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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

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Impact of electron trapping in degenerate quantum plasma on the ion-acoustic breathers and super freak waves S.A. El-Tantawy a,b,∗, Shaukat Ali Shan c, Naseem Akhtar c, A.T. Elgendy d a

Research Center for Physics (RCP), Department of Physics, Faculty of Science and Arts, Al-Makhwah, Al-Baha University, Saudi Arabia Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt c Theoretical Plasma Physics Division, PINSTECH, Nilore 44000, Islamabad, Pakistan d Department of Physics, Faculty of Science, Ain Shams University, Cairo, Egypt b

a r t i c l e

i n f o

Article history: Received 29 March 2018 Revised 24 April 2018 Accepted 26 April 2018

Keywords: Quantum plasma Modulational instability Ion-acoustic breathers Super freak waves

a b s t r a c t The propagation of nonlinear ion-acoustic (IA) structures in a two-component plasma consisting of ‘classical’ ions and temperature degenerate trapped electrons is investigated. Using the reductive perturbation method, a nonlinear Schrödinger equation (NLSE) is obtained and the modulational instability (MI) of the ion acoustic waves (IAWs) is investigated. The regions of the stability and instability of the modulated structures are deﬁned precisely depending on the MI criteria. The analytical solutions of the NLSE in the form of various types of freak waves, including the Peregrine soliton, the Akhmediev breather, and the Kuznetsov–Ma breather are examined. Moreover, the higher-order freak waves are presented. The characteristics of the rogue waves and their dependence on relevant parameters (the temperature of the degenerate trapped electrons and wavenumber) are investigated. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction The nonlinear wave propagation is one of the basic research ﬁeld in the plasma physics. Different types of nonlinear structures such as solitons, envelope holes, shocks, vortices, etc. have been investigated in many nonlinear medium [1–4,6] such as plasmas physics during the last few decades [3–5,7,8]. The soliton results due to the balance between nonlinearity and dispersion effects. This type of soliton is generally called as the Korteweg de Vries (KdV) soliton because its dynamics are governed by KdV equation [9]. On the other hand, envelope soliton is formed when wave group dispersion is in balance with nonlinearity of the medium. The envelope soliton is a localized modulated wavepacket whose dynamics are governed by the nonlinear Schrödinger equation (NLSE) [10]. The NLSE is one of the most relevant equations in physics and it is used to describe many nonlinear phenomena in various physical contexts such as the slow modulation of wave envelopes of the carrier waves [11]. Ion-acoustic wave (IAW) is a low-frequency mode in which the pressure of the inertialess species (electrons) provide restoring force, whereas inertia comes from the mass of ions [12]. The

∗ Corresponding author at: Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt. E-mail addresses: [email protected] (S.A. El-Tantawy), [email protected] (S. Ali Shan).

https://doi.org/10.1016/j.chaos.2018.04.037 0960-0779/© 2018 Elsevier Ltd. All rights reserved.

ﬁrst experimental observation of IA solitons was made by Ikezi et al. [13]. The existence of the electrostatic structures such as solitary waves in the magnetosphere with density depressions are observed by Viking spacecraft [14] and Freja satellite. In the context of the NLSE, the major mechanism of such wave creation is modulational instability (MI) which admits highintensity peaks and leads to the generation and interaction of the breathers, including the Peregrine solitons, i.e. ﬁrst-order rogue/freak waves, the Akhmediev breather, and the Kuznetsov– Ma breather [3,4]. The concept of the freak waves (FWs) was ﬁrst discussed in the studies of ocean waves [15]. After that the study of these waves was gradually extended to other ﬁelds e.g., optical ﬁbers, capillary water waves, Bose–Einstein condensates, superﬂuid helium, atmosphere, even in astrophysical environments, and recently in laboratory plasma physics as well [16–18]. Peregrine was the ﬁrst person who investigated the ﬁrst-order rogue wave solution of the generic NLSE. After that, the researchers carried out laboratory experiments to generate ﬁrst-order rogue waves (RWs) in the multi-component plasma in the presence of negative ions [16,17]. They observed that a slowly varying amplitude-modulated perturbation undergoes self-modulation and hence gives rise to localized pulses with huge amplitude. Moreover, they noticed that the measured amplitude of the ﬁrst-order rogue wave is three times the amplitude of the nearby carrier wave amplitude which agrees with the rational solution of the NLSE. Recently, the experimental observation of higher-order, i.e., second-order RWs in

