Nucleus-acoustic envelope solitons and their modulational instability in a degenerate quantum plasma system

Accepted Manuscript Nucleus-acoustic envelope solitons and their modulational instability in a degenerate quantum plasma system N.A. Chowdhury, M.M. Hasan, A. Mannan, A.A. Mamun PII:

S0042-207X(17)31149-1

DOI:

10.1016/j.vacuum.2017.10.004

Reference:

VAC 7633

To appear in:

Vacuum

Received Date: 24 August 2017 Revised Date:

5 October 2017

Accepted Date: 5 October 2017

Please cite this article as: Chowdhury NA, Hasan MM, Mannan A, Mamun AA, Nucleus-acoustic envelope solitons and their modulational instability in a degenerate quantum plasma system, Vacuum (2017), doi: 10.1016/j.vacuum.2017.10.004. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Nucleus-acoustic envelope solitons and their modulational instability in a degenerate quantum plasma system N. A. Chowdhury∗ , M. M. Hasan, A. Mannan, and A. A. Mamun

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Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh E-mail: [email protected]∗

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Abstract

The existence and the basic features of nucleus-acoustic (NA) envelope bright and dark solitons in a degenerate quantum plasma system (DQPSs) (containing non-relativistically degenerate nuclei and inertialess ultra-relativistically degenerate electrons and positrons), have been theoretically investigated by deriving the nonlinear Schr¨ odinger (NLS) equation. The reductive perturbation method, which is valid for a small but finite amplitude limit, is employed. The

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plasma parametric regimes for both modulationally stable and unstable NA waves (NAWs) are observed. It is found that the growth rate of the modulationally unstable NAWs is significantly modified by the number density of nucleus species. It is also observed that the modulationally stable (unstable)

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NAWs give rise to stable dark (bright) envelope solitons. The implications of our results obtained from our present investigation in astrophysical and laboratory DQPSs are briefly discussed.

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Keywords: Nucleus acoustic waves, envelope solitons, degenerate plasmas.

1. Introduction Nowadays, the nonlinear wave dynamics (viz. solitons [1, 2, 3], envelope

solitons [4], shock [5], vortices, and other nonlinear phenomena [6, 7, 8]) in degenerate quantum plasma systems (DQPSs) has received a renewed interest in understanding the localized electrostatic disturbances propagating in such a

Preprint submitted to Vacuum

October 11, 2017

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plasma system. The degenerate plasmas occur not only in astrophysical plasma

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situations (viz. white dwarfs, neutron stars, black holes, etc. [9, 10, 11, 12, 13]), but also in laboratory plasma conditions (viz. solid density plasmas [14, 15],

ultra-cold plasmas formed from laser-cooled atoms by the ionizing action of an

intense laser pulse [16, 17], laser produced plasmas formed from sold targets irradiating by intense laser [18], etc.). The particle number density of these

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compact objects is so high (order of 1030 cm−3 in white dwarfs, and order of 1036 cm−3 even more in neutron stars) that the de Broglie wavelength of particles is comparable to the inter-particle distance [11, 12]. Thus, the degenerate

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pressure and quantum effects, which play an important role in understanding the dynamics of the constituents charged particles; viz. electrons, positrons, and nuclei [19, 20, 21] of these compact objects, need to be considered. Chandrasekhar [9] first assumed that hydrogen/helium nuclei and degenerate electrons are the main constituents of compact objects like white dwarfs. On the other hand, Koester [11] noticed that instead of hydrogen/helium nuclei, white dwarfs unusually contain carbon/oxygen. The average density of particles in the core

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of white dwarfs is typically high (∼ 1030 cm−3 ) compared with outer mantle (∼ 1026 cm−3 ), due to this, the particles are relativistically degenerate in the inner core of white dwarfs, but are non-relativistically degenerate in the outer mantle [22]. Recently, Payload for Antimatter Matter Exploration and Light-

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nuclei Astrophysics (PAMELA) satellite has measured the cosmic-ray positron fraction [23, 24]. Kashiyama and Ioka [25] suggested that white dwarf pulsars can compete with neutron star pulsars for producing the excesses of cosmic ray

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positrons and electrons (e+ and e− ). This means that in addition to degenerate electrons and C or O nuclei, white dwarfs contains degenerate positrons [23, 24, 25].

