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Nonlinear excitations for the positron acoustic waves in auroral acceleration regions
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Nonlinear excitations for the positron acoustic waves in auroral acceleration regions Asit Saha a, Rustam Ali a,b, Prasanta Chatterjee b,⇑ a
Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Rangpo, East-Sikkim 737136, India b Department of Mathematics, Siksha-Bhavana, Visva-Bharati University, Santiniketan 731235, India Received 20 September 2016; received in revised form 18 May 2017; accepted 5 June 2017
Abstract Positron acoustic waves (PAWs) in an unmagnetized electron-positron-ion (e-p-i) plasma consisting of mobile cold positrons, immobile positive ions, q-nonextensive distributed electrons and hot positrons are studied. The standard reductive perturbation technique (RPT) is applied to derive the Kurteweg-de Vries (KdV) and modified Kurteweg-de Vries (mKdV) equations for PAWs. Variations of the total energy of the conservative systems corresponding to the KdV and mKdV equations are presented. Using numerical simulations, effect of the nonextensive parameter (q), temperature ratio (r) of electrons to hot positrons and speed (U) of the traveling wave are discussed on the positron acoustic solitary wave solutions of the KdV and mKdV equations. Considering an external periodic perturbation, the perturbed dynamical systems corresponding to the KdV and mKdV equations are analyzed by employing phase orbit analysis, Poincare section and Lyapunov exponent. The frequency (x) of the external periodic perturbation plays the role of the switching parameter in chaotic motions of the perturbed PAWs through quasiperiodic route to chaos. This work may be useful to understand the qualitative changes in the dynamics of nonlinear perturbations in auroral acceleration regions. Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Positron acoustic wave; Solitary wave; Quasiperiodicity; Chaos
1. Introduction During last few decades, systematized interactions in electron-positron-ion plasmas have received basic importance because of their tremendous applications in space (Popel et al., 1995; Chawla et al., 2013) and laboratory environments (Bulanov et al., 2005; Surko et al., 1989) depending on the fact that such plasmas exist in the polar regions of neutron stars (Michel, 1991), in the core of white dwarfs (Shapiro and Teukolsky, 1983), pulsar magnetosphere (Michel, 1982), in active galactic nuclei (Miller and Wiita, 1987), at the center of our own galaxy (Burns ⇑ Corresponding author.
E-mail addresses: asit_saha123@rediffmail.com (A. Saha), rustamali24@ gmail.com (R. Ali), prasantachatterjee1@rediffmail.com (P. Chatterjee).
et al., 1983) and in the inner region of accretion disc (Rees, 1971). Recently, a great deal attention has been paid toward the different features of the e-p-i plasmas (Verga and Fontan, 1984; Shukla et al., 1986; Rizzato, 1988; Tajima and Taniuti, 1990; Jammalamadaka et al., 1996; Nejoh, 1996; Rufai et al., 2015; Lakhina and Verheest, 1997; El-Taibany et al., 2008; Verheest et al., 2005; Tribeche et al., 2009; Pakzad, 2009; Mahmood and Ur-Rehman, 2009; Tribeche, 2010; El-Shamy and El-Bedwehy, 2010; Alinejad, 2010; El-Bedwehy and Moslem, 2011). Popel et al. (1995) observed that the presence of positron reduces the soliton amplitude for the ion acoustic solitons in a three component e-p-i plasma. Nejoh (1996) studied the nonlinear wave structures of large amplitude positron acoustic waves in an electron-positron plasma with an electron beam. Rufai et al. (2015)
http://dx.doi.org/10.1016/j.asr.2017.06.012 0273-1177/Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.
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investigated arbitrary amplitude nonlinear low frequency electrostatic structures in magnetized four component auroral plasmas composed of a cold singly charged oxygen-ion fluid, Boltzmann distribution of hot protons and two distinct group of electron species. Recently, El-Shamy and El-Bedwehy (2010) found that the inclusion of the background ions leads to a modification of the nonlinear characteristics of electrostatic solitary waves in e-p-i plasmas. Some authors (Verheest et al., 2005; Tribeche et al., 2009) reported that due to outflows of electronpositron plasmas from pulsars entering an interstellar cold, low density electron-ion plasma, may form two temperature positron electron ion plasma. Accordingly, Tribeche et al. (2009) studied nonlinear small amplitude PASWs involving the dynamics of cold positrons in a four component e-p-i plasma model containing two-temperature positron, isothermal electrons and immobile ions. Furthermore, Tribeche (2010) investigated the small amplitude positron double layer for the same model (Tribeche et al., 2009). Sahu (2010) investigated the PA shock waves in both planar and nonplanar geometries considering the same plasma model (Tribeche, 2010). Rahman et al. (2014) performed a theoretical investigation on positron-acoustic (PA) Gardner solitons (GSs) and double layers (DLs) in four-component plasma system consisting of immobile positive ions, mobile cold positrons, nonthermal hot positrons, and nonthermal electrons. Recently, ion acoustic solitary waves and periodic waves (Saha and Chatterjee, 2014a,b) in unmagnetized plasmas with non-Maxwellian electrons and positrons were investigated through a nonperturbative approach. Recently, Rufai et al. (2015) studied the effect of excess superthermal electrons on finite amplitude nonlinear ion-acoustic waves in a magnetized auroral plasma and observed the evolution of negative potential solitons and supersolitons at subsonic Mach numbers region. It has been reported that several spacecraft missions observed the electrostatic solitary structures in the auroral acceleration region (Temerin et al., 1982; Bostrom et al., 1988; Dubouloz et al., 1991; Mozer et al., 1997). Furthermore, Ghosh and Lakhina (2004) considered a four-component magnetized and unmagnetized plasma models consisting of heavy and light mass ions species, governed by the fluid dynamic equations with two temperature electrons using the Sagdeev pseudo-potential technique and they marked out the POLAR satellite observations of solitary waves in the auroral plasmas. Sabry (2009) studied the effect of positron concentration and temperature ratios on the features of the large and small amplitude ion-acoustic solitary waves and double layers in multicomponent plasma whose components were hot positrons, cold fluid ions, and two-electron temperature distributions. Recent FAST satellite measurements in the auroral acceleration region have revealed that the auroral plasmas are characterized by several ion species (Strangeway et al., 1998). Based on the spacecraft observations of nonlinear low frequency electrostatic structures in the auroral region of the Earths magnetosphere, the
measured parameters (Berthomier et al., 1998) were cool electron density and temperature nce ¼ 0:2 cm3 and T ce ¼ 1 eV, hot electron density and temperature nhe ¼ 1:8 cm3 and T he ¼ 26 eV, which gives the effective temperature T eff 7 eV with total electric field amplitude of less than 100 mV/m, width of about 100 m, about 20 ms of pulse duration and 10–50 km/s soliton velocities range respectively. Jilani et al. (2014) studied properties of fully nonlinear electron-acoustic solitary waves in an unmagnetized and collisionless electron-positron-ion plasma and considered cold electron density noc ð0:1–0:4Þ cm3 , positron density nop ð1:5–3Þ cm3 , hot electron density noh ð1:53Þ cm3 , temperature of hot electrons T h ð200–1000Þ eV, and temperature of positrons T p ð200–1000Þ eV to satisfy various plasma systems from laboratory level to astrophysical space plasmas. It is important to note that Maxwell distribution is valid for the macroscopic ergodic equilibrium state and it may be inadequate to investigate the long range interactions in unmagnetized collisionless plasma having the nonequilibrium stationary state. This type of state may exist due to a number of physical mechanisms, for example, external force field present in natural space plasma environments, wave-particle interaction, and turbulence. Furthermore, a great deal of attention was paid to nonextensive statistical mechanics based on deviations of the Boltzmann-Gibbs-Shannan (BGS) entropic measure. A suitable nonextensive generalization of BGS entropy for statistical equilibrium was first recognized by Renyi (1955) and subsequently proposed by Tsallis (1988), suitably extending the standard additivity of the entropies to the nonlinear, nonextensive case where one particular parameter, the entropy index q, characterizes the degree of nonextensivity of the considered system (Amour and Tribeche, 2010; Saha and Chatterjee, 2014a,b, 2015; Pakzad, 2011; Gill et al., 2010) (q ¼ 1 corresponds to the standard, extensive, BGS statistics). This non-additive entropy of Tsallis and the ensuring generalized statistics have been employed successfully in a wide range of phenomena characterized by nonextensivity (Silva et al., 1998; Lima et al., 2000; Du, 2004; Sattin, 2005; Muhoz, 2006; Gougam and Tribeche, 2011). The following evidences suggest that the q-entropy may provide a convenient frame for the analysis of many astrophysical sceneries, such as stellar polytopes, solar neutrino problem, and peculiar velocity distribution of galaxy clusters. Liyan and Du (2008) investigated ion acoustic waves in the plasma with power-law q-distribution in nonextensive statistics. They suggested that Tsallis (1988) statistics is suitable for the system being the non-equilibrium stationary-state with inhomogeneous temperature and containing plentiful supply of the superthermal or low velocity particles. Recently, using bifurcation theory of planar dynamical systems, Samanta et al. (2013) investigated bifurcations of nonlinear dust ion acoustic traveling waves in a
