Selfgravitating envelope solitons in a degenerate quantum plasma system
Abstract
The existence and the basic features of ionacoustic (IA) envelope solitons in a selfgravitating degenerate quantum plasma system (SGDQPS), containing inertial nonrelativistically degenerate light and heavy ion species as well as inertialess nonrelativistically degenerate positron and electron species, have been theoretically investigated by deriving the nonlinear Schrödinger (NLS) equation. The NLS equation, which governs the dynamics of the IA waves, has disclosed the modulationally stable and unstable regions for the IA waves. The unstable region allows to generate bright envelope solitons which are modulationaly stable. It is found that the stability and the growth rate dependent on the plasma parameters (like, mass and number density of the plasma species). The implications of our results in astronomical compact object (viz. white dwarfs, neutron stars, and black holes, etc.) are briefly discussed.
Keywords:
Envelope solitons Modulatioanal Insatability Reductive Perturbation Method∎ \thankstexte1email: nahmed93phy@gmail.com
1 Introduction
The field of selfgravitating degenerate quantum plasma (DQP) physics is one of the current interesting research field among the plasma physicists because of the painstaking observational evidence confirms the existence of such kind of extreme plasma in astronomical compact objects (viz. white dwarfs, neutron stars, and black holes, etc. Chandrasekhar1931 (); Koester1990 (); Fowler1994 (); Koester2002 (); Zaman2017 ()) and potential applications in modern technology (viz. metallic and semiconductor nanostructures, quantum xray freeelectron lasers, nanoplasmonic devices Atwater2007 (); Stockmann2011 (), metallic nanoparticles, spintronics Wolf2001 (), thin metal films, nanotubes, quantum dots, and quantum well Manfredi2007 (), etc.). The number density of the plasma species is extremely high in selfgravitating DQP system (SGDQPS) (order of in white dwarfs Koester1990 (); Shapiro1983 () and order of or even more in neutron stars Koester1990 (); Shapiro1983 ()) which leads to generate a strong gravitational field inside the plasma medium. Basically, the SGDQPS contains degenerate inertial light (viz. Fletcher2006 (); Killian2006 () or Chandrasekhar1931 (); Fowler1994 () or Koester1990 (); Koester2002 ()) and heavy (viz. Vanderburg2015 () or Witze2014 () or Witze2014 ()) ion species and inertialess degenerate electron and positron species. Heisenberg’s uncertainty principle established the relationship between the uncertainty to determine the position and momentum of a particle simultaneously, and mathematically it can be expressed as, (where is the uncertainty in position of the particle and is the uncertainty in momentum of the same particle, and is the reduced Planck constant). This indicates that the position of the plasma species are very certain (because of highly dense and compressed plasma species) inside the plasma system but the momenta of the plasma species are extremely uncertain. Therefore, these plasma species are uncertain (certain) in momentum (position) give rise to a very high pressure known as “degenerate pressure”. The expression for the degenerate pressure (degenerate plasma particle species ) as a function of number density () is given by Chandrasekhar1931 (); Fowler1994 (); Mamun2011 ()
(1) 