A number of works [26, 27, 28, 29, 30] have been done by considering the

degenerate electron-positron-ion plasma. Mehdipoor and Esfandyari-Kalejahi [31] studied the ion acoustic (IA) waves in a plasmas system consisting of degenerate electrons, positrons, and isothermal ions and found that higher order corrections significantly change the properties of the Korteweg-de Vries (K-dV) 2

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solitons, and under certain conditions, both compressive and rarefactive solitary

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waves exist in such a plasma system. The investigations of the modulational instability (MI) of waves, both theoretically and experimentally have been exponentially increasing day by day, due to their rigorous successful applications

in space, laboratory plasmas, ocean wave, and optics, etc [8, 32]. Eslami et al. [32] studied MI of IA waves in a plasma system consisting of inertial ions and

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inertialess nonextensive electrons and positrons. Chandra and Ghosh [33] ex-

amined the MI of electron-acoustic waves by using the quantum hydrodynamic model, and observed that the relativistic and degeneracy parameters modify the

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stability condition, and properties of the envelope solitons. El-Taibany [34] have investigated the nonlinear propagation of fast and slow magnetosonic perturbation modes in non-relativistic, ultra-cold, degenerate electron-positron plasma and found that the basic features of the electromagnetic solitary structure are significantly modified by the effects of degenerate electrons and positrons pressures. Nahar et al. [35] investigated IA K-dV and modified K-dV (m-KdV) solitons in a degenerate electron-ion dense plasma containing non-relativistically

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degenerate cold ions, and both non-relativistic and ultra-relativistic degenerate electrons, and found that the plasma system under consideration supports the propagation of electrostatic solitary structures. Recently, in a cold degenerate plasma, Mamun et al. [36] introduced a con-

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cept of new electro-acoustic (NA) mode in which the inertia is provided by the nucleus mass density, and the restoring force is provided by the degenerate pressures of inertialess particle species, which depends only on particle number

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density, but not on their thermal temperature. It is a new mode since it disappears if we neglect the effect of electron or positron degeneracy. It should be noted here that ion/electron/positron-acoustic waves do not exist in cold plasma limit, but the new NA waves (NAWs) exist in cold plasma limit. They studied the NA shock [36] and solitary [37] structures associated with this new mode. To the best knowledge of the authors no attempt has been made on MI and corresponding dark and bright envelope solitons associated with the NAWs in a DQPS containing non-relativistically degenerate nuclei and iner3

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tialess ultra-relativistically degenerate electrons and positrons. Therefore, in

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our present work, we will examine the conditions for the MI of the NAWs (in which the inertia is provided by the mass density of inertial, non-relativistically degenerate nuclei of helium/carbon/oxigen, and the restoring force is provided by inertialess, ultra-relativistically degenerate electrons and positrons), and we

identify the new basic features of the bright and dark envelope solitons formed

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in DQPSs.

The present paper is organized as follows. The basic governing equations of our plasma model are given in Sec. 2. The derivation of the nonlinear

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Schr¨ odinger (NLS) equation is derived in Sec. 3. The stability analysis of the NAWs is examined in Sec. 4. The basic features of envelope solitons are provided in Sec. 5. A brief discussion is presented in Sec. 6.

2. Governing Equations

We consider a DQPS containing inertial non-relativistically degenerate nu-

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cleus species n, and inertialess ultra-relativistically degenerate electron and positron species e and p, respectively. The dynamics of the NA waves propagating in such a DQPS are described by ∂ϕ ∂Pe − =0 ∂X ∂X ∂ϕ ∂Pp eNp + = 0. ∂X ∂X ∂Nn ∂ + (Nn Un ) = 0, ∂T ∂X ∂Un Zn e ∂ϕ 1 ∂Pn ∂Un + Un =− − , ∂T ∂x mn ∂X mn Nn ∂X ∂2ϕ = 4πe[Ne − Np − Zn Nn ], ∂X 2

