multi-dimensional instability of electrostatic solitary waves in ultra

Cent. Eur. J. Phys. • 10(5) • 2012 • 1168-1176DOI: 10.2478/s11534-012-0085-0

Central European Journal of Physics

Multi-dimensional instability of electrostatic solitarywaves in ultra-relativistic degenerateelectron-positron-ion plasmas

Research Article

M. Masum Haider1,2∗, Suraya Akter2,3† , Syed S. Duha2, Abdullah A. Mamun2

1 Department of Physics, Mawlana Bhashani Science and Technology University,Santosh, Tangail-1902, Bangladesh

2 Department of Physics, Jahangirnagar University,Savar, Dhaka-1342, Bangladesh

3 World University of Bangladesh,Dhanmondi, Dhaka-1205, Bangladesh

Received 26 November 2011; accepted 14 May 2012

Abstract: The basic features and multi-dimensional instability of electrostatic (EA) solitary waves propagating inan ultra-relativistic degenerate dense magnetized plasma (containing inertia-less electrons, inertia-lesspositrons, and inertial ions) have been theoretically investigated by reductive perturbation method andsmall-k perturbation expansion technique. The Zakharov-Kuznetsov (ZK) equation has been derived, andits numerical solutions for some special cases have been analyzed to identify the basic features (viz. am-plitude, width, instability, etc.) of these electrostatic solitary structures. The implications of our results insome compact astrophysical objects, particularly white dwarfs and neutron stars, are briefly discussed.

PACS (2008): 52.25.Xz; 52.35.Mw; 52.35.Fp; 94.05.Fg; 94.20.wf

Keywords: electrostatic solitary waves • degenerate plasma • ultra-relativistic limits • Zakharov-Kuznetsov equation© Versita sp. z o.o.

1. Introduction

Recently, there has been a great deal of interest in under-standing the basic properties of matter under extreme con-ditions [1–7], There are many interstellar compact objects,such as white dwarfs and neutron stars), where matters ex-∗E-mail: [email protected] (Corresponding author)†E-mail: [email protected]

ist in extreme conditions [1–7] not found in terrestrial en-vironments. One of the extreme conditions found in theseobjects is a high density of degenerate matter. These com-pact objects are relics of stars which have ceased burn-ing thermonuclear fuel and therefore no longer generatethermal pressure. These interstellar compact objects havesignificantly contracted, and as a result the density oftheir interiors becomes high enough to provide nonther-mal pressure via degenerate fermion/electron pressure andparticle-particle interactions. The observational evidenceand theoretical analysis imply that these compact objects,1168

M. Masum Haider, Suraya Akter, Syed S. Duha, Abdullah A. Mamun

which support themselves against gravitational collapseby cold degenerate fermion/electron pressure, are of twocategories. The interior of the first category is close toa dense solid (ion lattice surrounded by degenerate elec-trons, and possibly other heavy particles or dust). One ofthe examples of this kind of stars is a white dwarf, whichis supported by the pressure of degenerate electrons. Theinterior of the second category is close to a giant atomicnucleus: a mixture of interacting nucleons and electrons,and possibly other heavy elementary particles and con-densates or dust. One example of this kind of stars isa neutron star, which is supported by the pressure dueto a combination of nucleon degeneracy and nuclear in-teractions. These unique states (extreme conditions) ofmatter occur by significant compression of the interstel-lar medium. The degenerate plasma number density insuch a compact object is so high (e.g. the degenerateplasma number density can be of the order of 1030 cm−3and 1036 cm−3 or more in white dwarfs and neutron starsrespectively) that the electron Fermi energy is compara-ble to the electron mass energy and the electron speed iscomparable to the speed of light in vacuum [1–4].The equation of state for degenerate electrons in such in-terstellar compact objects were mathematically explainedby Chandrasekhar [5–7] for two limits, namely non-relativistic and ultra-relativistic limits. The degenerateelectron equation of state of Chandrasekhar is Pe ∝N5/3e for non-relativistic limit and Pe ∝ N4/3