Please cite this article in press as: Saha, A., et al. Nonlinear excitations for the positron acoustic waves in auroral acceleration regions. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.06.012
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magnetized dusty plasma for the first time. Sahu et al. (2012) studied the quasiperiodic behavior in quantum plasmas due to the presence of bohm potential. Saha et al. (2014) reported the dynamic behavior of ion acoustic waves in electron-positron-ion magnetoplasmas with superthermal electrons and positrons. Very recently, Selim et al. (2015) investigated bifurcations of nonlinear ion-acoustic traveling waves in a multicomponent magnetoplasma with superthermal electrons. But, the bifurcations, quasiperiodicity and chaotic behaviors of the positron acoustic waves are not reported in the literature. So, in the present work, our aim is to investigate the variations of the total energy of the conservative systems corresponding to the KdV and mKdV equations, effects of different physical parameters on the solitary wave solutions of the KdV and mKdV equations, and quasiperiodicity and chaotic behaviors of the positron acoustic waves corresponding to the perturbed systems. The organization of the paper is as follows: In Section 2 basic equations are considered and the KdV and mKdV equations are derived. The conservative systems corresponding to the KdV and mKdV equations are presented in Section 3. In Section 4, solitary wave solutions of the KdV and mKdV equations are considered and effects of different parameters are studied on these solutions. The quasiperiodic and chaotic motions of the positron acoustic waves corresponding to the perturbed systems are investigated in Section 5 and Section 6 is kept for conclusion. 2. Basic equations and derivation of the KdV and mKdV equations We consider a four-component plasma consisting of cold positrons, immobile positive ions, and q-nonextensive distributed electrons and hot positrons. The basic equations are given by @npc @npc upc þ ¼ 0; @t @x @upc @upc @/ ; þ upc ¼ @x @t @x @2/ ¼ npc þ ane bnph þ 1 ða bÞ: @x2
ð1Þ
kB T e , e
the time t is normalized to x1 pc ¼
ð2Þ ð3Þ
k B is the Boltzmann constant, mp is the positron mass, and e is the magnitude of the electron charge, r ¼ TTphe , e0 ; a ¼ nnpc0 , and ni0 =npc0 ¼ a b 1. / is normalized b ¼ nph0 pc0
mp 4pnpc0 e2
12
, where
ðxpc Þ cold positron plasma frequency and the space vari 12 kB T e able is normalized to the Debye length kD ¼ 4pn . 2 e pc0 In order to model an electron distribution with nonextensive particles, we consider the following nonextensive electron distribution function (Bains et al., 2011). 1 ðq1Þ m e v2 e/ f e ðvÞ ¼ C q 1 þ ðq 1Þ ; 2k B T e k B T e where / is the electrostatic potential and the other variables or parameters have their usual meaning. It is really important to note that f e ðvÞ is the special distribution which maximizes the Tsallis entropy and, thus, conforms to the laws of thermodynamics. Then, the constant of normalization is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 C 1q me ð1 qÞ C q ¼ ne0 for 1 < q < 1; 1 1 2pk B T e C 1q
2
and
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 þ q C q1 þ 2 me ðq 1Þ C q ¼ ne0 for q > 1: 1 2 2pk B T e C q1
Integrating f e ðvÞ over all velocity space, one can obtain the following nonextensive electron number density as: 1=ðq1Þþ1=2 e/ : ne ¼ ne0 1 þ ðq 1Þ kBT e Thus, the normalized electron number density (Bains et al., 2011) takes the form: 1
þ1
ne ¼ f1 þ ðq 1Þ/gq1 2 ;
ð4Þ
Similarly, the normalized number density of q-nonextensive hot positrons is given by nph ¼ f1 ðq 1Þr/gq1þ2 ; 1
where npc ; nph ; ne and ni respectively, are the cold positron, hot positron, electron and ion number densities, with ne0 ¼ npc0 þ nph0 þ ni0 at the equilibrium, where the subscript 000 stands for unperturbed quantity. upc is the velocity of the cold positron, / is the electrostatic potential. Here the following normalizations are used: npc ; nph ; ne and ni are normalized to its equilibrium value npc0 ; nph0 ; ne0 and 12 ni0 , respectively. upc is normalized to C pc ¼ kmB Tp e , in which
n
to
3
1
ð5Þ
where the parameter q is a real number greater than 1, and it stands for the strength of nonextensivity. To derive the KdV equation, the reductive perturbation technique is employed. The independent variables are stretched as n ¼ 1=2 ðx vtÞ;
ð6Þ
s¼
ð7Þ
3=2
t:
and the dependent variables are expanded as npc ¼ 1 þ npc1 þ 2 npc2 þ 3 npc3 þ ;
ð8Þ
upc ¼ upc1 þ upc2 þ upc3 þ ;
ð9Þ
2
3
/ ¼ /1 þ /2 þ /3 þ ; 2
3
ð10Þ
where is a small nonzero parameter proportional to the amplitude of the perturbation. Now, substituting
Please cite this article in press as: Saha, A., et al. Nonlinear excitations for the positron acoustic waves in auroral acceleration regions. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.06.012
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Eqs. (4)–(10) into Eqs. (1)–(3)and considering the lowest order of , we obtain the dispersion relation as
@/1 @/ @ 3 /1 þ D/21 1 þ B ¼ 0; @s @n @n3
1 v2 ¼ ; a
where D ¼ 156cv , with c ¼ ðqþ1Þð3qÞð53qÞ ða þ br3 Þ. 48 4v3
ð11Þ
where a ¼ qþ1 ða þ rbÞ. 2 Considering next higher order of and eliminating the perturbed quantities from a set of equations, one can obtain the KdV equation as @/1 @/ @ 3 /1 þ A/1 1 þ B ¼ 0; @s @n @n3
ð12Þ
where A ¼ 32bv ; B ¼ v2 , with b ¼ ðqþ1Þð3qÞ ða br2 Þ. 2v 8 The KdV Eq. (12) depends on A which is a function of a; b; r and q. In Fig. 1, it is shown that A may be positive or negative depending on different values of q with fixed value of a; b, and r. But, for q ¼ 0:57; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1, the coefficient A of the nonlinear term of Eq. (12) becomes zero, which can provide infinite growth to the amplitude of the traveling wave solutions of Eq. (12). To get the information of the traveling wave solutions around this critical values, the modified KdV equation is derived by considering the higher order coefficients of . To derive the modified KdV equation, the expansions of the dependent variables as Eqs. (8)–(10), together with the following stretched coordinates are considered 4
3
n ¼ ðx vtÞ;
ð13Þ
s ¼ t:
ð14Þ
3
Substituting Eqs. (8)–(10) along with the stretched coordinates (13), (14) into equations (1)–(3) and considering the coefficients smallest powers of , one can obtain the same dispersion relation as Eq. (11). Considering the coefficients next higher order of and eliminating the different perturbed quantities the mKdV equation is obtained as
ð15Þ
6
3. Conservative system To study the total energy of the conservative systems corresponding to the KdV and mKdV Eqs. (12) and (15) the transformation v ¼ ðn U sÞ is considered, where U is the speed of the positron acoustic traveling wave. Substituting wðvÞ ¼ /1 ðn; sÞ into Eqs. (12) and (15) and integrating once and neglecting the integrating constants, the KdV Eq. (12) and mKdV Eq. (15) transform to B
d 2w A 2 þ w U w ¼ 0; dv2 2
ð16Þ
and B
d 2w D 3 þ w U w ¼ 0; dv2 3
ð17Þ
respectively. The Eqs. (16) and (17), respectively, can be expressed as d 2 w dV 1 ¼ 0;ð18Þ þ dv2 dw d 2 w dV 2 ¼ 0;ð19Þ þ dv2 dw 1 1 where V 1 ¼ 6B ð3U AwÞw2 and V 2 ¼ 12B ð6U Dw2 Þw2 , are the corresponding potential energies. Therefore the total energy corresponding to the conservative systems (18) and (19) are given by
_2 _ ¼ w 1 ð3U AwÞw2 ; E1 ðw; wÞ 2 6B
ð20Þ
and _2 _ ¼ w 1 ð6U Dw2 Þw2 : E2 ðw; wÞ 2 12B
ð21Þ
2 1.5 1
A
0.5 0
−0.5 q = − 0.57
−1 −1.5 −2 −1
−0.5
0
0.5
q
1
1.5
2
Fig. 1. Plot of A vs. q for a ¼ 1:1; b ¼ 0:3; r ¼ 1:1.