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multi-component plasma with negative ions has been investigated by Pathak et al. [18]. They noticed that the wave energy concentrates to a smaller localized area with amplitude ampliﬁcation up to 5 times of the background carrier wave. Also, they compared the experimental results with second-order RWs of the NLSE and found that there was full consensus among them. Quantum plasmas are common in planetary interiors, compact astrophysical objects [19], the cores of giant planets, the crusts of old stars [20,21], semiconductors [22], quantum X-ray free-electron lasers [23,24], and intense laser-solid density plasma experiments. This ﬁeld is growing widely these days because of its wide ranging potential applications in semiconductors, metals, microelectronics [25], thin metal ﬁlms and modern technology [26–28]. Furthermore, a Fermi-degenerate dense plasma may also arise when a pellet of hydrogen is compressed to many times the solid density in the fast ignition scenario for inertial conﬁnement fusion [29,30]. A substantial research investigations on quantum degenerate plasmas have been carried out by different authors, taking into account various important effects like magnetic ﬁeld quantization, relativistic effects, degeneracy and trapping [31–33]. For instance, Shukla and Eliasson [34] have recapitulated the linear and nonlinear investigation in quantum degenerate plasmas. The nonlinear propagation of ion-acoustic freak waves in an unmagnetized plasma consisting of cold positive ions and superthermal electrons subjected to cold positrons beam has been investigated in which it has been found that the region of the modulational stability is enhanced with the increase of positron beam speed and positron population [35]. Moreover, the linear and nonlinear (solitary structures) propagation of quantum drift IAWs have been studied in an inhomogeneous degenerate quantum plasma taking into account the effect of electron trapping [36]. The authors used a reductive perturbation method to obtain the drift the KdV and KP equations for ion drift and coupled drift-ion acoustic solitary structures. We would like to point here that relatively little work has been done on trapping as microscopic phenomena in quantum plasmas. One of the ﬁrst investigations in this area was carried out by Luque et al. [37] who considered quantum corrected electron holes by solving the Wigner–Poisson system perturbatively. Gurevich [38] introduced the effect of adiabatic trapping at the microscopic level and observed that when trapping was absent, the adiabatic trapping produced a 3/2 power nonlinearity instead of the usual quadratic one. Experimental analysis [39] and computer simulations [40] conﬁrmed the presence of trapping as a microscopic phenomenon. Demeio [41] explored the effects of trapping on Bernstein, Greene, and Kruskal equilibria and solved the Wigner–Poisson system employing the perturbative technique in order to study the effect of trapping in quantum phase space. Recently, Shah et al. [42] studied the effect of trapping in quantum plasma using Gurevich approach and investigated the formation of one-dimensional ion acoustic solitary structures in both partially and fully degenerate plasma with small temperature effects. Waqas et al. [43] investigated the propagation of linear and nonlinear electrostatic waves in a dense magneto plasma with trapped electrons. Later on, the investigation on the solitary structure in the presence of a quantizing magnetic ﬁeld via Landau quantization was carried out in Refs. [44,45]. In the present work, we extend the study [42] to derive a NLSE using a reductive perturbation technique (the derivative expansion method), and examined the effects of degenerate trapped electrons on the modulational instability of ion-acoustic waves (IAWs) and the proﬁles of breathers waves. We use the Fermi–Dirac distribution function for the electrons with arbitrary degeneracy and obtain an expression for the number density for the electrons trapped in a potential well. We point out here that in contrast with Refs. [37,41], where the Wigner–Poisson equation is used and thus quantum diffraction effects are taken into account. Here, the quan-