where () for the electron (positron) species, and () for the light (heavy) ion species, respectively. The is the relativistic factor ( stands for nonrelativistic case and stands for ultrarelativistic case) and is the mass. It is clear from equation (1) that the degenerate pressure is independent on thermal temperature but depends on degenerate particle number density and mass . Finally, the strong gravitational field (degenerate pressure) of the SGDQPS wants to squeeze (stretch) the plasma system but they are counterbalanced to each other.
During the last few years, a large number of authors have studied the propagation of nonlinear waves in DQP by considering selfgravitational or without selfgravitational field. Asaduzzaman et al. Asaduzzaman2017 () have investigated the linear and nonlinear propagation of selfgravitational perturbation mode in a SGDQPS and found that selfgravitational perturbation mode becomes unstable when the wavelength of the perturbation mode is minimum. Mamun Mamun2017 () examined the selfgravito shock structures in a SGDQPS. Chowdhury et al. Chowdhury2018 () have studied the modulational instability (MI) of nucleusacoustic waves in a DQP system and found that the bright and dark envelope solitons are modulationally stable. But to the best of our knowledge, no attempt has been made to study MI of the ionacoustic waves (IAWs) by deriving a nonlinear Schrödinger equation (NLS) and formation of the envelope solitons in any kind of SGDQPS. Therefore, in the present work, a SGDQPS (containing inertialess degenerate electron and positron species, inertial degenerate light as well as heavy ion species) has been considered to obtain the conditions of MI of the IAWs and the formation of the envelope solitons, and also to identify their basic features.
The manuscript is organized as follows. The basic governing equations for the dynamics of the SGDQPS are descried in Sec. 2. The derivation of the NLS equation is provided in Sec. 3. The stability of the IAWs and envelope solitons are examined in Sec. 4. A brief discussion is finally presented in Sec. 5.
2 Governing equation
We consider a SGDQPS containing inertialess degenerate electrons (mass ; number density ), positrons (mass ; number density ), inertial degenerate light ions (mass ; number density ), and heavy ions (mass ; number density ). The detail information about the light and heavy nuclei is provided in Table 1. The nonlinear dynamics of the SGDQPS is described by
(2)  
(3)  
(4)  
(5)  
(6) 
where , , and are the degenerate pressure of the degenerate electrons, positrons, and light ions, respectively; () is the space (time) variable; is the light ion fluid speed; is the selfgravitational potential; is the universal gravitational constant. Now, the charge neutrality condition for the electrostatic wave potential is
(7) 
where and are the charge state of light and heavy ions, respectively. Here, it may be noted that the effect of the electrostatic wave potential has been neglected. Now, we consider normalized variables, namely, , , , , , , (where and are the equilibrium number densities of the light ion and electron species, respectively). After normalization, Eqs. ()() can be taken in the following form
(8)  
(9)  
(10)  
(11)  
(12) 
where , , , , , , (which is greater than for any set of heavy and light ion species), (here, , where varies from to , and varies from to , and this means that ), , . For inertialess degenerate electron and positron, the number densities can be expressed as
(13)  
(14) 
Now, we substitute Eqs. (13) and (14) into Eq. (12) and extend the resulting equation up to third order, we get
(15) 
where
Light ion species  Heavy ion species  