e for ultra-relativistic limit, where Pe is the degenerate electronpressure and Ne is the degenerate electron number den-sity. We note that the degenerate electron pressure de-pends only on the electron number density, but not on theelectron temperature. These interstellar compact objects,therefore, provide us cosmic laboratories for studying theproperties of the medium (matter), as well as waves andinstabilities [8–23] in such a medium at extremely highdensities (degenerate state) for which quantum as well asrelativistic effects become important [22, 23]. The quan-tum effects on linear [11–16, 18, 19, 21] and nonlinear[17, 20] propagation of electrostatic and electro-magneticwaves have been investigated by using the quantum hy-drodynamic (QHD) model [21, 23], which is an extensionof classical fluid model in a plasma, and by using thequantum magneto-hydrodynamic (QMHD) model [11–20],which involve spin- 12 and one-fluid MHD equations.Recently a number of theoretical investigations have alsobeen made of the nonlinear propagation of electrostaticwaves in degenerate quantum plasma by a number ofauthors, e.g. Hass [24], Misra and Samanta [25], etc.These investigations are, however, based on the elec-tron equation of state Pe ∝ N5/3e , which is valid for thenon-relativistic limit. Very recently, Mamun and Shukla

[26, 27] have considered an unmagnetized ultra-relativisticdegenerate dense plasma containing cold ion fluid andultra-relativistic electrons and have studied the basic fea-tures of the solitary structure. It has been known thatthe effects of obliqueness and external magnetic field,which have not been considered in the work of Mamunand Shukla [26, 27], drastically modify the propertiesof obliquely propagating electrostatic solitary structures[27–29]. Therefore, in our present work, we consider anultra-relativistic degenerate dense plasma in the presenceof an external static magnetic field and study the obliquelypropagating electrostatic solitary structures in such anultra-relativistic degenerate dense magnetized plasma.To the best of our knowledge no investigation has beenmade of the nonlinear propagation of the electrostaticwaves in such a degenerate dense electron-positron-ionplasma based on equation of state Ps ∝ N4/3s [where,

Ps represents the degenerate electron (positron) pres-sure Pe(Pp) and Ns represents the degenerate electron(positron) number density Ne(Np)] which is valid in theultra-relativistic limit. Therefore, in our present work,we consider an inertial ultra-cold ion fluid and ultra-relativistic degenerate ultra-cold inertia less electron-positron fluid following the equation of state Ps ∝ N4/3s ,and study the basic features of the arbitrary amplitudesolitary waves (SWs) in such an ultra-relativistic degen-erate dense plasma. In this work we derive the Zakharov-Kuznetsov (ZK) Equation [30] by reductive perturbationmethod and solution [31] of ZK equation. We also usesmall-k perturbation expansion technique for instabilityanalysis.The manuscript is organized as follows: The basic equa-tions governing the magnetized multi-ion dusty plasmasystem are given in Sec. 2. The Zakharov-Kuznetsov (ZK)equation is derived by employing a reductive perturbationmethod in Sec. 3. The stationary solitary wave solution ofZK equation and its basic features are studied in Sec. 4.The instability criterion is analyzed in Sec. 5. Finally, abrief discussion is presented in Sec. 6.

2. Basic equations

We consider the nonlinear propagation of electrostaticperturbation in an ultra-relativistic degenerate denseplasma (containing ultra-relativistic degenerate ultra-coldinertia less electron-positron fluid and inertial ultra-coldion fluid) in the presence of an external static magneticfield B0 acting along the z-direction (B0 = zB0 ), wherez is the unit vector along the z-direction. The electronand positron fluid is assumed to follow the equation ofstate Ps ∝ N4/3

s . We assume that the magnetic field is1169

Multi-dimensional instability of electrostatic solitary waves in ultra-relativistic degenerate electron-positron-ion plasmas