357

tum statistical effects are taken into account via the calculations of the electron number density by using the Fermi–Dirac distribution function. The layout of the manuscript is summarized as follows: In Section 2, the basic equations describing the IAWs in a plasma comprising of warm positive ions and temperature degenerate electrons are presented. The RPT is employed to derive a NLSE which describes the evolution of the wave packet envelope in Section 3. In Section 4, criteria for modulational instability (MI) of the IAWs are examined. Within the MI region, a random perturbation of the amplitude grows enormously and thus creates breathers waves. The analytical solutions of the NLSE in the form of the various types of freak waves (FWs), including the fundamental rogue waves (RWs)/Peregrine soliton, the Akhmediev breather (AB), and the Kuznetsov–Ma (KM) breather are investigated. The variation of the structural properties of the breathers with relevant plasma parameters is discussed as well. Summary of the research work is presented in Section 5. 2. Set of dynamic equation and derivation of a NLSE We consider an homogeneous quantum plasma comprising of cold positive nondegenerate ions and temperature degenerate trapped electrons. The ions are considered to be classical, whereas the electrons are assumed to follow the Fermi–Dirac distribution. Therefore, we shall adopt the adiabatictrapped degenerate for electrons, by relying on a similar notations in Ref. [42] wherein the fundamental algebra is expressed in detail. Therefore, the normalized number density of electrons is accordingly expressed as

ne = (1 + )2/3 + T 2 (1 + )−1/2 ,

(1)

where T and are, respectively, the normalized degenerate electron temperature and the electrostatic potential. Note that the ﬁrst term of Eq. (1) is responsible for the effect of trapping while second term represents the temperature effects for partially degenerate plasma. The ions are taken to be cold and non degenerate due to their mass as compared to degenerate electrons. Therefore, the normalized ions ﬂuid equations can be expressed as [42]

∂ ni ∂ ( ni vi ) + = 0, ∂t ∂x

(2)

∂vi ∂vi ∂ + vi =− , ∂t ∂x ∂x

(3)

∂ 2 = (ne − ni ). ∂ x2

(4)

Here, ni and ne are, respectively, the normalized number densities of the ion and electron, respectively while vi is the re-scaled ions ﬂuid speed and is the normalized electrostatic potential. For small-amplitude waves, i.e. under the approximation 1 and using binomial series expansion, Eq. (1) can be expanded as

ne =

∞

α j j,

(5)

j=0

with α0 = (1 + T 2 ), α1 = (3 − T 2 )/2, α2 = 3(1 + T 2 )/8, and α3 = − ( 1 + 5T 2 )/6. In order to investigate the modulational instability (MI) of the ion-acoustic waves (IAWs), the governing equations is reduced to a NLSE using the standard derivative expansion method. According to this method, the independent (slow) variables are stretched as,

ξ = ε (x − vgt ) and τ = ε 2t ,

(6)

where ε is a real parameter (ε < < 1) and vg is the group velocity of the wavepackets that is deﬁned by the compatibility condition.

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The dependent variables are expanded around their corresponding equilibrium states as,

ni (x, t ) vi (x, t ) (x, t )

⎛

1 0 0

=

+

+∞

εm

m=1

l=m

nm (ξ , τ ) l

⎞

⎝ vm (ξ , τ ) ⎠eil θ l m l (ξ , τ )

(7)

l=−m

with θ = (kx − ωt ), where k and ω,respectively, are the normalized wavenumber and

frequency of the carrier wave. The vecm , m must satisfy the condition (m ) = ( (m ) )∗ , tors ≡ nm , v l l l −l l where the asterisk “∗ ” denotes complex conjugate. Inserting Eqs. (6) and (7) into Eqs. (2)–(5), and collecting terms in the ﬁrst harmonics with the ﬁrst-order (m = l = 1 ), we get the ﬁrst-order quantities as,

ni(11)

=

vi(11)

(α1 + k2 ) 1(1) , ω/k

(8)

with the following linear dispersion relation for the IAW,

ω=

k2 . α1 + k2

(9)

For the second-order mode and ﬁrst harmonic (m = 2 and l = 1 ), we have

ni(12)

vi(12)

k2 /ω2

=

k/ω

(2 )

1

2k/ω +i 1

kvg − ω

ω2

∂ 1(1) ∂ξ

(10)

Fig. 1. The stability and instability regions according to the conditions PQ < 0 and PQ > 0, respectively, determined in the kc − T plane. The yellow (white) color represents the region where the unstable (stable) pulses set in.