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eNe

(1) (2) (3) (4) (5)

where Ne , Np , and Nn is the number densities of the electrons, positrons, and

nuclei, respectively; Un is the is nucleus fluid speed and ϕ is the electrostatic

wave potential; mn (Zn ) is the mass (charge state) of the nucleus, and e is the charge of the positron/proton; T (X) is the time (space) variable; Pe , Pp , and

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Pn are the degenerate pressure associated with degenerate electrons, positrons,

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and nuclei pressure, respectively, and are given [9] by Ps = Ks Nsγ ,

(6)

where s represents electron or positron or nucleus species, i.e. s = e for the

electron species, s = p for the positron species, and s = n for the nucleus

γ=

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species;

5 3  π  13 π~2 3 ; Ks = ' Λcs ~c, 3 5 3 ms 5

(7)

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for the non-relativistic limit [36, 37] (with Λcs = π~/ms , ~ is the Planck constant (h) divided by 2π, ms is the mass of the species s, and c is the speed of light in vacuum), and γ=

3 4 ; K= 3 4



π2 9

 13

~c '

3 ~c, 4

(8)

for the ultra-relativistic limit [36, 37]. We note that for ultra-relativistic limit

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the subscript s on K is dropped out since it is independent on species s. It may be noted here that the term ‘inertial’ is used before the species which provides the inertia in the NAWs, and that the non-relativistically degenerate nucleus species are

1 1H

or

4 2 He

or

12 6C

or

16 8O

[9, 10, 11, 12]. It should also be

added here that to satisfy the plasma condition we have neglected the effects

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of annihilation, pair production, recombination, ionization, any kind of reaction among different nuclear species. Now, introducing normalized variables, namely, ns = Ns /ns0 , un = Un /Cn ,

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φ = ϕ/E0 , t = T ωph , x = X/λDm [where ns0 is the equilibrium number 1/3

1/3

density of the species s, Cn = (3~cZn ne0 /mn )1/2 , E0 = 3~cne0 /e, ωpn = (4πZn2 e2 nn0 /mn )1/2 , and λDm = (E0 /4πZn e2 nn0 )1/2 ], and using Eqs. (6) −

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(8), (1)− (5) can be expressed in normalized form as (9)

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∂φ γKe0 ∂nγ−1 e = , ∂x (γ − 1) ∂x γKp0 ∂nγ−1 ∂φ e =− , ∂x (γ − 1) ∂x ∂nn ∂ + (nn un ) = 0, ∂t ∂x 2/3 ∂un ∂un ∂φ ∂nn + un =− −β , ∂t ∂x ∂x ∂x µe ∂2φ 1 = ne − np − nn , ∂x2 (µe − 1) (µe − 1) where Ke nγ−1 eo

(11) (12) (13)

Kp nγ−1 po

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Ke0 =

(10)

, 1/3 3~cne0

Kp0 =

, 1/3 3~cne0   2/3 Λcn nn0 ne0 β= , and µ = . e 1/3 np0 2Zn n e0

We note that in order to get the right side of Eq. (13), the quasi-neutrality condition at equilibrium, viz. Zn nn0 + np0 = ne0 has been used. To make our mathematical steps simple, we now substitute Eqs. (9) and (10) into Eq. (13),

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expand the resulting equation, and reduce Eqs. (9), (10), and (13) to ∂2φ + nn − (1 + γ1 φ + γ2 φ2 + γ3 φ3 + · · ··) = 0, ∂x2

where

(14)

γ2 = 3(µe − µ2/3 e )/(µe − 1), γ3 = 2µe /(µe − 1),

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γ1 = 3(µe + µ1/3 e )/(µe − 1),

We note that the last term on the left hand side is the contribution of electron and positron species. Thus, the coupled Eqs. (11), (12), and (14) describe the dynamics of the NAWs in the DQPS under consideration. 3. NLS equation There are different methods to derive the NLS equation. However, to study the MI of the small but finite amplitude NAWs and corresponding bright and 6

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dark solitons, one can derive the NLS equation by employing the reductive

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perturbation method [4, 38, 39]. So, we first introduce the stretched co-ordinates ξ = (x − vg t),

(15)

τ = 2 t,

(16)

where vg is the envelope group velocity to be determined later and  (0 <  < 1)