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Depending on the physical parameters q; a; b; r and U, we have presented all possible graphs of the energy function and corresponding phase orbits of the Eqs. (16) and (17)
5
homo-clinic orbits at ð0; 0Þ in Fig. 5 of the Eq. (17) are given by rffiffiffiffiffiffiffi rffiffiffiffi ! 6U U sech v : ð23Þ w¼ D B
on the w; dw plane in Figs. 2–5. dv
It can be easily verified that E1 has a local minima at the ; 0 . Therefore, all trajectories sufficiently close to point 2U 2U A ; 0 are closed and hence, it is a center (See Figs. 2–4). A Similarly, E attains its local minima at the points qffiffiffiffiffi 2 qffiffiffiffiffi 3U ; 0 , and therefore the points 3U ; 0 are surD D
The behavior of the compressive positron acoustic solitary wave profile of Eq. (12) for several values of r is shown in Fig. 6. It is observed that amplitude and width of the solitary wave decrease as r increases. Thus, the positron acoustic solitary wave diminishes as the temperature ratio of electrons to hot positrons is enhanced. Fig. 7 shows the behavior of the rarefactive positron acoustic solitary wave profile of Eq. (12) for several values of the parameter q in the range 1 < q < 0 with same fixed values of the other parameters as Fig. 6. It is observed that the amplitude of the positron acoustic solitary wave decreases rapidly with decrease in q while width of the positron acoustic solitary waves increases with decrease in q. In Fig. 8, the behavior of the compressive positron acoustic solitary wave profile of Eq. (12) for different values of q is presented in the range 0 < q < 1. The amplitude and width of the positron acoustic solitary waves decrease with increase in q. In Fig. 9, the behavior of the compressive positron acoustic solitary wave profile of Eq. (12) for different values of q is presented in the range q > 1. The amplitude and width of the positron acoustic solitary waves decrease with increase in q. It is important to note that if q varies in 1 < q < 0:57 with a ¼ 1:1; b ¼ 0:3; r ¼ 1:1 and U ¼ 0:1, then the KdV Eq. (12) has rarefactive solitary wave solution, where as for 0:57 < q < 1 with same values of other parameters, the KdV Eq. (12) has compressive solitary wave solution. Thus, the parameter q plays a crucial role for the solitary wave solution of the KdV Eq. (12). Fig. 10 shows the behavior of the compressive positron acoustic solitary wave profile of Eq. (12) for several values of the parameter U with some fixed values of the other parameters. It is observed that the amplitude of the
rounded by a family of small closed orbits (see Fig. 5). In Fig. 2, we present the graph of the energy function and level contour of the Eq. (16) for q ¼ 0:6; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1 and U ¼ 0:1 in the range 1 < q < 0 of the nonextensive parameter q. Similarly, in Fig. 3, we depict the graph of the energy function and level contour of the Eq. (16) for q ¼ 0:55 with same value of other parameters as Fig. 2 in the range 0 < q < 1 of the nonextensive parameter q. Furthermore, considering q ¼ 1:1 in the range q > 1 with same value of parameters as Fig. 2, we illustrate the graph of the energy function and level contour of the Eq. (16) in Fig. 4. In Fig. 5, we present the graph of the energy function and level contour of the Eq. (17) for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1. 4. Effect of the different parameters The compressive solitary wave solution of the KdV Eq. (12) corresponding to the homo-clinic orbit at ð0; 0Þ in Fig. 3 of the Eq. (16) is given by rffiffiffiffi ! 3U U 2 1 w¼ sech v : ð22Þ A 2 B The compressive and rarefactive solitary wave solutions of the mKdV Eq. (15) corresponding to the pair of
E1
18 16 14 12 10 8 6 4 2 0 −2 −4 6 4 2 0
ψ
−2 −4 −6
−6
−4
−2
0
2
4
6
Fig. 2. Graph of the energy function and level contours of the Eq. (16) for q ¼ 0:6; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:1.
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0.25 0.2 0.15 0.1
E1
0.05 0
−0.05 −0.1 −0.15 −0.2 0.5 0.4 0.3 0.2 0.1
ψ
0 −0.1 −0.2
−0.3 −0.4
−0.4
−0.5
0
−0.2
0.2
0.4
0.6
Fig. 3. Graph of the energy function and level contours of the Eq. (16) for q ¼ 0:55; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:1.
0.4 0.3 0.2 0.1
E1
0
−0.1 −0.2 −0.3 −0.4 0.5 0.4 0.3 0.2 0.1
ψ
0.2 0 −0.1 −0.2 −0.3 −0.4
−0.4
−0.2
0.4
0.6
0
−0.5
Fig. 4. Graph of the energy function and level contours of the Eq. (16) for q ¼ 1:1; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:1.
35 30 25 20
E
2
15 10 5 0
−5 −10 8
6
4
2
ψ
0
−2
−4
−6
−8
−8
−6
−4
−2
0
2
4
6
8
Fig. 5. Graph of the energy function and level contours of the Eq. (17) for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1.
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7
0.25
0.2
ψ
0.15
0.1
0.05
0 −15
−10
−5
0
χ
5
10
15
Fig. 6. Variation of the solitary wave solutions of Eq. (12) for different values of r ¼ 1:1 (dashed red curve), r ¼ 1:2 (black curve) and r ¼ 1:3 (dotted blue curve) with b ¼ 0:3; q ¼ 0:55; a ¼ 1:1 and U ¼ 0:1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
0 −0.1 −0.2 −0.3
ψ
−0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 −60
−40
−20
0
χ
20
40
60
Fig. 7. Variation of the solitary wave solutions of Eq. (12) for different values of q = 0.7 (dashed red curve), q = 0.8 (black curve) and q = 0.9 (dotted blue curve) with r ¼ 1:1 and other parameters are same as Fig. 6. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
positron acoustic solitary wave increases rapidly with increase in U while width of the positron acoustic solitary waves decreases with increase in U. In Fig. 11, the variation of positron acoustic compressive and rarefactive solitary wave profiles of Eq. (15) is presented for different values of r ¼ 0:5 (dashed red curve), r ¼ 0:6 (black curve), and r ¼ 0:7 (dotted blue curve) with b ¼ 0:35; q ¼ 0:4836; a ¼ 0:5 and U ¼ 0:1. The amplitude and width of the positron acoustic compressive and rarefactive solitary waves decrease with increase in r. Thus, the positron acoustic compressive and rarefactive solitary waves diminish as the parameter r is enhanced. In Fig. 12, the variation of positron acoustic compressive and rarefactive solitary wave profiles of Eq. (15) is presented for different values of q = 0.4836 (dashed red curve),
q = 0.5836 (black curve), and q = 0.6836 (dotted blue curve) with r ¼ 0:5 and other parameters are same as Fig. 11. The amplitude and width of the positron acoustic compressive and rarefactive solitary waves decrease with increase in q. Thus, the positron acoustic compressive and rarefactive solitary waves diminish as the parameter q is enhanced. In Fig. 13, the variation of positron acoustic compressive and rarefactive solitary wave profiles of Eq. (15) is presented for different values of U = 0.03 (dashed red curve), U = 0.07 (black curve), and U = 0.11 (dotted blue curve) with q ¼ 0:4836 and other parameters are same as Fig. 12. The amplitude of the positron acoustic solitary waves increases and width decreases with increase in U. Thus, the positron acoustic solitary wave grows rapidly as speed U of the solitary wave is enhanced in the subsonic
Please cite this article in press as: Saha, A., et al. Nonlinear excitations for the positron acoustic waves in auroral acceleration regions. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.06.012
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A. Saha et al. / Advances in Space Research xxx (2017) xxx–xxx 0.25
0.2
ψ
0.15
0.1
0.05
0 −15
−10
−5
0
5
10
15
χ
Fig. 8. Variation of the solitary wave solutions of Eq. (12) for different values of q = 0.55 (dashed red curve), q = 0.65 (black curve) and q = 0.75 (dotted blue curve) with other parameters are same as Fig. 7. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
0.2 0.18 0.16 0.14
ψ
0.12 0.1 0.08 0.06 0.04 0.02 0 −10
−8
−6
−4
−2
0
χ
2
4
6
8
10
Fig. 9. Variation of the solitary wave solutions of Eq. (12) for different values of q = 1.1 (dashed red curve), q = 1.9 (black curve) and q = 2.7 (dotted blue curve) with other parameters are same as Fig. 7. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
region. The positron acoustic compressive and rarefactive solitary wave profiles of the present work are supported by the work (Uddin et al., 2015).
5. Quasiperiodic and chaotic motions In order to study the quasiperiodic and chaotic behavior of the perturbed systems given by B
d 2w A 2 þ w U w ¼ f 0 cosðxvÞ; dv2 2
and
ð24Þ
B
d 2w D 3 þ w U w ¼ f 0 cosðxvÞ; dv2 3
ð25Þ
different numerical tools are applied: (i) phase portrait plots, (ii) Poincare section and (iii) Lyapunov exponent. They all are very useful tools to examining quasiperiodic and chaotic features (Lakshmanan and Rajasekar, 2003). The perturbed systems (24) and (25) contain seven independent parameters q; a; r; b; U ; x and f 0 . Because of large number of parameters, it is really difficult to investigate the systems for the complete range of parametric space. To simplify the task, we focus on some special values of the parameters q; a; r; b; U varying the frequency parameter (x) of the external periodic perturbation. Thus, x plays the critical role for the dynamic behavior of the positron acoustic waves. We
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9
0.8 0.7 0.6
ψ
0.5 0.4 0.3 0.2 0.1 0
−10
−5
0
5
χ
10
Fig. 10. Variation of the solitary wave solutions of Eq. (12) for different values of U = 0.1 (dashed red curve), U = 0.2 (black curve) and U = 0.3 (dotted blue curve) with q ¼ 0:55 and other parameters are same as Fig. 8. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1 0.8 0.6 0.4
ψ
0.2 0
−0.2 −0.4 −0.6 −0.8 −1 −20
−15
−10
−5
0
χ
5
10
15
20
Fig. 11. Variation of the solitary wave solutions of compressive and rarefactive types of Eq. (15) for different values of r ¼ 0:5 (dashed red curve), r ¼ 0:6 (black curve), and r ¼ 0:7 (dotted blue curve) with b ¼ 0:35; q ¼ 0:4836; a ¼ 0:5 and U ¼ 0:1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
could vary any of the other parameters, but this will not give any significant different qualitative feature. 5.1. 3D phase portrait analysis In Fig. 14, 3D phase orbits of the perturbed system (24) are presented for q ¼ 0:6; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:125; f 0 ¼ 0:2 and x ¼ 7:5. In Fig. 15, we depict z vs. v of the Eq. (24) for same value of parameters as Fig. 14. In Fig. 16, 3D phase orbits of the perturbed system (24) are presented for q ¼ 1:1; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:125; f 0 ¼ 0:2 and x ¼ 7:5. In Fig. 17, we depict z vs. v of the Eq. (24) for same value of parameters as
Fig. 16. In Fig. 18, 3D phase orbits of the perturbed system (25) are presented for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62 and x ¼ 15:77. In Fig. 19, we illustrate z vs. v of the Eq. (25) for same value of parameters as Fig. 18. The orbits presented in Figs. 14, 16 and 18, ignore the periodic motions and exhibit aperiodic oscillations i.e., quasiperiodic motions of the systems (24) and (25) are found with incommensurable periodic motions and the trajectory in the phase space winds around torus filling its surface densely. In Fig. 20, 3D phase orbits of the perturbed system (25) are presented for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62 and x ¼ 0:77. The orbits in Fig. 20 ignore the periodic
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10
A. Saha et al. / Advances in Space Research xxx (2017) xxx–xxx 1 0.8 0.6 0.4
ψ
0.2 0
−0.2 −0.4 −0.6 −0.8 −1 −20
−15
−10
−5
0
χ
5
10
15
20
Fig. 12. Variation of the solitary wave solutions of compressive and rarefactive types of Eq. (15) for different values of q = 0.4836 (dashed red curve), q = 0.5836 (black curve), and q = 0.6836 (dotted blue curve) with r ¼ 0:5 and other parameters are same as Fig. 11. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1 0.8 0.6 0.4
ψ
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −30
−20
−10
0
χ
10
20
30
Fig. 13. Variation of the solitary wave solutions of compressive and rarefactive types of Eq. (15) for different values of U = 0.03 (dashed red curve), U = 0.07 (black curve), and U = 0.11 (dotted blue curve) with q ¼ 0:4836 and other parameters are same as Fig. 12. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
motions and exhibit chaotic behavior. In Fig. 21, we depict z vs. v of the Eq. (25) for same value of parameters as Fig. 20.