and the compatibility condition gives the group velocity as

ω

Q=

∂ω vg = α1 3 ≡ . ∂k k 3

For the second harmonic mode (m = l = 2 ) of the carrier wave, we obtain

⎛

(2 )

ni2

⎞

⎛ ⎞ C1

⎜ (2 ) ⎟ ⎝ ⎠ (1 ) 2 ⎝ vi2 ⎠ = C2 (1 ) ,

(11)

C3

2(2)

2

where C1 = (α1 + 4k2 )C3 − α2 , C2 = ω C1 − (α1 + k2 ) /k, and

C3 = (α1 + k2 )2 / 2k2 + α2 / 3k2 . The zeroth-harmonic mode (m = 2, l = 0) arises due to the selfinteraction of the modulated carrier wave. To determine the expression of the zeroth-harmonic mode, the second- and thirdorders equations are considered and after tedious calculations we get

⎛

ni(02)

⎞

⎛ ⎞

D1 ⎜ (2) ⎟ ⎝ ⎠ (1) 2 D2 1 = , ⎝ vi0 ⎠

(12)

D3

0(2)

where D1 = (α1 D3 − 2c2 ), D2 = −2( ω/k )(α1 + k2 )2 + vg D1 , and 2 2 2 D 3 = 2 α2 v g + ( 3 α 1 + k ) / α1 v g − 1 . Proceeding to the third-order in ε with the ﬁrst harmonic (m = 3 and l = 1 ), by eliminating of secularity producing resonant terms, leads to a NLSE for the slowly varying amplitude as follows [46,47]:

i

∂ P ∂ 2 + + Q 2 = 0. 2 ∂τ 2 ∂ξ

(13)

Here, ≡ 1(1 ) for simplicity and Eq. (13) describes the MI of the IAWs by including a degenerate trapped electrons. The coeﬃcients of the dispersion and the nonlinear interaction terms, respectively, are given by

P = −3α1

ω5 k4

,

ω3 2k2

3α3 − 2α2 (D3 + C3 ) − 2

k

ω

(C2 + D2 )(α1 + k2 )

− (α1 + k2 )(C1 + D1 ) .

(14)

It is noted here that the condition P = ∂ 2 ω/∂ k2 ≡ ∂vg /∂ k holds. 3. Modulational instability and breathers solution The conditions that cause breather waves to grow extremely are a topic of great interest with many different hypotheses being proposed. An important one is the nonlinear mechanism of the self wave interactions, such as the MI of the envelope IAWs, and has been found under certain conditions to explain the vagueness of creating rogue wave. It is already pointed out in the literature that the evolution of a wave whose amplitude satisﬁes the NLSE depends on the product of the coeﬃcients of the dispersion and the nonlinear interaction terms, i.e. PQ [48–50]. It has been documented in the literature that a negative sign for PQ is required for wave amplitude modulation stability. For negative PQ, the carrier wave may propagate in the form of a “dark” envelope wavepacket. On the contrary, for a positive sign of PQ, the carrier wave grows modulationally “unstable”. This modulational unstable carrier wave may either “collapse,” due to (possibly random) external perturbations, or lead to the formation of “bright” envelope modulated wave pulses (solitons) [51–53]. Here we focus our attention on studying a special modulational unstable solutions for a positive sign of PQ > 0, which is localized both in space and time (this solution called freak waves) or localized with time only (Akhmediev breather (AB)) or localized with space only (Kuznetsov–Ma (KM) breather). The product PQ (whose sign determines the wave is modulationally stable or unstable) is plotted versus the critical wavenumber kc for which P Q = 0 and electron temperature T in Fig. 1. It is shown that the MI occurs only for the colored region (PQ > 0) while the white region shows that the wave is stable (PQ < 0). It is noted that the critical wave number kc decreases with the enhancement of T which means that the unstable region extends while the stable region shrinks with the increase of T. Therefore,

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359

Fig. 2. (a) The contour of the Akhmediev breather (AB) proﬁle is plotted against ξ and τ for T = 0.5 and k = 3, (b) The breather solution (15) is plotted against ξ , τ , and the transformer parameter R. Here, T = 0.5 and k = 3, (c) The proﬁle of the AB is plotted against ξ for different values of R in the acceptable range of the AB (0 < R < 0.5).