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is a small (real) parameter. Then we can expand the dependent variables in power series of  as ∞ X m=1

un = φ=

∞ X

(m)

m=1 ∞ X

(m)

nnl (ξ, τ ) exp[il(kx − ωt)],

l=−∞

∞ X

(m)

l=−∞ ∞ X

(m)

m=1

∞ X

(m)

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nn = 1 +

(17)

unl (ξ, τ ) exp[il(kx − ωt)],

(18)

(m)

(19)

φl

(ξ, τ ) exp[il(kx − ωt)],

l=−∞

where k (ω) is the fundamental carrier wave number (frequency). The derivative operators in (11), (12), and (14) are treated as

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∂ ∂ ∂ ∂ → − vg + 2 , ∂t ∂t ∂ξ ∂τ ∂ ∂ ∂ → + . ∂x ∂x ∂ξ

(20) (21)

Now, substituting Eqs. (17)−(21) into Eqs. (11), (12), and (14), and collecting

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the terms containing , the first order (m = 1 with l = 1) equations can be

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expressed as

(1)

(1)

iku1 − iωn1 = 0, (1)

(22)

ikβ1 n1 − iωu1 + ikφ1 = 0,

(1)

(23)

(1) n1

= 0,

(24)

−

(1)

(1) k 2 φ1

−

(1) γ1 φ1

where β1 = 2β/3. These equations reduce to (1)

k 2 (1) φ , S 1 kω (1) = φ , S 1

n1 = (1)

u1

7

(25) (26)

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where S = ω 2 − β1 k 2 . We thus obtain the dispersion relation for the NAWs: (k 2

k2 + β1 k 2 . + γ1 )

(27)

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ω2 =

On the other hand, the second-order (m = 2 with l = 1) equations are given by (1)

(2)

n1 =

k 2 (2) 2ikω(vg k − ω) ∂φ1 φ + , S 1 S2 ∂ξ

(28)

(1)

kω (2) i(ω 2 + β1 k 2 )(vg k − ω) ∂φ1 φ + , S 1 S2 ∂ξ

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(2)

u1 =

with the compatibility condition

∂ω ω 2 − (ω 2 − β1 k 2 )2 = . ∂k kω

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vg =

(29)

(30)

The second-harmonic mode of the carrier comes from nonlinear self interaction caused by the components (l = 2) for the second order (m = 2) reduced equations in the form

(2)

(1)

(31)

(2)

(1)

(32)

(1)

(33)

n2 = C1 |φ1 |2 ,

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u2 = C2 |φ1 |2 , (2)

φ2 = C3 |φ1 |2 ,

where

3ω 2 k 4 + 2C3 S 2 k 2 , 2S 3 2 4 ωC1 S − ωk C2 = , kS 2 3ω 2 k 4 − 2γ2 S 3 C3 = . 3 2S (4k 2 + γ1 ) − 2S 2 k 2

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C1 =

Now, we consider the expression for (m = 3, l = 0) and (m = 2, l = 0), which

leads the zeroth harmonic modes. Thus, we obtain (2)

(1)

(34)

(2)

(1)

(35)

(2)

(1)

(36)

n0 = C4 |φ1 |2 , u0 = C5 |φ1 |2 , φ0 = C6 |φ1 |2 ,

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C4 =

2vg ωk 3 + ω 2 k 2 − β2 k 4 + C6 S 2 , S 2 (vg2 − β1 )

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where

β vg C4 S 2 − 2ωk 3 , β2 = S2 9 2vg ωk 3 + k 2 ω 2 − β2 k 4 − 2γ2 S 2 (vg2 − β1 ) C6 = . γ1 S 2 (vg2 − β1 ) − S 2

C5 =

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The third harmonic (m = 3) with (l = 1) modes can be obtained with the help of Eqs. (25)−(36). This system of third harmonic equations can finally be reduced to the NLS equation in the form

(1)

∂Φ ∂2Φ + P 2 + Q|Φ|2 Φ = 0, ∂τ ∂ξ

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i

(37)

where Φ = φ1 for simplicity, and P (Q) is the dispersion (nonlinear) coefficient, and is given by

vg β12 k 5 − 3vg kω 4 + 4β1 k 2 ω 3 + 2vg β1 ω 2 k 3 − 4ωβ12 k 4 , 2ω 2 k 2 S2 Q= [2γ2 (C3 + C6 ) + 3γ3 − Fk ] , 2ωk 2