very strong and electrons and positrons are moving alongthe magnetic field direction so fast that the electron andpositron response are the same as that in the unmagne-tized plasma. The nonlinear dynamics (equation of conti-nuity for ion fluid, equation of momentum for ion, electronand positron, and Poisson’s equation) of the electrostaticwaves propagating in such an ultra-relativistic degenerateplasma is governed by∂ni∂t +∇ · (niui) = 0, (1)∂ui∂t + (ui · ∇)ui = −∇ψ + ωciui × z, (2)0 = j∇ψ − 3β4ns∇n4/3

s , (3)∇2ψ = (µp + 1)ne − µpnp − ni, (4)

where ni is the ion number density normalized by its equi-librium value ni0, ui is the ion fluid speed normalized byCi = (mec2/mi)1/2 with ms(mp and me) being the rest massof species (positron and electron respectively), mi beingthe ion rest mass, c being the speed of light in vacuum, ψis the electrostatic wave potential normalized by mec2/ewith e being the magnitude of the charge of an electron,β = (n0/72π)1/3(h/mec) having h be the Planck’s con-stant, n0 is the plasma number density and j = +1 forpositron and j = −1 for electron. The time variable (t)is normalized by ωpi−1 = (mi/4πni0e2)1/2, the space vari-ables are normalized by λs = (mec2/4πne0e2)1/2, and ωciis the ion cyclotron frequency (eB0/mic) normalized byωpi. At equilibrium we have ne0 = np0 + zini0; where,µp = np0/zini0. Now, integrating (3) we have

0 = −jψ − 3βn1/3s + C0,

where, C0 is the integration constant. Since at equilibriumns = 1 and ψ = 0, we have C0 = 3β. Thus we can expressnp and ne as

np = (1− ψ3β )3, (5)ne = (1 + ψ3β )3. (6)

3. Derivation of ZK equationWe now follow the reductive perturbation technique andconstruct a weakly nonlinear theory for the electrostaticwaves with a small but finite-amplitude, which leads to ascaling of the independent variables through the stretched

coordinates [32–38]X = ε1/2x,Y = ε1/2y,Z = ε1/2(z − Vpt),τ = ε3/2t,

where ε is a small parameter measuring the weakness ofthe dispersion and Vp is the unknown wave phase speed(to be determined later). It may be noted here that X,Y, and Z are all normalized by the Debye radius (λi), τis normalized by the ion plasma period (ω−1pi ), and Vp isnormalized by the ion-acoustic speed (Ci). We can nowexpand the perturbed quantities about their equilibriumvalues in powers of ε as [32–38]

ni = 1 + εn(1)i + ε2n(2)

i + · · ·, (7)uiz = εu(1)

iz + ε2u(2)iz + · · ·, (8)

uix = ε3/2u(1)ix + ε2u(2)

ix + · · ·, (9)uiy = ε3/2u(1)

iy + ε2u(2)iy + · · ·, (10)

ψ = εψ (1) + ε2ψ (2) + · · ·. (11)We now use the stretched coordinates and (7)-(11) in (1)-(4), and develop equations in various powers of ε. Tothe lowest order in ε, i.e. equating the coefficients ofε3/2 from the continuity and momentum equation, one canobtain the first-order continuity equation, and the x-, y-,and z-components of the momentum equation as

u(1)iz = 1

Vpψ (1), (12)

n(1)i = 1

V 2pψ (1), (13)

u(1)ix = − 1

ωci∂ψ (1)∂Y , (14)

u(1)iy = 1

ωci∂ψ (1)∂X , (15)

and equating the coefficients of ε from Poisson’s equation,we get0 = 2µp + 1

β ψ (1) − n(1)i . (16)

Using the value of n(1)i from (13) into (16), we get the lineardispersion relationVp =√ β2µp + 1 . (17)

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To the next higher order of ε, i.e. equating the coeffi-cients of ε2, we can express x- and y-components of themomentum equation, and Poisson’s equation asu(2)ix = Vp

ω2ci

∂2ψ (1)∂Z∂X , u(2)

iy = Vpω2ci

∂2ψ (1)∂Y∂Z , (18)