increasing the electron temperature T leads to the enhancement of the chance of creating breathers waves. 3.1. Localized solution of the NLSE As mentioned earlier, various types of localized solutions of the NLSE are known as breathers, namely Akhmediev breather (AB), Kuznetsov–Ma breather (KM) soliton, and Peregrine solitons/or rogue waves (RWs). Kuznetsov–Ma breather was originally proposed by Kuznetsov [54] and three years later it was suggested by Ma [55]. This kind of breather, in contrast to the Akhmediev breather, shows a periodic waveform in time only but localized in space. The MI is believed to result in a high probability of the outcrop of RWs in the unstable region (PQ > 0). The physical reason behind this assumption is that the nonlinear growth of this instability causes accumulation of wave energy into a small spatial region. The general ﬁrst-order breathers solution of Eq. (13) can be written in implicit form as follows [56–61]:

2(1−2R ) cosh [bP τ ] + ib sinh [bP τ ] (ξ , τ )=0 1+ exp(iP τ ), √ 2R cos (cξ ) − cosh [bP τ ]

(15)

where, 0 = P/Q is the background amplitude. The free parameter “R” determines of the solution through the physical behavior √ the relations c = 4(1 − 2R ) and b = 2Rc. This solution has three types of breather solutions; next, we summarize and categorize these three types as follow: 3.1.1. Space periodic solution For 0 < R < 0.5, i.e. the spatial frequency of initial modulation “c” and the parameter “b” have real values 0 < b, c < 2 , and in this case the solution describes the AB which is localized in time

and exhibits a periodic modulation in space ξ with period 2π /c as shown in Fig. 2(a). In this solution the amplitude increases, either exponentially or according to a power law in τ , till reaching its maximum value and ﬁnally decays symmetrically and disappear forever. Moreover, the maximum peak height of the AB is given by

√

||max −AB = 0 1 + 2 2R , R ∈ (0, 0.5 ).

(16)

This solution reduces to the localized (in both space and time) solution with the increase of the converter parameter R up to the limit R → 0.5 as illustrated in Fig. 2(b). Also, Fig. 2(c) demonstrates the effect of the converter parameter R on the proﬁle of the AB. It is found that the increase of R from 0 → 0.5 leads to an increase in the maximum amplitude of the AB from the background height 0 and smoothly matches the freak wave peak height 30 . 3.1.2. Time periodic solution For 0.5 < R < ∞, the parameters “c” and “b” become imaginary and the hyperbolic functions in solution (15) are converted to the ordinary circular functions and vice-versa, i.e. the circular trigonometric functions become hyperbolic. In this case, solution (15) reduces to the KM soliton (see Fig. 3(a)) as

2(1−2R ) cos [bP τ ]−ib sin [bP τ ] (ξ , τ )|KM =0 1+ √ exp(iP τ ). 2R cosh (cξ ) − cos [bP τ ]

(17) This solution is periodic in time only with temporal period 2π /(bP). The maximum peak height of the KM soliton is given by

√

||max −KM = 0 1 + 2 2R , R ∈ (0.5, ∞ ).

(18)

Note that the KM soliton maximum peak height (18) differs from the AB maximum amplitude (16) due to the value of the transformer parameter “R”, i.e. ||max −KM > ||max −AB . It is obvious

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Fig. 3. (a) The KM soliton proﬁle is plotted against ξ and τ , (b)The breather solution (15) is plotted against ξ , τ , and the transformer parameter R in the acceptable range of the KM soliton (0.5 < R < ∞). The other parameters are the same as Fig. 2.

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361

Fig. 4. The breather solution (15) according to the RWs (R → 0.5) is plotted versus ξ and τ . The other parameters are the same as Fig. 2.

from Fig. 3(b) that the periodic of the KM soliton solution decreases with the decrease of “R” and this solution becomes localized in both space and time when R → 0.5, which gives the freak wave solution. 3.1.3. Localized solution Solution (15) reduces to the space-time localized solution in the limit R → 0.5 (see Fig. 4) as

(ξ , τ )|RW = lim1 (ξ , τ )=0 1− R→ 2

4(1 + 2iP τ )

1 + 4ξ 2 + (2P τ )2

exp(iP τ ). (19)