5.2. Poincare return map analysis A very useful tool for exploring chaotic attractors is Poincare sections (Lorenz, 1963; Molenaar et al., 2005). Poincare sections provide us with images of the attractor in a ðn 1Þ-dimensional hyperplane of n-dimensional phase space. Thus, the Poincare return map converts problems about closed orbits into problems about fixed points of a mapping. Moreover, by looking at the behavior of the Poincare return map near a fixed point we can deter-
mine the stability of the closed orbit. The Poincare return map of the systems (24) and (25) are shown in Figs. 22–25. A close inspection of Fig. 22 clearly shows that the points are densely populated and lie on a smooth closed curve, which indicates two-frequency oscillation and confirms the existence of quasi-periodic structure of the nonlinear PASWs for the system (24) for the parameter values q ¼ 0:6; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:125; f 0 ¼ 0:2; x ¼ 7:5. Fig. 23 shows that the points are densely populated and lie on a smooth closed curve, which indicates two -frequency oscillation and confirms the existence of quasi-periodic structure of the nonlinear PASWs for the system (24) for the parameter values q ¼ 1:1; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:125; f 0 ¼ 0:2; x ¼ 7:5. Fig. 24 shows that the points are densely populated and lie on a smooth curve, which also indicates
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900 800 700 600
χ
500 400 300 200 100 0 0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
−4.6
−4.4
−4.2
−3.8
−4
−3.6
−3.4
−3
−3.2
ψ
Fig. 14. 3 D plot of the phase orbits for the Eq. (24) for q ¼ 0:6; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:125; f 0 ¼ 0:2; x ¼ 7:5.
0.4 0.3 0.2
z
0.1 0
−0.1 −0.2 −0.3 −0.4
0
10
20
30
40
50
χ
60
70
80
90
100
Fig. 15. Plot of z vs. v of the Eq. (24) for same values of parameters as Fig. 14.
350 300 250
χ
200 150 100 50 0 0.08
0.06
0.04
0.02
z
0
−0.02 −0.04
−0.06 −0.08
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
ψ
Fig. 16. 3 D plot of the phase orbits for the Eq. (24) for q ¼ 1:1 and other parameters are same as Fig. 14.
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A. Saha et al. / Advances in Space Research xxx (2017) xxx–xxx 0.08 0.06 0.04 0.02
z
0 −0.02 −0.04 −0.06 −0.08 0
20
40
60
80
100
120
χ
140
160
180
200
Fig. 17. Plot of z vs. v of the Eq. (24) for same values of parameters as Fig. 16.
900 800 700
χ
600 500 400 300 200 100 0
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
1.5
1
2
2.5
3
ψ
Fig. 18. 3 D plot of the phase orbits for the Eq. (25) for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62; x ¼ 15:77.
0.4 0.3 0.2
z
0.1 0
−0.1 −0.2 −0.3 −0.4
0
10
20
30
χ
40
50
60
70
Fig. 19. Plot of z vs. v of the Eq. (25) for same values of parameters as Fig. 18.
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800 700 600
χ
500 400 300 200 100 0 6
4
2
0
−2
−4
−6
−10
−8
−6
−4
−2
0
2
4
6
10
8
ψ
Fig. 20. 3 D plot of the phase orbits for the Eq. (25) for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62; x ¼ 0:77. 15
10
z
5
0
−5
−10
−15 0
100
200
300
400
χ
500
600
700
800
Fig. 21. Plot of z vs. v of the Eq. (25) for same values of parameters as Fig. 20.
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2
−4.6
−4.4
−4.2
−4
−3.8
ψn
−3.6
−3.4
−3.2
−3
Fig. 22. Poincare section of the Eq. (24) for the same value of the parameters as Fig. 14.
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A. Saha et al. / Advances in Space Research xxx (2017) xxx–xxx
0.06
0.04
0.02
0
−0.02
−0.04
−0.06 0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
ψ
n
Fig. 23. Poincare section of the Eq. (24) for the same value of the parameters as Fig. 16.
0.3
0.2
0.1
ψ˙ n
0
−0.1
−0.2
−0.3 1.2
1.4
1.6
1.8
2
2.2
ψn
2.4
2.6
2.8
3
3.2
Fig. 24. Poincare section of the Eq. (25) for the same parameters value as Fig. 18.
10
ψ˙ n
5
0
−5
−10 −10
−5
0
ψn
5
10
15
Fig. 25. Poincare section of the Eq. (25) for the same parameters value as Fig. 20.
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Lyapunov Exponents (λ(χ))
0.1 0.08
λ =
0.086612
0.06
λ2=
−0.086612
0.04
λ =
0
1
3
0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1
50
100
150
200
χ
250
300
350
400
Fig. 26. Dynamics of the Lyapunov exponents of the Eq. (25) for the same parameters value as Fig. 14.
two-frequency oscillation and confirms the existence of quasi-periodic behavior of the nonlinear PASWs for the system (25) with parameter values q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62 and x ¼ 15:77. In return map plot, Fig. 25 demonstrates that the points are densely scattered and exhibit irregular distributions without any known pattern for the system (25) with parameter values q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62 and x ¼ 0:77. Such return maps are characteristics of chaotic oscillations. 5.3. Lyapunov exponents Lyapunov exponents provide a quantitative measure of the divergence or convergence of nearby trajectories for a dynamical system. If we consider a small hypersphere of initial conditions in the phase space, for sufficiently short time scales, the effect of the dynamics will be distort this set into a hyper-ellipsoid, stretched along some directions and contracted along others. The asymptotic rate of expansion of the largest axis, corresponds to the most unstable direction of the flow, is measured by the largest Lyapunov exponent. For a periodic orbit, one of the Lyapunov components is zero. For a quasi-periodic orbit, two of the Lyapunov exponents are zero and others are negative. For chaotic behavior at least one of the Lyapunov exponents must be positive so that neighboring trajectories diverge. When more than one of the Lyapunov exponents are positive, then the motion is referred as hyper-chaos. In Fig. 26, we have presented the dynamics of the Lyapunov exponents of the system (25) for the parameter values q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62; x ¼ 0:77, which confirms the existence of choas in the system (25). 6. Conclusions In this work, we have studied the positron acoustic waves in an unmagnetized e-p-i plasmas whose constituents
are mobile cold positrons, immobile positive ions, q-nonextensive distributed electrons and hot positrons. The standard RPT has been applied to derive the KdV and mKdV equations for PAWs. Variations of the total energy of the conservative systems for the KdV and mKdV equations are presented. Applying numerical computations, effects of physical parameters q; r and U have been presented on the positron acoustic solitary wave solutions of the KdV and mKdV equations. It is observed that when nonextensive particles moves far away from Maxwellian distribution, the positron acoustic solitary waves are flourished. Considering an external periodic perturbation the perturbed dynamical systems corresponding to the KdV and mKdV equations have been analyzed by employing phase orbit analysis, Poincare return map analysis and Lyapunov exponent. The frequency (x) of the external periodic perturbation has played the role of the switching parameter in chaotic motions of the perturbed PAWs through quasiperiodic route to chaos. The results of this study may be applied to understand the qualitative changes in the dynamics of the positron acoustic waves in auroral acceleration regions. Acknowledgments The authors are grateful to the reviewers and the Editor for their useful comments and suggestions which helped to improve the paper. References Popel, S.I., Vladimirov, S.V., Shukla, P.K., 1995. Ion acoustic solitons in electronpositronion plasmas. Phys. Plasmas 2, 716. Chawla, J.K., Mishra, M.K., Tiwari, R.S., 2013. Modulational instability of ion-acoustic waves in electronpositronion plasmas. Astrophys. Space Sci. 347, 283–292. Bulanov, S.S., Fedotov, A.M., Pegoraro, F., 2005. Damping of electromagnetic waves due to electron-positron pair production. Phys. Rev. E. 71, 016404. Surko, C.M., Leventhal, M., Passner, A., 1989. Positron plasma in the laboratory. Phys. Rev. Lett. 62, 901–904.