(38) (39)

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P =

in which Fk = k 2 /S 2 [ω 2 (C1 + C4 ) + 2ωk(C2 + C5 ) + (β3 k 6 /S 2 )] and β3 = 4β/81. It is important to mention few more points: The reductive perturbation method can also be used to derive the K-dV equation for describing the evolution

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of a non-modulated waves, i.e. a bare pulse with no fast oscillations inside the packet. However, the well known nonlinear mechanism involved in plasma wave dynamics is amplitude modulation, which may be due to parametric wave

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coupling, interaction between high and low frequency modes or simply to the nonlinear self-interaction of the carrier waves. The standard method to study this mechanism adopts a multiple scales perturbation technique (also known as reductive perturbation method [4, 38, 39]), which generally leads to a NLS equation describing the evolution of a slowly varying wave packet or envelope. The wave packet may undergo a BenjaminFeir-type MI under certain conditions. The MI of wave packets in plasmas acts as a precursor for the formation of bright and dark envelope solitons. 9

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4. Stability of NAWs

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To study the MI of NAWs, we consider the linear solution of the NLS

˜ 2τ ˜ i Q|Φ| ˜ = Φ ˜ 0 + Φ ˜ 1 and equation (37) in the form Φ = Φe + c. c., where Φ

˜1 = Φ ˜ 1,0 ei(k˜ ξ−˜ω τ ) + c. c. We note that the amplitude depends on the freΦ

quency, and that the perturbed wave number k˜ and frequency ω ˜ are different from k and ω. Now, substituting these into Eq. (37), one can easily obtain the

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following nonlinear dispersion relation

  Q ˜ 2 | ω ˜ 2 = P 2 k˜2 k˜2 − 2 |Φ . 0 P

(40)

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It is observed here that the NAWs will be stable (unstable) for that range of values of k˜ in which the product P Q is negative (positive) P Q < 0 (P Q > 0). The regimes for P/Q < 0 (P/Q > 0) are depicted in Fig. 1(a)−1(b). It is HaL

HbL

0.0004

0.0015

Μe =2.0

Β=0.03

0.0010

Β=0.05

0.0002

PQ

PQ

Μe =2.5 Μe =3.0

0.0005

Β=0.07 0.0000

0.0000

-0.0005

-0.0004 1.0

1.5

-0.0010

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-0.0002

2.0

2.5

3.0

3.5

-0.0015

4.0

1.0

1.5

2.0

2.5

k

HcL

0.8 Β=0.03 Β=0.05

HdL 0.20 Μe =2.0 Μe =2.5

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0.2

Μe =3.0

G

G

Β=0.07

0.4

2

0.10

0.05

0.0

0

3.5

0.15

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0.6

3.0

k

0.00 4

6

8

0

1

2

3

~

~

k

k

4

Figure 1: The variation of P/Q with k (a) for different values of β and µe = 2; (b) for different

˜ (c) for different values values of µe and β = 0.03. The variation o the growth rate (Γ) with k ˜ 0 = 0.05; (d) for different values of µe , β = 0.03, k = 3, and of β, µe = 2, k = 3, and Φ

˜ 0 = 0.05. Φ

obvious from Eq. (40) that the NAWs becomes nodulationally unstable when 10

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k˜c > k˜ in the regime P/Q > 0 or P Q > 0, where k˜c =

p ˜ 0 |. The 2(Q/P )|Φ

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growth rate (Γ) of the modulationally unstable of the NAWs is given by s k˜c2 Γ = |P | k˜2 − 1. k˜2

(41)

We have numerically found the parametric degenerate plasma regime for which the NAWs are modulationally stable and unstable as shown in Figs. 1(a) and

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1(b). We have also graphically shown how the growth rate (Γ) varies with k˜ for

different values of β and µ (in the modulationally unstable regime) as shown in Figs. 1(c) and 1(d). It is obvious from Figs. 1(c) and 1(d) that (i) the

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growth rate decreases (increases) with the increase of the wavelength for it’s lower (upper) range; (ii) the growth rate increases with the increase of nn0 and with the decrease of Zn for fixed mn and ne0 ; (iii) the growth rate increases (decreases) with the ne0 (np0 ).