∂2ψ (1)∂X 2 + ∂2ψ (1)

∂Y 2 + ∂2ψ (1)∂Z 2

= 2µp + 1β ψ (2) + 13β2 [ψ (1)]2 − n(2)

i . (19)Again, following the same procedure, one can obtain thenext higher order continuity equation and z-component ofmomentum equation as

∂n(1)i

∂τ − Vp∂n(2)

i∂Z + ∂u(2)

ix∂X + ∂u(2)

iy

∂Y

+ ∂u(2)iz

∂Z + ∂∂Z

[n(1)i u

(1)iz

] = 0, (20)∂u(1)

iz∂τ − Vp

∂u(2)iz

∂Z = −u(1)iz∂u(1)

iz∂Z − ∂ψ (2)

∂Z . (21)Now, using (12) - (21), we can readily obtain

∂ψ (1)∂τ + ABψ (1) ∂ψ (1)

∂Z + 12A ∂∂Z

×[∂2∂Z 2 +D

(∂2∂X 2 + ∂2

∂Y 2)]

ψ (1) = 0, (22)where

A = V 3p = ( β2µp + 1)3/2,

B = 32V 4p− 13β2 = 6µp(µp + 1)

β2 + 76β2 ,D = 1 + 1

ω2ci. (23)

The Equation (22) is known as the Zakharov-Kuznetsov(ZK) equation or the Korteweg-de Vries (K-dV) equationin three dimensions.4. SW solution of the ZK equationTo study the properties of the solitary waves propagatingin a direction making an angle δ with the Z-axis, i.e. with

the external magnetic field and lying in the (Z-X) plane,we first rotate the coordinate axes (X, Z) through an an-gle δ , keeping the Y-axis fixed. Thus, we transform ourindependent variables toζ = X cos δ − Z sin δ, η = Y ,ξ = X sin δ + Z cos δ, τ = t. (24)

Using the transformation we get,∂∂X = cos δ ∂∂ζ + sin δ ∂∂ξ , ∂

∂Y = ∂∂η , (25)

∂∂Z = cos δ ∂∂ξ + sin δ ∂∂ζ , ∂

∂t = ∂∂τ . (26)

This transformation of these independent variables allowsus to write the ZK equation in the form∂ψ (1)∂t + δ1ψ (1) ∂ψ (1)

∂ξ + δ2 ∂3ψ (1)∂ξ3

+ δ3ψ (1) ∂ψ (1)∂ζ + δ4 ∂3ψ (1)

∂ζ3 + δ5 ∂3ψ (1)∂ξ2∂ζ

+ δ6 ∂3ψ (1)∂ξ∂ζ2 + δ7 ∂3ψ (1)

∂ξ∂η2 + δ8 ∂3ψ (1)∂ζ∂η2 = 0, (27)

whereδ1 = AB cos δ,δ2 = 12A(cos3 δ +D sin2 δ cos δ),δ3 = −AB sin δ,δ4 = −12A(sin3 δ +D sin δ cos2 δ),δ5 = A

[D(sin δ cos2 δ − 12 sin3 δ

)− 32 sin δ cos2 δ

],

δ6 = −A [D(sin2 δ cos δ − 12 cos3 δ)− 32 sin2 δ cos δ] ,

δ7 = 12AD cos δ,δ8 = −12AD sin δ. (28)

We now look for a steady state solution of this ZK equa-tion in the formψ (1) = ψ0(Z), (29)

where Z = ξ −U0T and t = T , in which U0 is a constantspeed normalized by the ion-acoustic speed (Ci). Usingthis transformation, we get∂∂ξ = ∂

∂Z ,∂∂t = ∂

∂T− U0 ∂

∂Z ,

∂∂ζ → 0, ∂

∂η → 0. (30)1171


0.10.2

0.30.4

0.5

Μp

0

20

40

60

∆

1234

Ψm

0.10.2

0.30.4Μp

Figure 1. Variation of the amplitude of the solitary wave potential(ψm) with µp and δ for U0 = 1 and β = 0.5.