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ScienceDirect Advances in Space Research xxx (2017) xxx–xxx www.elsevier.com/locate/asr
Nonlinear excitations for the positron acoustic waves in auroral acceleration regions Asit Saha a, Rustam Ali a,b, Prasanta Chatterjee b,⇑ a
Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Rangpo, East-Sikkim 737136, India b Department of Mathematics, Siksha-Bhavana, Visva-Bharati University, Santiniketan 731235, India Received 20 September 2016; received in revised form 18 May 2017; accepted 5 June 2017
Abstract Positron acoustic waves (PAWs) in an unmagnetized electron-positron-ion (e-p-i) plasma consisting of mobile cold positrons, immobile positive ions, q-nonextensive distributed electrons and hot positrons are studied. The standard reductive perturbation technique (RPT) is applied to derive the Kurteweg-de Vries (KdV) and modified Kurteweg-de Vries (mKdV) equations for PAWs. Variations of the total energy of the conservative systems corresponding to the KdV and mKdV equations are presented. Using numerical simulations, effect of the nonextensive parameter (q), temperature ratio (r) of electrons to hot positrons and speed (U) of the traveling wave are discussed on the positron acoustic solitary wave solutions of the KdV and mKdV equations. Considering an external periodic perturbation, the perturbed dynamical systems corresponding to the KdV and mKdV equations are analyzed by employing phase orbit analysis, Poincare section and Lyapunov exponent. The frequency (x) of the external periodic perturbation plays the role of the switching parameter in chaotic motions of the perturbed PAWs through quasiperiodic route to chaos. This work may be useful to understand the qualitative changes in the dynamics of nonlinear perturbations in auroral acceleration regions. Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Positron acoustic wave; Solitary wave; Quasiperiodicity; Chaos
1. Introduction During last few decades, systematized interactions in electron-positron-ion plasmas have received basic importance because of their tremendous applications in space (Popel et al., 1995; Chawla et al., 2013) and laboratory environments (Bulanov et al., 2005; Surko et al., 1989) depending on the fact that such plasmas exist in the polar regions of neutron stars (Michel, 1991), in the core of white dwarfs (Shapiro and Teukolsky, 1983), pulsar magnetosphere (Michel, 1982), in active galactic nuclei (Miller and Wiita, 1987), at the center of our own galaxy (Burns ⇑ Corresponding author.
E-mail addresses: asit_saha123@rediffmail.com (A. Saha), rustamali24@ gmail.com (R. Ali), prasantachatterjee1@rediffmail.com (P. Chatterjee).
et al., 1983) and in the inner region of accretion disc (Rees, 1971). Recently, a great deal attention has been paid toward the different features of the e-p-i plasmas (Verga and Fontan, 1984; Shukla et al., 1986; Rizzato, 1988; Tajima and Taniuti, 1990; Jammalamadaka et al., 1996; Nejoh, 1996; Rufai et al., 2015; Lakhina and Verheest, 1997; El-Taibany et al., 2008; Verheest et al., 2005; Tribeche et al., 2009; Pakzad, 2009; Mahmood and Ur-Rehman, 2009; Tribeche, 2010; El-Shamy and El-Bedwehy, 2010; Alinejad, 2010; El-Bedwehy and Moslem, 2011). Popel et al. (1995) observed that the presence of positron reduces the soliton amplitude for the ion acoustic solitons in a three component e-p-i plasma. Nejoh (1996) studied the nonlinear wave structures of large amplitude positron acoustic waves in an electron-positron plasma with an electron beam. Rufai et al. (2015)
http://dx.doi.org/10.1016/j.asr.2017.06.012 0273-1177/Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.
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investigated arbitrary amplitude nonlinear low frequency electrostatic structures in magnetized four component auroral plasmas composed of a cold singly charged oxygen-ion fluid, Boltzmann distribution of hot protons and two distinct group of electron species. Recently, El-Shamy and El-Bedwehy (2010) found that the inclusion of the background ions leads to a modification of the nonlinear characteristics of electrostatic solitary waves in e-p-i plasmas. Some authors (Verheest et al., 2005; Tribeche et al., 2009) reported that due to outflows of electronpositron plasmas from pulsars entering an interstellar cold, low density electron-ion plasma, may form two temperature positron electron ion plasma. Accordingly, Tribeche et al. (2009) studied nonlinear small amplitude PASWs involving the dynamics of cold positrons in a four component e-p-i plasma model containing two-temperature positron, isothermal electrons and immobile ions. Furthermore, Tribeche (2010) investigated the small amplitude positron double layer for the same model (Tribeche et al., 2009). Sahu (2010) investigated the PA shock waves in both planar and nonplanar geometries considering the same plasma model (Tribeche, 2010). Rahman et al. (2014) performed a theoretical investigation on positron-acoustic (PA) Gardner solitons (GSs) and double layers (DLs) in four-component plasma system consisting of immobile positive ions, mobile cold positrons, nonthermal hot positrons, and nonthermal electrons. Recently, ion acoustic solitary waves and periodic waves (Saha and Chatterjee, 2014a,b) in unmagnetized plasmas with non-Maxwellian electrons and positrons were investigated through a nonperturbative approach. Recently, Rufai et al. (2015) studied the effect of excess superthermal electrons on finite amplitude nonlinear ion-acoustic waves in a magnetized auroral plasma and observed the evolution of negative potential solitons and supersolitons at subsonic Mach numbers region. It has been reported that several spacecraft missions observed the electrostatic solitary structures in the auroral acceleration region (Temerin et al., 1982; Bostrom et al., 1988; Dubouloz et al., 1991; Mozer et al., 1997). Furthermore, Ghosh and Lakhina (2004) considered a four-component magnetized and unmagnetized plasma models consisting of heavy and light mass ions species, governed by the fluid dynamic equations with two temperature electrons using the Sagdeev pseudo-potential technique and they marked out the POLAR satellite observations of solitary waves in the auroral plasmas. Sabry (2009) studied the effect of positron concentration and temperature ratios on the features of the large and small amplitude ion-acoustic solitary waves and double layers in multicomponent plasma whose components were hot positrons, cold fluid ions, and two-electron temperature distributions. Recent FAST satellite measurements in the auroral acceleration region have revealed that the auroral plasmas are characterized by several ion species (Strangeway et al., 1998). Based on the spacecraft observations of nonlinear low frequency electrostatic structures in the auroral region of the Earths magnetosphere, the
measured parameters (Berthomier et al., 1998) were cool electron density and temperature nce ¼ 0:2 cm3 and T ce ¼ 1 eV, hot electron density and temperature nhe ¼ 1:8 cm3 and T he ¼ 26 eV, which gives the effective temperature T eff 7 eV with total electric field amplitude of less than 100 mV/m, width of about 100 m, about 20 ms of pulse duration and 10–50 km/s soliton velocities range respectively. Jilani et al. (2014) studied properties of fully nonlinear electron-acoustic solitary waves in an unmagnetized and collisionless electron-positron-ion plasma and considered cold electron density noc ð0:1–0:4Þ cm3 , positron density nop ð1:5–3Þ cm3 , hot electron density noh ð1:53Þ cm3 , temperature of hot electrons T h ð200–1000Þ eV, and temperature of positrons T p ð200–1000Þ eV to satisfy various plasma systems from laboratory level to astrophysical space plasmas. It is important to note that Maxwell distribution is valid for the macroscopic ergodic equilibrium state and it may be inadequate to investigate the long range interactions in unmagnetized collisionless plasma having the nonequilibrium stationary state. This type of state may exist due to a number of physical mechanisms, for example, external force field present in natural space plasma environments, wave-particle interaction, and turbulence. Furthermore, a great deal of attention was paid to nonextensive statistical mechanics based on deviations of the Boltzmann-Gibbs-Shannan (BGS) entropic measure. A suitable nonextensive generalization of BGS entropy for statistical equilibrium was first recognized by Renyi (1955) and subsequently proposed by Tsallis (1988), suitably extending the