5. Envelope solitons

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The envelope solitonic solutions of the NLS equation (37), which can be obtained by a number of straightforward mathematical steps, are available in a large number of existing literature [40, 41, 42, 43, 44]. Thus, the envelope solitonic solution of Eq. (37) can be directly given by [44]

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Φ(ξ, τ ) =

p

ψ(ξ, τ ) eiθ(ξ,τ ) ,

(42)

where ψ(= |Φ|) represents the envelope amplitude and θ is the nonlinear phase

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shift due to self-interaction. Now, depending on the sign of the coefficients P and Q, two types of envelope solitonic solutions, which gives rise to bright and envelope solitons, can be obtained [40, 41, 42, 43]. 5.1. Bright envelope solitons We already mentioned that the condition for bright envelope solitons is P Q >

0. Thus, the exact bright envelope solitonic solutions of Eq. (37) can be written

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  ξ − Uτ ψ = ψb0 sech2 , Wb     1 U2 θ= U ξ + Ω0 − τ , 2P 2

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in the form [40, 41, 42, 43, 44]

(43)

where U is travelling speed of the localized pulse in the case of bright envelope solitons, and Ω0 is the oscillating frequency for U = 0.

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The relation between the pulse width (Wb ) and the amplitude ψb0 is Wb = p

2P/Qψb0 . This means that for P Q > 0 the NAWs are modulationally un-

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stable in the sense that the finite amplitude plane wave breaks up into a train of solitons, and that the NLS equation (37) has an envelope solitonic solution, which satisfies the boundary condition, Φ = 0 and ∂Φ/∂ξ = 0 at ξ = ±∞. The stable bright envelope solitons are formed in P Q > 0 region due to the balance between the nonlinearity and the dispersion. On the other hand, in the same region (P Q > 0), the unstable bright envelope solitons are formed when the nonlinearity is so high that the dispersion can not balance it.

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The bright envelope solitons [obtained from the numerical analysis of Eq. (43)] are depicted in Figs. 2(a), 2(c), 2(d), 3(a), and 4(a), which clearly indicate that (i) the number of oscillations of the bright envelope solitons increases with the increase of the localized pulse speed U as shown in Figs. 2(a) and 2(c); (ii) the width of the bright envelope solitons decreases with the increase of nn0

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and with the decrease of Zn for fixed mn and ne0 ; (iii) the width of the bright envelope solitons decreases (increases) as we increase the value of ne0 (np0 ); (iv)

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the bright envelope solitons remain unchanged, but they shift towards the left as time (τ ) passes as shown in Fig. 4(a), i.e. the bright envelope solitons are modulationally stable [45, 46, 47]. 5.2. Dark envelope solitons The condition for the dark envelope solitons is P Q < 0 (as mentioned be-

fore). Thus, the exact dark envelope solitonic solutions of Eq. (37) can be

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0.02

0.02

0.01

0.01

0.00

0.00 -0.01

-0.01

-0.02

-0.02 -2

0

2

4

-15

-10

-5

0

5

10

15

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-4

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HbL

Re HFL

Re HFL

HaL

Ξ

Ξ

HdL

HcL 0.025

Β=0.030

0.02

Β=0.035

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0.020

Re HFL

Β=0.040

 F¤

0.01

0.015 0.010

0.00

0.005

-0.01

0.000

-4

-2

0

2

4

Ξ

-4

0

-2

2

4

Ξ

Figure 2: The variation of Re(Φ) with ξ for (a) bright envelope solitons, U = 0.3 and k = 3; (b) dark envelope solitons, V = 0.3 and k = 1; (c) bright envelope solitons, U = 0.1 and

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k = 3; (d) The variation of the |Φ| of bright envelope solitons with ξ for β, U = 0.3, and k = 3; along with ψb0 = ψd0 = 0.0008, τ = 0, and Ω0 = 0.4.