0.20.4

0.60.8

1Β

0.1

0.2

0.3

0.4

0.5

Μp

11.52

2.53

Ψm

0.20.4

0.60.8

1Β

Figure 2. Variation of the amplitude of the solitary wave potential(ψm) with β and µp for U0 = 1 and δ = 45.

Therefore, from (27), we can writedψ0dT− U0 dψ0

dZ + δ1ψ0 dψ0dZ + δ2 d3ψ0

dZ3 = 0. (31)At stationary state (dψ0/dT ) → 0. So, we can write theZK equation in steady state form as

−U0 dψ0dZ + δ1ψ0 dψ0

dZ + δ2 d3ψ0dZ3 = 0. (32)

Now, using the appropriate boundary conditions, viz.,ψ (1) → 0, dψ (1)/dZ → 0, d2ψ (1)/dZ2 → 0 as Z → ±∞,

0.10.2

0.30.4

0.5

Μp

0.2

0.4

0.6

0.8

1

Β

2

4

6D

0.10.2

0.30.4Μp

Figure 3. Variation of the width (∆) of the solitary wave with µp andβ for U0 = 1, ωci = 0.1 and δ = 45.

020

40

60∆

0.1

0.2

0.3

0.4

0.5

Ωci

01234

D

020

40

60∆

Figure 4. Variation of the width (∆) of the solitary wave with δ andωci for U0 = 1, β = 0.5 and µp = 0.1.

we can writedψ0dZ =√U0

δ2 ψ0√1− δ13U0ψ0, (33)

after integrating and rearrangingψ0 = 3U0

δ1[ 2e√U0/4δ2 + e−

√U0/4δ2

]2, (34)

the solitary wave solution is given byψ0(Z) = ψmsech2(κZ), (35)

1172


where ψm = 3U0/δ1 is the amplitude and κ = √U0/4δ2 isthe inverse of the width (∆) of the solitary waves. It is clearfrom (17), (23), and (28) that as A > 0, B > 0, the solitarywaves will be associated with positive potential (ψm >0). Therefore, there exists solitary waves associated withpositive potential.Figure 1 and Figure 2 show the variation of the ampli-tude of the positive solitary potential (ψm > 0). Am-plitude increases with decreasing positron number den-sity and increasing propagating angle with magnetic field(Figure 1) and also increases with increasing degenerateplasma number density (Figure 2).Figure 3 shows the variation of the width (∆) of the soli-tary wave with µp and β for U0 = 1, ωci = 0.1 andδ = 45. Figure 4 shows the variation of the width (∆) ofthe solitary wave with δ and ωci for U0 = 1, µp = 0.1 andβ = 0.5. The width increases with decreasing positronnumber density and increasing plasma number density.The width decreases with increasing ion cyclotron fre-quency and it also increases with direction of propagationfor the lower range, (i.e. from δ = 0-50) but decreasesfor its higher range, (i.e. δ = 50 to higher).5. Instability analysisWe now study the instability of the obliquely propagatingsolitary waves, discussed in the previous section, by themethod of small-k perturbation expansion [32–38]. Wefirst assume that

ψ (1) = ψ0(Z) + φ(Z, ζ, η, t), (36)where ψ0 is defined by (35), and for a long-wavelengthplane wave perturbation in a direction with directioncosines (lζ , lη, lξ ), φ is given by

φ = φ(Z)ei[k(lζζ+lηη+lξZ)−ωt], (37)in which l2ζ + l2η + l2ξ = 1, and for small k , φ(Z) and ω canbe expanded as

φ(Z) = φ0(Z) + kφ1(Z) + k2φ2(Z) + · · ·, (38)ω = kω1 + k2ω2 + · · ·. (39)

Now, substituting (36) into (27), and linearizing with re-spect to ψ, we can express the linearized ZK equation inthe form∂ψ∂t − U0 ∂ψ∂Z + δ1ψ0 ∂ψ∂Z + δ1ψ ∂ψ0