standard additivity of the entropies to the nonlinear, nonextensive case where one particular parameter, the entropy index q, characterizes the degree of nonextensivity of the considered system (Amour and Tribeche, 2010; Saha and Chatterjee, 2014a,b, 2015; Pakzad, 2011; Gill et al., 2010) (q ¼ 1 corresponds to the standard, extensive, BGS statistics). This non-additive entropy of Tsallis and the ensuring generalized statistics have been employed successfully in a wide range of phenomena characterized by nonextensivity (Silva et al., 1998; Lima et al., 2000; Du, 2004; Sattin, 2005; Muhoz, 2006; Gougam and Tribeche, 2011). The following evidences suggest that the q-entropy may provide a convenient frame for the analysis of many astrophysical sceneries, such as stellar polytopes, solar neutrino problem, and peculiar velocity distribution of galaxy clusters. Liyan and Du (2008) investigated ion acoustic waves in the plasma with power-law q-distribution in nonextensive statistics. They suggested that Tsallis (1988) statistics is suitable for the system being the non-equilibrium stationary-state with inhomogeneous temperature and containing plentiful supply of the superthermal or low velocity particles. Recently, using bifurcation theory of planar dynamical systems, Samanta et al. (2013) investigated bifurcations of nonlinear dust ion acoustic traveling waves in a
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magnetized dusty plasma for the first time. Sahu et al. (2012) studied the quasiperiodic behavior in quantum plasmas due to the presence of bohm potential. Saha et al. (2014) reported the dynamic behavior of ion acoustic waves in electron-positron-ion magnetoplasmas with superthermal electrons and positrons. Very recently, Selim et al. (2015) investigated bifurcations of nonlinear ion-acoustic traveling waves in a multicomponent magnetoplasma with superthermal electrons. But, the bifurcations, quasiperiodicity and chaotic behaviors of the positron acoustic waves are not reported in the literature. So, in the present work, our aim is to investigate the variations of the total energy of the conservative systems corresponding to the KdV and mKdV equations, effects of different physical parameters on the solitary wave solutions of the KdV and mKdV equations, and quasiperiodicity and chaotic behaviors of the positron acoustic waves corresponding to the perturbed systems. The organization of the paper is as follows: In Section 2 basic equations are considered and the KdV and mKdV equations are derived. The conservative systems corresponding to the KdV and mKdV equations are presented in Section 3. In Section 4, solitary wave solutions of the KdV and mKdV equations are considered and effects of different parameters are studied on these solutions. The quasiperiodic and chaotic motions of the positron acoustic waves corresponding to the perturbed systems are investigated in Section 5 and Section 6 is kept for conclusion. 2. Basic equations and derivation of the KdV and mKdV equations We consider a four-component plasma consisting of cold positrons, immobile positive ions, and q-nonextensive distributed electrons and hot positrons. The basic equations are given by @npc @npc upc þ ¼ 0; @t @x @upc @upc @/ ; þ upc ¼ @x @t @x @2/ ¼ npc þ ane bnph þ 1 ða bÞ: @x2
ð1Þ
kB T e , e
the time t is normalized to x1 pc ¼
ð2Þ ð3Þ
k B is the Boltzmann constant, mp is the positron mass, and e is the magnitude of the electron charge, r ¼ TTphe , e0 ; a ¼ nnpc0 , and ni0 =npc0 ¼ a b 1. / is normalized b ¼ nph0 pc0
mp 4pnpc0 e2
12
, where
ðxpc Þ cold positron plasma frequency and the space vari 12 kB T e able is normalized to the Debye length kD ¼ 4pn . 2 e pc0 In order to model an electron distribution with nonextensive particles, we consider the following nonextensive electron distribution function (Bains et al., 2011). 1 ðq1Þ m e v2 e/ f e ðvÞ ¼ C q 1 þ ðq 1Þ ; 2k B T e k B T e where / is the electrostatic potential and the other variables or parameters have their usual meaning. It is really important to note that f e ðvÞ is the special distribution which maximizes the Tsallis entropy and, thus, conforms to the laws of thermodynamics. Then, the constant of normalization is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 C 1q me ð1 qÞ C q ¼ ne0 for 1 < q < 1; 1 1 2pk B T e C 1q
2
and
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 þ q C q1 þ 2 me ðq 1Þ C q ¼ ne0 for q > 1: 1 2 2pk B T e C q1
Integrating f e ðvÞ over all velocity space, one can obtain the following nonextensive electron number density as: 1=ðq1Þþ1=2 e/ : ne ¼ ne0 1 þ ðq 1Þ kBT e Thus, the normalized electron number density (Bains et al., 2011) takes the form: 1
þ1
ne ¼ f1 þ ðq 1Þ/gq1 2 ;
ð4Þ
Similarly, the normalized number density of q-nonextensive hot positrons is given by nph ¼ f1 ðq 1Þr/gq1þ2 ; 1
where npc ; nph ; ne and ni respectively, are the cold positron, hot positron, electron and ion number densities, with ne0 ¼ npc0 þ nph0 þ ni0 at the equilibrium, where the subscript 000 stands for unperturbed quantity. upc is the velocity of the cold positron, / is the electrostatic potential. Here the following normalizations are used: npc ; nph ; ne and ni are normalized to its equilibrium value npc0 ; nph0 ; ne0 and 12 ni0 , respectively. upc is normalized to C pc ¼ kmB Tp e , in which
n
to
3
1
ð5Þ
where the parameter q is a real number greater than 1, and it stands for the strength of nonextensivity. To derive the KdV equation, the reductive perturbation technique is employed. The independent variables are stretched as n ¼ 1=2 ðx vtÞ;
ð6Þ
s¼
ð7Þ
3=2
t:
and the dependent variables are expanded as npc ¼ 1 þ npc1 þ 2 npc2 þ 3 npc3 þ ;
ð8Þ
upc ¼ upc1 þ upc2 þ upc3 þ ;
ð9Þ
2
3
/ ¼ /1 þ /2 þ /3 þ ; 2
3
ð10Þ
where is a small nonzero parameter proportional to the amplitude of the perturbation. Now, substituting
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Eqs. (4)–(10) into Eqs. (1)–(3)and considering the lowest order of , we obtain the dispersion relation as
@/1 @/ @ 3 /1 þ D/21 1 þ B ¼ 0; @s @n @n3
1 v2 ¼ ; a
where D ¼ 156cv , with c ¼ ðqþ1Þð3qÞð53qÞ ða þ br3 Þ. 48 4v3
ð11Þ
where a ¼ qþ1 ða þ rbÞ. 2 Considering next higher order of and eliminating the perturbed quantities from a set of equations, one can obtain the KdV equation as @/1 @/ @ 3 /1 þ A/1 1 þ B ¼ 0; @s @n @n3
ð12Þ
where A ¼ 32bv ; B ¼ v2 , with b ¼ ðqþ1Þð3qÞ ða br2 Þ. 2v 8 The KdV Eq. (12) depends on A which is a function of a; b; r and q. In Fig. 1, it is shown that A may be positive or negative depending on different values of q with fixed value of a; b, and r. But, for q ¼ 0:57; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1, the coefficient A of the nonlinear term of Eq. (12) becomes zero, which can provide infinite growth to the amplitude of the traveling wave solutions of Eq. (12). To get the information of the traveling wave solutions around this critical values, the modified KdV equation is derived by considering the higher order coefficients of . To derive the modified KdV equation, the expansions of the dependent variables as Eqs. (8)–(10), together with the following stretched coordinates are considered 4
3
n ¼ ðx vtÞ;
ð13Þ
s ¼ t:
ð14Þ
3
Substituting Eqs. (8)–(10) along with the stretched coordinates (13), (14) into equations (1)–(3) and considering the coefficients smallest powers of , one can obtain the same dispersion relation as Eq. (11). Considering the coefficients next higher order of and eliminating the different perturbed quantities the mKdV equation is obtained as
ð15Þ
6
3. Conservative system To study the total energy of the conservative systems corresponding to the KdV and mKdV Eqs. (12) and (15) the transformation v ¼ ðn U sÞ is considered, where U is the speed of the positron acoustic traveling wave. Substituting wðvÞ ¼ /1 ðn; sÞ into Eqs. (12) and (15) and integrating once and neglecting the integrating constants, the KdV Eq. (12) and mKdV Eq. (15) transform to B
d 2w A 2 þ w U w ¼ 0; dv2 2
ð16Þ
and B
d 2w D 3 þ w U w ¼ 0; dv2 3
ð17Þ
respectively. The Eqs. (16) and (17), respectively, can be expressed as d 2 w dV 1 ¼ 0;ð18Þ þ dv2 dw d 2 w dV 2 ¼ 0;ð19Þ þ dv2 dw 1 1 where V 1 ¼ 6B ð3U AwÞw2 and V 2 ¼ 12B ð6U Dw2 Þw2 , are the corresponding potential energies. Therefore the total energy corresponding to the conservative systems (18) and (19) are given by
_2 _ ¼ w 1 ð3U AwÞw2 ; E1 ðw; wÞ 2 6B
ð20Þ
and _2 _ ¼ w 1 ð6U Dw2 Þw2 : E2 ðw; wÞ 2 12B
ð21Þ
2 1.5 1
A
0.5 0
−0.5 q = − 0.57
−1 −1.5 −2 −1
−0.5
0
0.5
q
1
1.5
2
Fig. 1. Plot of A vs. q for a ¼ 1:1; b ¼ 0:3; r ¼ 1:1.
