 
, 
, and 
. The lowest power of ε gives the first order perturbed quantities as 
, 
, and 
.[8] The next higher power of ε provides the second order perturbed quantities 
, 
, 
, and 
that are presented in Appendix A (Eqs. (A6) and (A7)). Finally, the next higher order of ε gives the following mKdV equation by taking the first and the second order perturbed quantities into account:[8]
where
In order to obtain the rogue wave, substitute Eqs. (
36) and (
37) to the mKdV Eq. (
40) and collect terms having the same order of
ε. The lowest order approximation for
m = 1 with the first harmonic
l = 1 gives the dispersion relation of the electrostatic waves as
ω = −
Bk3. The second order approximation for
m = 2 with
l = 1 predicts
vg = −3
Bk2. Finally, the compatibility condition for the equation that is obtained from the next order (
m = 3,
l = 1) provides the NLSE as mentioned in Eq. (
38). The nonlinearity (
Q) and dispersive coefficients (
P) are obtained as
5. Results and discussionTwo-sided KdV and NLSE equations are derived to investigate the unrevealed physical issues in the plasma considered, such as temporal evolution of electrostatic resonances and phase shifts due to the interaction of single and multi-solitons, and modulus instability. The effects of the plasma parameters on the temporal evolution of the electrostatic resonances, phase shifts, and rogue waves are investigated considering helium ion mass mi = 6.68 × 10−24 kg, ion density ni0 = 3.0 × 1031−3, electron density ne0 = 9.11 × 1029 cm−3, positron density np0 = 1.5 × ne0, quantum parameters β = 0.01–0.5 and λ = 0.01–0.5 which are for the relativistic degenerate astrophysical plasmas.[4,12,45] The results obtained from this study are described below.
When two or more solitary waves propagate toward each other, they will interact and exchange their energies among themselves, and then separate off, regain their original wave forms. During the whole process of the interactions, the solitary waves are remarkably stable entities, preserving their identities through interaction. Each soliton gains two phase shifts, one is due to the head-on interaction and the other one is due to overtaking one soliton by the other one. On the other hand, the unique effect is due to the interactions that change their phase shifts. The phase shifts become either positive or negative due to head-on and overtaking interactions of the solitons, which is independent of the wave modes.[31–38] Figures 1(a)–1(f) illustrate the effects of σ and μ on the phase shift ∇Y0 due to the interaction among two, four, and six equal amplitude IA solitons in both ultra-relativistic and non-relativistic cases, respectively, taking the remaining parameters constant. It is seen that the changes of the phase shifts are decreasing with the increase of positron to ion density ratio σ = (np0/ni0). This means that with the increase of the positrons density, their interaction with electrons increases having opposite charges due to the contribution to the higher restoring forces. On the other hand, the changes of the phase shifts are increasing with the increase of μ = (ne0/ni0) from 0.2 to 0.35 for ultra-relativistic and from 0.2 to 0.25 for non-relativistic cases due to the interaction of oppositely propagating single and multi-solitons, and then decreasing significantly. It is provided that the electrons can contribute to the restoring force due to their small number density in the range 0.2 ≤ μ ≤ 0.35 and then the restoring force increases because the electrostatic interaction between electrons and positrons increases in the range μ > 0.35. Figures 2(a)–2(b) show the changes of phase shifts ∇Y0 with β for the interaction between equal amplitude single and multi-solitons in both cases, respectively, considering the fixed values of the remaining parameters. The quantum parameter β is mainly arisen due to the Bohm potential, which is solely responsible for the tunneling effect of the corresponding plasma components. It is clear from Fig. 2 that the Bohm quantum potential significantly affects the phase shifts in which the changes of the phase shifts are decreasing with the increase of quantum parameter β. This phenomenon indicates that the electrons interact more actively with the helium ions, causing the reduction in the magnitude changes of the phase shift. Thus, it can be concluded that the phase shifts due to the interaction of two-sided single and multi-solitons are strongly dependent on the plasma parameters and the wave numbers.
Figures 3(a)–3(d), 4(a)–4(d), and 5(a)–5(d) display the two-sided equal amplitude electrostatic potential structures 
for single, double, and tripple-soliton against ξ and η, respectively, with different values of β and τ taking the remaining parameters constant for both ultra-relativistic and non-relativistic cases. It is seen from these figures that the amplitudes and widths of both right and left moving solitons are decreasing with increasing β. The quantum parameter β is mainly arisen due to the influence of the Bohm potential and exclusively related to the tunneling effect of the corresponding plasma component. The tunneling effect is also increasing with the increase of β. It is ensured that the electrons interact more actively with the ions, causing the reduction in amplitude and width of solitons, which is in agreement with the theoretical finding of Ref. [8]. Furthermore, the potential profiles 
of the solitons are shifted towards the right, while 
are shifted towards the left with increasing τ. This dictates that the position of solitons R is at ξ = 0, η → −∞ and that of L is at η = 0, ξ → + ∞ before interaction and they collide at time t → 0 and then the soliton R is at ξ = 0, η → + ∞ and L is at η = 0, ξ → +∞ after interaction, such interaction phenomena are displayed in Figs. 3–5.
On the other hand, it is found that the NLSE obtained from the KdV equation does not support rogue wave solutions due to the modulational stability of the quasi-monochromatic wave packets. But, the NLSE obtained from the mKdV equation supports the rogue wave solutions for the considered plasma parameters as well as the critical value for which the nonlinear coefficient A of the KdV equation is zero. Figures 6(a) and 6(b) show the influences of σ and β on the rogue waves, respectively, taking different values of μ and remaining parameters constant for ultra-relativistic degenerate electrons and positrons. It is seen that the amplitudes of the rogue waves are decreasing with increasing β, σ and μ in the plasmas. The quantum parameter β arises due to the Bohm potential which is solely responsible for the tunneling effect of the corresponding plasma component. The tunneling effect becomes pronounced with the increase of β, which dictates that the electrons interact more actively with the ions, causing the reduction of amplitude of rogue waves. Furthermore, the amplitude of the rogue waves decreases significantly due to the increase of σ and μ in the aforementioned plasma system, which predicts that the electrostatic interaction between electrons and positrons increases, and thus their contribution to the restoring force increases in the plasmas. Finally, figures 7(a)–7(d) display the existence regions (red color) of the rouge waves with repect to μ, β, σ, and η1 along with k, respectively, taking the remaining parameters as constant for ultra-relativistic degenerate electrons and positrons.
6. SummaryThe interactions between the IA solitons, their phase shifts, and the production of rogue waves are investigated by considering the soliton solution of the two-sided KdV equations and the rational function solution of the NLSE, respectively. It is found that the quantum parameters become prominent due to the Bohm potential, which significantly modifies the propagation characteristics due to the interactions of the small amplitude long-lived solitons as well as large amplitude short-lived rogue waves in the plasmas. The results obtained in this study might be useful for the understanding of the effects of electrostatic resonance and phase shifts after weak interaction between multi-solitons and rogue waves for astrophysical compact objects, e.g., white dwarfs, neutron stars, etc., and for laboratory plasmas like intense laser–solid matter interaction experiments.