∂Z + δ2 ∂3ψ∂Z3

+ δ3ψ0 ∂ψ∂ζ + δ4 ∂3ψ∂ζ3 + δ5 ∂3ψ

∂Z2∂ζ + δ6 ∂3ψ∂Z∂ζ2

+ δ7 ∂3ψ∂Z∂η2 + δ8 ∂3ψ

∂ζ∂η2 = 0. (40)

Now substituting (36) into ZK equation and linearizingwith respect to ψ we can express the linear ZK equation[−ikω − iU0klξ + iδ1klξψ0 + δ1 ∂ψ0

∂Z − iδ2k3l3ξ+ δ3klζψ0 − iδ4k3l3ζ − iδ5k3l2ξ lζ − iδ6k3lξ l2ζ− iδ7k3lξ l2η − iδ8k3lζ l2η ]φ(Z)+ [−U0 + δ1ψ0 − 3δ2k2l2ξ − 2δ5lξ lζ − δ6k2l2ζ − δ7k2l2η ]× ∂φ(Z)

∂Z + [3iδ2klξ + iklζ ]∂2φ(Z)∂Z2 + δ2 ∂3φ(Z)

∂Z3 = 0.(41)Our main object is to find ω1 by solving the zeroth-, first-,and second-order equations obtained from (37)-(41).Equating the co-efficient of k0 for zeroth order equation,we get

δ1 ∂ψ0∂Z φ0(Z)− U0 ∂φ0(Z)

∂Z + δ1ψ0 ∂φ0(Z)∂Z+ δ2 ∂3φ0(Z)

∂Z3 = 0, (42)the zeroth-order equation can be written, after integration,as

(−U0 + δ1ψ0)φ0 + δ2 d2φ0dZ2 = C, (43)

where C is an integration constant. It is clear from (32)that the homogeneous part of this equation has two lin-early independent solutions, namelyf = dψ0

dZ , g = f∫ Z dZ

f2 . (44)Therefore, the general solution of this zeroth-order equa-tion can be written asφ0 = C1f + C2g− Cf

∫ Z gδ2 dZ + Cg

∫ Z fδ2 dZ, (45)

where C1 and C2 are two integration constants, and δ2 isdefined by (28). Now, evaluating all integrals, the generalsolution of this zeroth-order equation, for φ0 not tendingto ±∞ as Z → ±∞, can finally be simplified toφ0 = C1f . (46)

The first-order equation, i.e. the equation with terms lin-ear in k , obtained from (37)-(41) and (46), can be ex-pressed as−iω1φ0(Z)− iU0lξφ0(Z) + iδ1lξψ0φ0(Z)

+ δ1 ∂ψ0∂Z φ1(Z) + iδ3lζψ0φ0(Z)− U0 ∂φ1(Z)

∂Z+ δ1ψ0 ∂φ(Z)∂Z + 3iδ2lξ ∂2φ0(Z)

∂Z2 + iδ5lζ ∂2φ0(Z)∂Z2

+ δ2 ∂3φ1(Z)∂Z3 = 0, (47)

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05

10

15∆

0.1

0.2

0.3

0.4

0.5

lΗ0

0.10.20.3

Ωci

05

10

15∆

Figure 5. Plot of Si = 0 (variation of ωci with δ and lη for lζ = 0.5).

after integration, first-order equation becomes(−U0 + δ1ψ0)φ1 + δ2 d2φ1

dZ2= iC1(α1 + β1 tanh2 κZ)ψ0 + K, (48)where K is another integration constant, and α1 and β1are given by

α1 = (ω1 + lξU0)− 12ψmµ1 + 2κ2µ2,β1 = 12ψmµ1 − 6κ2µ2,µ1 = δ1lξ + δ3lζ , µ2 = 3δ2lξ + δ5lζ . (49)

Now, following the same procedure, the general solutionof this first-order equation, for φ1 not tending to ±∞ asZ → ±∞, can be written asφ1 = K1f + iC18δ2κ2