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Depending on the physical parameters q; a; b; r and U, we have presented all possible graphs of the energy function and corresponding phase orbits of the Eqs. (16) and (17)
5
homo-clinic orbits at ð0; 0Þ in Fig. 5 of the Eq. (17) are given by rffiffiffiffiffiffiffi rffiffiffiffi ! 6U U sech v : ð23Þ w¼ D B
on the w; dw plane in Figs. 2–5. dv
It can be easily verified that E1 has a local minima at the ; 0 . Therefore, all trajectories sufficiently close to point 2U 2U A ; 0 are closed and hence, it is a center (See Figs. 2–4). A Similarly, E attains its local minima at the points qffiffiffiffiffi 2 qffiffiffiffiffi 3U ; 0 , and therefore the points 3U ; 0 are surD D
The behavior of the compressive positron acoustic solitary wave profile of Eq. (12) for several values of r is shown in Fig. 6. It is observed that amplitude and width of the solitary wave decrease as r increases. Thus, the positron acoustic solitary wave diminishes as the temperature ratio of electrons to hot positrons is enhanced. Fig. 7 shows the behavior of the rarefactive positron acoustic solitary wave profile of Eq. (12) for several values of the parameter q in the range 1 < q < 0 with same fixed values of the other parameters as Fig. 6. It is observed that the amplitude of the positron acoustic solitary wave decreases rapidly with decrease in q while width of the positron acoustic solitary waves increases with decrease in q. In Fig. 8, the behavior of the compressive positron acoustic solitary wave profile of Eq. (12) for different values of q is presented in the range 0 < q < 1. The amplitude and width of the positron acoustic solitary waves decrease with increase in q. In Fig. 9, the behavior of the compressive positron acoustic solitary wave profile of Eq. (12) for different values of q is presented in the range q > 1. The amplitude and width of the positron acoustic solitary waves decrease with increase in q. It is important to note that if q varies in 1 < q < 0:57 with a ¼ 1:1; b ¼ 0:3; r ¼ 1:1 and U ¼ 0:1, then the KdV Eq. (12) has rarefactive solitary wave solution, where as for 0:57 < q < 1 with same values of other parameters, the KdV Eq. (12) has compressive solitary wave solution. Thus, the parameter q plays a crucial role for the solitary wave solution of the KdV Eq. (12). Fig. 10 shows the behavior of the compressive positron acoustic solitary wave profile of Eq. (12) for several values of the parameter U with some fixed values of the other parameters. It is observed that the amplitude of the
rounded by a family of small closed orbits (see Fig. 5). In Fig. 2, we present the graph of the energy function and level contour of the Eq. (16) for q ¼ 0:6; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1 and U ¼ 0:1 in the range 1 < q < 0 of the nonextensive parameter q. Similarly, in Fig. 3, we depict the graph of the energy function and level contour of the Eq. (16) for q ¼ 0:55 with same value of other parameters as Fig. 2 in the range 0 < q < 1 of the nonextensive parameter q. Furthermore, considering q ¼ 1:1 in the range q > 1 with same value of parameters as Fig. 2, we illustrate the graph of the energy function and level contour of the Eq. (16) in Fig. 4. In Fig. 5, we present the graph of the energy function and level contour of the Eq. (17) for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1. 4. Effect of the different parameters The compressive solitary wave solution of the KdV Eq. (12) corresponding to the homo-clinic orbit at ð0; 0Þ in Fig. 3 of the Eq. (16) is given by rffiffiffiffi ! 3U U 2 1 w¼ sech v : ð22Þ A 2 B The compressive and rarefactive solitary wave solutions of the mKdV Eq. (15) corresponding to the pair of
E1
18 16 14 12 10 8 6 4 2 0 −2 −4 6 4 2 0
ψ
−2 −4 −6
−6
−4
−2
0
2
4
6
Fig. 2. Graph of the energy function and level contours of the Eq. (16) for q ¼ 0:6; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:1.
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0.25 0.2 0.15 0.1
E1
0.05 0
−0.05 −0.1 −0.15 −0.2 0.5 0.4 0.3 0.2 0.1
ψ
0 −0.1 −0.2
−0.3 −0.4
−0.4
−0.5
0
−0.2
0.2
0.4
0.6
Fig. 3. Graph of the energy function and level contours of the Eq. (16) for q ¼ 0:55; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:1.
0.4 0.3 0.2 0.1
E1
0
−0.1 −0.2 −0.3 −0.4 0.5 0.4 0.3 0.2 0.1
ψ
0.2 0 −0.1 −0.2 −0.3 −0.4
−0.4
−0.2
0.4
0.6
0
−0.5
Fig. 4. Graph of the energy function and level contours of the Eq. (16) for q ¼ 1:1; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:1.
35 30 25 20
E
2
15 10 5 0
−5 −10 8
6
4
2
ψ
0
−2
−4
−6
−8
−8
−6
−4
−2
0
2
4
6
8
Fig. 5. Graph of the energy function and level contours of the Eq. (17) for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1.
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7
0.25
0.2
ψ
0.15
0.1
0.05
0 −15
−10
−5
0
χ
5
10
15
Fig. 6. Variation of the solitary wave solutions of Eq. (12) for different values of r ¼ 1:1 (dashed red curve), r ¼ 1:2 (black curve) and r ¼ 1:3 (dotted blue curve) with b ¼ 0:3; q ¼ 0:55; a ¼ 1:1 and U ¼ 0:1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
0 −0.1 −0.2 −0.3
ψ
−0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 −60
−40
−20
0
χ
20
40
60
Fig. 7. Variation of the solitary wave solutions of Eq. (12) for different values of q = 0.7 (dashed red curve), q = 0.8 (black curve) and q = 0.9 (dotted blue curve) with r ¼ 1:1 and other parameters are same as Fig. 6. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
positron acoustic solitary wave increases rapidly with increase in U while width of the positron acoustic solitary waves decreases with increase in U. In Fig. 11, the variation of positron acoustic compressive and rarefactive solitary wave profiles of Eq. (15) is presented for different values of r ¼ 0:5 (dashed red curve), r ¼ 0:6 (black curve), and r ¼ 0:7 (dotted blue curve) with b ¼ 0:35; q ¼ 0:4836; a ¼ 0:5 and U ¼ 0:1. The amplitude and width of the positron acoustic compressive and rarefactive solitary waves decrease with increase in r. Thus, the positron acoustic compressive and rarefactive solitary waves diminish as the parameter r is enhanced. In Fig. 12, the variation of positron acoustic compressive and rarefactive solitary wave profiles of Eq. (15) is presented for different values of q = 0.4836 (dashed red curve),
q = 0.5836 (black curve), and q = 0.6836 (dotted blue curve) with r ¼ 0:5 and other parameters are same as Fig. 11. The amplitude and width of the positron acoustic compressive and rarefactive solitary waves decrease with increase in q. Thus, the positron acoustic compressive and rarefactive solitary waves diminish as the parameter q is enhanced. In Fig. 13, the variation of positron acoustic compressive and rarefactive solitary wave profiles of Eq. (15) is presented for different values of U = 0.03 (dashed red curve), U = 0.07 (black curve), and U = 0.11 (dotted blue curve) with q ¼ 0:4836 and other parameters are same as Fig. 12. The amplitude of the positron acoustic solitary waves increases and width decreases with increase in U. Thus, the positron acoustic solitary wave grows rapidly as speed U of the solitary wave is enhanced in the subsonic
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A. Saha et al. / Advances in Space Research xxx (2017) xxx–xxx 0.25
0.2
ψ
0.15
0.1
0.05
0 −15
−10
−5
0
5
10
15
χ
Fig. 8. Variation of the solitary wave solutions of Eq. (12) for different values of q = 0.55 (dashed red curve), q = 0.65 (black curve) and q = 0.75 (dotted blue curve) with other parameters are same as Fig. 7. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
0.2 0.18 0.16 0.14
ψ
0.12 0.1 0.08 0.06 0.04 0.02 0 −10
−8
−6
−4
−2
0
χ
2
4
6
8
10
Fig. 9. Variation of the solitary wave solutions of Eq. (12) for different values of q = 1.1 (dashed red curve), q = 1.9 (black curve) and q = 2.7 (dotted blue curve) with other parameters are same as Fig. 7. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
region. The positron acoustic compressive and rarefactive solitary wave profiles of the present work are supported by the work (Uddin et al., 2015).
5. Quasiperiodic and chaotic motions In order to study the quasiperiodic and chaotic behavior of the perturbed systems given by B
d 2w A 2 þ w U w ¼ f 0 cosðxvÞ; dv2 2
and
ð24Þ
B
d 2w D 3 þ w U w ¼ f 0 cosðxvÞ; dv2 3
ð25Þ
different numerical tools are applied: (i) phase portrait plots, (ii) Poincare section and (iii) Lyapunov exponent. They all are very useful tools to examining quasiperiodic and chaotic features (Lakshmanan and Rajasekar, 2003). The perturbed systems (24) and (25) contain seven independent parameters q; a; r; b; U ; x and f 0 . Because of large number of parameters, it is really difficult to investigate the systems for the complete range of parametric space. To simplify the task, we focus on some special values of the parameters q; a; r; b; U varying the frequency parameter (x) of the external periodic perturbation. Thus, x plays the critical role for the dynamic behavior of the positron acoustic waves. We
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9
0.8 0.7 0.6
ψ
0.5 0.4 0.3 0.2 0.1 0
−10
−5
0
5
χ
10
Fig. 10. Variation of the solitary wave solutions of Eq. (12) for different values of U = 0.1 (dashed red curve), U = 0.2 (black curve) and U = 0.3 (dotted blue curve) with q ¼ 0:55 and other parameters are same as Fig. 8. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1 0.8 0.6 0.4
ψ
0.2 0
−0.2 −0.4 −0.6 −0.8 −1 −20
−15
−10
−5
0
χ
5
10
15
20
Fig. 11. Variation of the solitary wave solutions of compressive and rarefactive types of Eq. (15) for different values of r ¼ 0:5 (dashed red curve), r ¼ 0:6 (black curve), and r ¼ 0:7 (dotted blue curve) with b ¼ 0:35; q ¼ 0:4836; a ¼ 0:5 and U ¼ 0:1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
could vary any of the other parameters, but this will not give any significant different qualitative feature. 5.1. 3D phase portrait analysis In Fig. 14, 3D phase orbits of the perturbed system (24) are presented for q ¼ 0:6; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:125; f 0 ¼ 0:2 and x ¼ 7:5. In Fig. 15, we depict z vs. v of the Eq. (24) for same value of parameters as Fig. 14. In Fig. 16, 3D phase orbits of the perturbed system (24) are presented for q ¼ 1:1; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:125; f 0 ¼ 0:2 and x ¼ 7:5. In Fig. 17, we depict z vs. v of the Eq. (24) for same value of parameters as
Fig. 16. In Fig. 18, 3D phase orbits of the perturbed system (25) are presented for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62 and x ¼ 15:77. In Fig. 19, we illustrate z vs. v of the Eq. (25) for same value of parameters as Fig. 18. The orbits presented in Figs. 14, 16 and 18, ignore the periodic motions and exhibit aperiodic oscillations i.e., quasiperiodic motions of the systems (24) and (25) are found with incommensurable periodic motions and the trajectory in the phase space winds around torus filling its surface densely. In Fig. 20, 3D phase orbits of the perturbed system (25) are presented for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62 and x ¼ 0:77. The orbits in Fig. 20 ignore the periodic
Please cite this article in press as: Saha, A., et al. Nonlinear excitations for the positron acoustic waves in auroral acceleration regions. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.06.012
10
A. Saha et al. / Advances in Space Research xxx (2017) xxx–xxx 1 0.8 0.6 0.4
ψ
0.2 0
−0.2 −0.4 −0.6 −0.8 −1 −20
−15
−10
−5
0
χ
5
10
15
20
Fig. 12. Variation of the solitary wave solutions of compressive and rarefactive types of Eq. (15) for different values of q = 0.4836 (dashed red curve), q = 0.5836 (black curve), and q = 0.6836 (dotted blue curve) with r ¼ 0:5 and other parameters are same as Fig. 11. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1 0.8 0.6 0.4
ψ
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −30
−20
−10
0
χ
10
20
30
Fig. 13. Variation of the solitary wave solutions of compressive and rarefactive types of Eq. (15) for different values of U = 0.03 (dashed red curve), U = 0.07 (black curve), and U = 0.11 (dotted blue curve) with q ¼ 0:4836 and other parameters are same as Fig. 12. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
motions and exhibit chaotic behavior. In Fig. 21, we depict z vs. v of the Eq. (25) for same value of parameters as Fig. 20.