Vanderburg2015 ()  2.16  
Killian2006 (); Fletcher2006 () 
Witze2014 ()  2.30 
Witze2014 ()  2.28  
Vanderburg2015 ()  1.08  
Chandrasekhar1931 () 
Witze2014 ()  1.15 
Witze2014 ()  1.14  
Vanderburg2015 ()  1.08  
Koester1990 (); Koester2002 () 
Witze2014 ()  1.15 
Witze2014 ()  1.14 
3 Derivation of NLS equation
To investigate the MI of the IA waves in SGDQPS, we will derive the NLS equation by employing the reductive perturbation method Taniuti1969 (); Chowdhury2017a (). So, we first introduce the stretched coordinates for independent variables and in terms of and as follows:
(16) 
where is the envelope group velocity and is a small dimensionless expansion parameter. Then we can expand all dependent physical variables , , and in power series of as
(17)  
(18)  
(19) 
where and () is the real variable representing the fundamental carrier wave number (frequency). The derivative operators in Eqs. (10), (11), and (15) are regarded as
(20)  
(21) 
Now, by substituting Eqs. (17)(21) into Eqs. (10), (11), and(15) and collecting the different powers of . Now, the first order () reduced equations with can be expressed as
(22)  
(23) 
where and . The compatibility of the system leads to the linear dispersion relation as
(24) 
The dispersion characteristics of the wave are depicted in Fig. 2 [obtained from Eq. (24)], which indicates that (a) the angular wave frequency () of the IAWs exponentially decreases with the increase of ; (b) the value of increases with the increase of for the fixed value of (via ). The second order () reduced equations with are given by,
(25)  
(26) 
thus, the expression for is obtained as
(27) 
The amplitude of the secondorder harmonics is found to be proportional to and these are expressed as
(28) 
where the coefficients are
Finally, by substituting all the Eqs. (22)(28) into the third order part () and and simplifying them, we can obtain the following NLS equation:
(29) 
where for simplicity. The coefficient of dispersion and nonlinear terms & are given by
(30)  
(31) 
where
,
, and .
4 Stability analysis and envelope solitons
Let us now analysis the MI of IAWs by considering the linear solution of the NLS equation (29) in the form (c. c denotes the complex conjugate), where and + c. c. Now, by substituting these values into Eq. (29), one readily obtains the following nonlinear dispersion relation Chowdhury2018 (); Sultana2011 (); Schamel2002 (); Kourakis2005 (); Fedele2002 (); Chowdhury2017b ()
(32) 
Here, the perturbed wave number and the perturbed frequency are different from the carrier wave number and frequency . It is observed from Eq. (32) that the IAWs will be modulationally stable (unstable) in SGDQPS for that range of values of in which is negative (positive), i.e., (). When , the corresponding value of () is known as the critical or threshold wave number () for the onset of MI. The variation of with for is shown in Fig. 2 and which clearly indicates that (a) the IAWs are modulatonally stable (unstable) in SGDQPS for small (long) wavelength; (b) the increases with the increase of for constant value of (via ). In the modulationally unstable () region and under this condition , the MI growth rate can be written [from Eq. (32)] as
(33) 
The effect of and on the growth rate are presented in Figs. 4 and 4, where is plotted against and it is observed that (a) the growth rate () increases with the increase in the value of positron number density , but decreases with increase of the electron number density (via ); (b) the maximum value of increases (decreases) with the decrease of () for the fixed value of and (via ); (c) on the other hand, the maximum value of increases (decreases) with the decrease of () for the fixed value of and (via ). So, the charge state and mass of the light and heavy ion plays an opposite role to manifest the in SGDQPS. The physics of this result is that the nonlinearity of the SGDQPS increases (decreases) with the increase of the value of or ( or ) which enhance (suppress) the maximum value of the .
The selfgravitating bright envelop solitons are generated in the modulationally unstable region (when ) and the solitonic solution of Eq. (29) for the selfgravitating bright envelope solitons can be written as Chowdhury2018 (); Sultana2011 (); Schamel2002 (); Kourakis2005 (); Fedele2002 ()
(34) 
where is the propagation speed of the localized pulse, is the pulse width which can be written as ( is the constant amplitude), and is the oscillating frequency for . The selfgravitating bright envelop solitons which are obtained from the numerical analysis of Eq. (34), are depicted in Figs. 6 and 6. The bright envelop solitons remain same as time() passes, i.e., the selfgravitating bright envelop solitons are modulationally stable (please see Fig. 6).
5 Discussion
In our above analysis, we have considered an unmagnetized realistic laboratory or astrophysical SGDQPS consisting of inertialess nonrelativistically degenerate electron and positron species, inertial nonrelativistically degenerate light ion species as well as heavy ion species. The NLS equation has been derived by employing the wellknown reductive perturbation method, which governs the evolution of nonlinear IAWs. The notable informations that have been found from our theoretical investigation, can be pinpointed as follows:

The angular wave frequency () of the IAWs exponentially decreases with the increase of . On the other hand, the value of increases with the increase of for the fixed value of (via ).

The IAWs will be modulationally stable (unstable) for that range of values of in which is negative (positive), i.e., ().

The growth rate () increases with the increase in the value of positron number density , but decreases with increase of the electron number density (via ). On the other hand, the maximum value of increases (decreases) with the decrease of () for the fixed value of and (via ). Furthermore, the maximum value of increases (decreases) with the decrease of () for the fixed value of and (via ).

The selfgravitating bright envelop solitons remain same (modulationally stable) as time passes.
The findings of this theoretical investigation may be useful for understanding the nonlinear structure (bright envelope solitons) of a SGDQPS in space (viz. neutron stars and white dwarf Chandrasekhar1931 (); Koester1990 (); Fowler1994 (); Koester2002 (); Zaman2017 ()).
Footnotes
 journal: Eur. Phys. J. D
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