[(α1 + β1)Zf + 23(3α1 + β1)ψ0]. (50)

The second-order equation, i.e. the equation with termsinvolving k2, obtained from (40) after substituting (37)-(39), can be written as[−U0 d

dZ + δ1 ddZψ0 + δ2 d3

dZ3]φ2 = iω2φ0

+ i(ω1 + lξU0)φ1 − iµ1ψ0φ1 + µ3 dφ0dZ − iµ2 d2φ1

dZ2 ,(51)where

µ3 = 3δ2l2ξ + 2δ5lζ lξ + δ6l2ζ + δ7l2η. (52)

0.10.2

0.30.4

0.5lΗ

0.5

0.6

0.7

0.8

0.9

lΖ

0.060.080.1

Ωci

0.10.2

0.30.4

0.5lΗ

Figure 6. Plot of Si = 0 (variation of ωci with lη and lζ for δ = 5.)

02

46

8∆

0.5

0.6

0.7

0.8

0.9

lΗ

0

2

4G

02

46

8∆

Figure 7. Variation of the growth rate (Γ) with δ and lη for U0 = 1.0,ωci = 0.1 and lζ = 0.5.

The solution of this second-order equation exists if theright-hand side is orthogonal to a kernel of the operatoradjoint to the operator−U0 d

dZ + δ1 ddZψ0 + δ2 d3

dZ3 . (53)This kernel, which must tend to zero as Z → ±∞, is ψ0 =ψmsech2(κZ). Thus, we can write the following equationdetermining ω1:∫ ∞

−∞ψ0

[iω2φ0 + i(ω1 + lξU0)φ1 − iµ1ψ0φ1

+µ3 dφ0dZ − iµ2 d2φ1

dZ2]dZ = 0. (54)

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0.2

0.4Ωci

0.5

0.6

0.7

0.8

0.9

lΖ

1

1.5

2G

0.2

0.4Ωci

Figure 8. Variation of the growth rate (Γ) with ωci and lζ for U0 = 1.0,lη = 0.5 and δ = 5.

Now, substituting the expressions for φ0 and φ1 given by(46) and (50), respectively, and then performing the inte-gration, we arrive at the following dispersion relation:ω1 = Ω− lξU0 + (Ω2 − Υ)1/2, (55)

whereΩ = 23(ψmµ1 − 2µ2κ2), (56)

Υ = 1645 (ψ2mµ21 − 3ψmµ1µ2κ2 − 3µ22κ4 + 12δ2µ3κ4). (57)

It is clear from the dispersion relation (55) that there isalways instability if (Υ− Ω2) > 0. Thus, using (23), (28),(49), (52), (56), and (57), we can express the instabilitycriterion asSi > 0, (58)

withSi = l2η [ω2

ci + sin2 δ ] + l2ζ[ω2ci −

53 (ω2ci + 1) tan2 δ

]. (59)

Figures 5, 6 show the variation of the parametric regimeswhich play important roles for the instability of the soli-tary waves. Figures 5, 6 indicate that for the parametersabove the surface, the solitary waves become unstable.Figure 5 represents the Si = 0 surface plot showing thevariation of ωci with δ and lη for lζ = 0.5 and also Figure 6represents the Si = 0 surface plot showing the variation

of ωci with lη and lζ for δ = 5. According to these figureswe found that ion-cyclotron frequency (for which the soli-tary waves become unstable) increases with direction ofpropagation that makes an angle δ with Z-direction. Andalso ion-cyclotron frequency decreases (increases) with lη(lζ ).If this instability criterion Si > 0 is satisfied, the growthrate Γ = (Υ−Ω2)1/2 of the unstable perturbation of thesesolitary waves is given byΓ = 2√15 U0(ω2