5.2. Poincare return map analysis A very useful tool for exploring chaotic attractors is Poincare sections (Lorenz, 1963; Molenaar et al., 2005). Poincare sections provide us with images of the attractor in a ðn 1Þ-dimensional hyperplane of n-dimensional phase space. Thus, the Poincare return map converts problems about closed orbits into problems about fixed points of a mapping. Moreover, by looking at the behavior of the Poincare return map near a fixed point we can deter-
mine the stability of the closed orbit. The Poincare return map of the systems (24) and (25) are shown in Figs. 22–25. A close inspection of Fig. 22 clearly shows that the points are densely populated and lie on a smooth closed curve, which indicates two-frequency oscillation and confirms the existence of quasi-periodic structure of the nonlinear PASWs for the system (24) for the parameter values q ¼ 0:6; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:125; f 0 ¼ 0:2; x ¼ 7:5. Fig. 23 shows that the points are densely populated and lie on a smooth closed curve, which indicates two -frequency oscillation and confirms the existence of quasi-periodic structure of the nonlinear PASWs for the system (24) for the parameter values q ¼ 1:1; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:125; f 0 ¼ 0:2; x ¼ 7:5. Fig. 24 shows that the points are densely populated and lie on a smooth curve, which also indicates
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900 800 700 600
χ
500 400 300 200 100 0 0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
−4.6
−4.4
−4.2
−3.8
−4
−3.6
−3.4
−3
−3.2
ψ
Fig. 14. 3 D plot of the phase orbits for the Eq. (24) for q ¼ 0:6; a ¼ 1:1; b ¼ 0:3; r ¼ 1:1; U ¼ 0:125; f 0 ¼ 0:2; x ¼ 7:5.
0.4 0.3 0.2
z
0.1 0
−0.1 −0.2 −0.3 −0.4
0
10
20
30
40
50
χ
60
70
80
90
100
Fig. 15. Plot of z vs. v of the Eq. (24) for same values of parameters as Fig. 14.
350 300 250
χ
200 150 100 50 0 0.08
0.06
0.04
0.02
z
0
−0.02 −0.04
−0.06 −0.08
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
ψ
Fig. 16. 3 D plot of the phase orbits for the Eq. (24) for q ¼ 1:1 and other parameters are same as Fig. 14.
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A. Saha et al. / Advances in Space Research xxx (2017) xxx–xxx 0.08 0.06 0.04 0.02
z
0 −0.02 −0.04 −0.06 −0.08 0
20
40
60
80
100
120
χ
140
160
180
200
Fig. 17. Plot of z vs. v of the Eq. (24) for same values of parameters as Fig. 16.
900 800 700
χ
600 500 400 300 200 100 0
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
1.5
1
2
2.5
3
ψ
Fig. 18. 3 D plot of the phase orbits for the Eq. (25) for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62; x ¼ 15:77.
0.4 0.3 0.2
z
0.1 0
−0.1 −0.2 −0.3 −0.4
0
10
20
30
χ
40
50
60
70
Fig. 19. Plot of z vs. v of the Eq. (25) for same values of parameters as Fig. 18.
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800 700 600
χ
500 400 300 200 100 0 6
4
2
0
−2
−4
−6
−10
−8
−6
−4
−2
0
2
4
6
10
8
ψ
Fig. 20. 3 D plot of the phase orbits for the Eq. (25) for q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62; x ¼ 0:77. 15
10
z
5
0
−5
−10
−15 0
100
200
300
400
χ
500
600
700
800
Fig. 21. Plot of z vs. v of the Eq. (25) for same values of parameters as Fig. 20.
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2
−4.6
−4.4
−4.2
−4
−3.8
ψn
−3.6
−3.4
−3.2
−3
Fig. 22. Poincare section of the Eq. (24) for the same value of the parameters as Fig. 14.
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A. Saha et al. / Advances in Space Research xxx (2017) xxx–xxx
0.06
0.04
0.02
0
−0.02
−0.04
−0.06 0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
ψ
n
Fig. 23. Poincare section of the Eq. (24) for the same value of the parameters as Fig. 16.
0.3
0.2
0.1
ψ˙ n
0
−0.1
−0.2
−0.3 1.2
1.4
1.6
1.8
2
2.2
ψn
2.4
2.6
2.8
3
3.2
Fig. 24. Poincare section of the Eq. (25) for the same parameters value as Fig. 18.
10
ψ˙ n
5
0
−5
−10 −10
−5
0
ψn
5
10
15
Fig. 25. Poincare section of the Eq. (25) for the same parameters value as Fig. 20.
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Lyapunov Exponents (λ(χ))
0.1 0.08
λ =
0.086612
0.06
λ2=
−0.086612
0.04
λ =
0
1
3
0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1
50
100
150
200
χ
250
300
350
400
Fig. 26. Dynamics of the Lyapunov exponents of the Eq. (25) for the same parameters value as Fig. 14.
two-frequency oscillation and confirms the existence of quasi-periodic behavior of the nonlinear PASWs for the system (25) with parameter values q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62 and x ¼ 15:77. In return map plot, Fig. 25 demonstrates that the points are densely scattered and exhibit irregular distributions without any known pattern for the system (25) with parameter values q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62 and x ¼ 0:77. Such return maps are characteristics of chaotic oscillations. 5.3. Lyapunov exponents Lyapunov exponents provide a quantitative measure of the divergence or convergence of nearby trajectories for a dynamical system. If we consider a small hypersphere of initial conditions in the phase space, for sufficiently short time scales, the effect of the dynamics will be distort this set into a hyper-ellipsoid, stretched along some directions and contracted along others. The asymptotic rate of expansion of the largest axis, corresponds to the most unstable direction of the flow, is measured by the largest Lyapunov exponent. For a periodic orbit, one of the Lyapunov components is zero. For a quasi-periodic orbit, two of the Lyapunov exponents are zero and others are negative. For chaotic behavior at least one of the Lyapunov exponents must be positive so that neighboring trajectories diverge. When more than one of the Lyapunov exponents are positive, then the motion is referred as hyper-chaos. In Fig. 26, we have presented the dynamics of the Lyapunov exponents of the system (25) for the parameter values q ¼ 0:4836; a ¼ 0:5; b ¼ 0:05; r ¼ 0:5; U ¼ 0:1; f 0 ¼ 0:62; x ¼ 0:77, which confirms the existence of choas in the system (25). 6. Conclusions In this work, we have studied the positron acoustic waves in an unmagnetized e-p-i plasmas whose constituents
are mobile cold positrons, immobile positive ions, q-nonextensive distributed electrons and hot positrons. The standard RPT has been applied to derive the KdV and mKdV equations for PAWs. Variations of the total energy of the conservative systems for the KdV and mKdV equations are presented. Applying numerical computations, effects of physical parameters q; r and U have been presented on the positron acoustic solitary wave solutions of the KdV and mKdV equations. It is observed that when nonextensive particles moves far away from Maxwellian distribution, the positron acoustic solitary waves are flourished. Considering an external periodic perturbation the perturbed dynamical systems corresponding to the KdV and mKdV equations have been analyzed by employing phase orbit analysis, Poincare return map analysis and Lyapunov exponent. The frequency (x) of the external periodic perturbation has played the role of the switching parameter in chaotic motions of the perturbed PAWs through quasiperiodic route to chaos. The results of this study may be applied to understand the qualitative changes in the dynamics of the positron acoustic waves in auroral acceleration regions. Acknowledgments The authors are grateful to the reviewers and the Editor for their useful comments and suggestions which helped to improve the paper. References Popel, S.I., Vladimirov, S.V., Shukla, P.K., 1995. Ion acoustic solitons in electronpositronion plasmas. Phys. Plasmas 2, 716. Chawla, J.K., Mishra, M.K., Tiwari, R.S., 2013. Modulational instability of ion-acoustic waves in electronpositronion plasmas. Astrophys. Space Sci. 347, 283–292. Bulanov, S.S., Fedotov, A.M., Pegoraro, F., 2005. Damping of electromagnetic waves due to electron-positron pair production. Phys. Rev. E. 71, 016404. Surko, C.M., Leventhal, M., Passner, A., 1989. Positron plasma in the laboratory. Phys. Rev. Lett. 62, 901–904.
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