HaL

HbL

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0.025

0.025

Μe =2.0 Μe =2.5

0.020

0.020

 F¤

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 F¤

Μe =3.0

0.015

0.015

0.010

0.010

0.005

0.005

0.000

0.000

-4

-2

Μe =2.0 Μe =2.5

0

2

4

Μe =3.0

-3

Ξ

-2

-1

0

1

2

3

Ξ

Figure 3: The variation of |Φ| of bright envelope solitons with ξ for (a) µe and k = 3; (b) µe

and k = 2; along with ψb0 = ψd0 = 0.0008, U = V = 0.3, τ = 0, and Ω0 = 0.4.

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Figure 4: The variation of the |Φ| with ξ and τ for (a) bright envelope soliton and k = 3; (b) dark envelope solitons and k = 2; along with ψb0 = ψd0 = 0.0008, U = V = 0.3, and

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Ω0 = 0.4.

written in the form [40, 41, 42, 43]



 ξ−Vτ ψ = ψd0 tanh , Wd   2   V 1 Vξ− − 2P Qψd0 τ , θ= 2P 2 2

(44)

where V is the speed of the localized pulse in the case of dark envelope soli-

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tons, and Ω0 is the oscillating frequency for V = 0. The relation between the p pulse width (Wd ) and the amplitude (ψd0 ) is Wd = (2|P/Q|)/ψd0 . The dark envelope solitonic profiles [obtained from the numerical analysis of Eq. (44)] are displayed in Figs. 2(b), 3(b), and 4(b), which imply that (i) as we increase

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the value of ne0 (np0 ), the width of dark envelope soliton increases (decreases), but their amplitude remains constant; (ii) the dark envelope solitons remain unchanged, but they shift towards the left as time (τ ) passes as shown in Fig.

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4(b), i.e. the dark envelope solitons are modulationally stable [45, 46, 47].

6. Discussion

We have investigated the amplitude modulation of NAWs in an unmagne-

tized multi-component plasma system consisting of inertial non-relativistically degenerate nuclei as well as inertialess ultra-relativistically degenerate electrons and positrons. The NLS equation, which governs the evolution of nonlinear

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NAWs, is derived by employing the reductive perturbation method. The results,

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which have been found from this theoretical investigation, can be pinpointed as follows:

1. The NAWs will be stable (unstable) for that range of values of k in which the ratio P/Q is negative (positive) P/Q < 0 (P/Q > 0).

2. The growth rate increases with the increase of nn0 and with the decrease

rate increases (decreases) with ne0 (np0 ).

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of Zn for the fixed values of mn and ne0 . On the other hand, the growth

3. The width of the bright envelope solitons decreases with the increase of

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nn0 and with the decrease of Zn for the fixed values of mn and ne0 . 4. The width of the bright envelope solitons decreases (increases) as we increase the value of ne0 (np0 ). On the other hand, the width of dark envelope solitons increases (decreases) as we increase the value of ne0 (np0 ), but their amplitude remains constant.

5. The bright and dark envelope solitons remain unchanged, but they shift

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towards the left as time (τ ) passes.

We hope that the results obtained from this theoretical investigation should be helpful for understanding the nonlinear electrostatic structures in astrophysical (viz. white dwarf, neutron star [9, 12]) and laboratory (viz. quantum wells

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[48], spintronics [49], plasmonics [50, 51], etc.) DQPSs containing inertial nonrelativistically degenerate nuclei and ultra-relativistically degenerate electrons

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along with positrons.

Acknowledgements N. A. Chowdhury is grateful to the Bangladesh Ministry of Science and Tech-

nology for awarding the National Science and Technology (NST) Fellowship. The authors are also grateful to the anonymous reviewers for their constructive suggestions which have significantly improved the quality of the manuscript.

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Highlights

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1. The modulational instability of new kind of electro-acoustic waves (named here nucleus-acoustic waves) in degenerate quantum plasma system has been investigated for the first time. 2. The new basic features of the nucleus-acoustic bright and dark envelope solitons are identified. 3. The investigation is applicable not only to astrophysical quantum plasma systems (viz. white dwarf and neutron star), but also to laboratory ones (viz. quantum wells, spintronics, plasmonics, etc.).