ci + sin2 δ)√(ω2

ci + 1)Si. (60)The Equation (60) represents that the growth rate Γ ofthe unstable perturbation is a linear function of EA wavespeed U0, but a nonlinear function of propagating angleδ , ion-cyclotron frequency ωci and direction cosine (lζ ,and lη). The nonlinear variations of Γ with δ , ωci, lζ , andlη are shown in Figures 7, 8. Figure 7 shows how thevalue of Γ changes with δ and lη for U0 = 1.0, ωci = 0.1and lζ = 0.5 and figure 8 shows the variation of Γ withωci and lζ for U0 = 1.0, δ = 5 and lη = 0.5. Thesefigures represent that the growth rate Γ of the unstableperturbation changes proportionally with direction cosines(lζ and lη) but inversely with ion-cyclotron frequency anddirection of propagation.6. DiscussionWe have studied EA solitary waves associated with bothpositive potential, containing ultra-relativistic degenerateultra-cold inertia less electron-positron fluid and inertialultra-cold ion fluid in the presence of an external staticmagnetic field by the reductive perturbation method. Wehave then analyzed their multi-dimensional instability bythe small-k perturbation expansion method. The resultscan be summarized as follows:

1. From (35) we can say that solitary waves associatewith positive potential, as A > 0 and B > 0 issatisfied in this plasma system.2. The amplitude of the solitary waves are sig-nificantly affected by the plasma number den-sity, direction of propagation and concentration ofpositrons. The amplitude of the solitary waves in-creases with plasma number density and propagat-ing angle with magnetic field but decreases withpositron number density.3. The width of the solitary waves increases withplasma number density, but decreases with positron

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Table 1. A table for approximate mass density ρ0 (in gm cm−3) andcorresponding n0 (in gm cm−3) and β for white dwarfs andneutron stars [2, 3]. We note that to estimate the plasmanumber density (n0) we have used n0 = zini0 = ρ0/2mp.

Objects ρ0 n0 βWhite dwarfs 106 − 108 3× (1029 − 1031) 0.27− 1.24Neutron stars 1012 − 1014 3× (1035 − 1037) 26.6− 123number density and ion-cyclotron frequency. It alsoincreases with propagating angle for its lower range(0 - 55), but decrease for its upper range (55 -90). It may also be mentioned that as δ → 90,the width goes to zero, and the amplitude goes to∞. It is likely that, for large angles, the assump-tion that the waves are electrostatic is no longervalid, and we should look for fully electromagneticstructures.

4. The magnitude of the external magnetic field B0 hasno direct effect on the SW amplitude. However, itdoes have a direct effect on the width of the SWsand we have found that, as the magnitude of B0increases, the width of the waves decreases, i.e. themagnetic field makes the solitary structures morespiky.5. The parametric regimes for which the solitary wavesbecome stable and unstable are identified. Theseare ion cyclotron frequency, direction of propaga-tion and direction cosine.6. Direction of propagation, ion cyclotron frequencyand direction cosine (lζ and lη) are the dependingfactors which can significantly modify the growthrate (Γ) of the unstable solitary structures.7. These results may be useful for understanding thelocalized electrostatic disturbances in astrophysi-cal compact object, particularly in white dwarfs andneutron stars.

The ranges of the plasma parameters for astrophysicalcompact objects like white dwarfs and neutron stars arevery wide. The plasma parameters for white dwarfs andneutron stars are shown in Table I. The plasma parame-ters used in our present theoretical analysis correspondto white dwarfs. However, our present theoretical anal-ysis can be applied to neutron stars. Therefore, ourpresent results may be useful for understanding the lo-calized electrostatic disturbance in astrophysical compactobjects, particularly, in white dwarf and neutron stars. Itmay be mentioned here that we have used a reductive per-turbation method and small-k perturbation expansion that

are valid for small but finite-amplitude solitary waves andlong-wavelength perturbation modes. Since in many as-trophysical situations there are extremely large-amplitudesolitary waves and short-wavelength perturbation modes,we propose to develop a more exact theory for insta-bility analysis of arbitrary-amplitude solitary waves andarbitrary-wavelength perturbation modes, through a gen-eralization of our present work to such waves and modes.References

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multi-dimensional instability of electrostatic solitary waves in ultra

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