Without loss of generality, solution (19) can be multiplied by a negative sign to match the conventional solution [62]. RWs are characterized by some properties such as: (i) they always have two or even more times higher amplitude than their surrounding waves and are accompanied by deep troughs, which occurs before and/or after the large crest and generally they are formed in a short time, (ii) unpredictability, therefore, they appear obviously from nowhere and die out without a trace, and (iii) has a speciﬁc shape of probability distribution function of the wave amplitudes [63]. The scenario of RWs generation in many experiments can be demonstrated as follows: Firstly, modulational instability of a moderate amplitude monochromatic wave produces a chain of solitons which combine afterwards due to inelastic interaction consequently giving one huge amplitude soliton. The colliding soliton sucks energy from neighboring waves, then it becomes unstable and collapses thus producing a RW [64]. The RW maximum amplitude (0, 0 ) ≡ ||max −RW = 30 is shown in Fig. 4. Finally, the maximum amplitude of breathers can be summarized as

⎫ ||max −AB < 30 for R < 1/2, ⎬ √

||max −KM = 0 1 + 2 2R > 30 for R > 1/2, ⎭ ||max −RW = 30 for R → 1/2,

(20)

it should be noted here that the last relationship is related to the physical system through the pump carrier wave 0 which it is a function

4. Super freak waves Recently, the super RW solutions in NLSE were studied theoretically and conﬁrmed experimentally by many researchers [18,65– 71]. It is found that the super RWs can be treated as superpositions of several ﬁrst-order/or fundamental RWs, and the superpositions can create higher amplitudes that still keep located both of time and space [71]. The super RW structures in different types of plasmas have been investigated in many articles [18,66–70]. For any order, the RW solutions of the NLSE can be written in the following form

G (ξ , P τ )+iτ H j (ξ , P τ ) j ≡ j (ξ , τ )=0 (−1 ) j + j exp(iP τ ), Fj (ξ , P τ )

(21) where index j is related to the order of the solution in the hierarchy and it is called “the order of the RW solution”, while Gj (ξ , τ ), Hj (ξ , τ ), and Fj (ξ , τ ) = 0 are polynomials in (ξ , τ ) only and do not contain exponential and they are given in Ref. [72] in details. The higher-order solutions are progressively complicated and we will focus only on the ﬁrst three-order solutions of this series, i.e., for j ≤ 3. Corresponding to j = 1, the ﬁrst-order/fundamental solution is obtained and superposition of 1 generates the super rogue wave solutions, i.e. 2 and 3 and so on. The comparison between the ﬁrst three-order solutions of the NLSE, i.e. 1 , 2 , and 3 is illustrated in Fig. 5. It is noted that the ﬁrst-order solution of this hierarchy is characterized by a maximum amplitude with ampliﬁcation (at ξ = 0 and τ = 0) 3 times of the surrounding wave amplitude [16,73,74]. The ampliﬁcation of the second-order solution equals to 5 times of the background wave height and so on. Finally, the general formula for the maximum value of the main peak of the jth order rational solution reads: j (0, 0 )=(2 j + 1 )0 . It is shown that all order solutions of the RWs have similar dynamics, with the particular property to generate much higher maximal peak amplitudes, compared to background [15]. Moreover, localized waves modelled by the higher-order solutions can cause the

362

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Fig. 5. Comparison between the ﬁrst-, second-, and third-order RW solutions of the NLSE. Here, T = 0.5 and k = 3.5.

formation of super RWs with taller amplitude and narrower width as shown in Fig. 5. Physically, higher-order solutions are able to concentrate great amounts of energy into a relatively small area in space which lead to the teller amplitude and narrower width. It can be clearly seen from Fig. 4 that (i) the third-order solution 3 has treble structures compared with 2 , and 2 has double structures as compared with 1 , (ii) the third-order solution 3 has energy E3 larger than E2 , and E2 has energy larger than E1 , i.e. E3 > E2 > E1 , (iii) the third-order solution 3 has amplitude higher than 2 and 1 , i.e. 3 (0, 0) > 2 (0, 0) > 1 (0, 0), and (iv) moreover, the third-order solution |3 (ξ , 0)|, the secondorder solution |2 (ξ , 0)|, and the ﬁrst-order solution |1 (ξ , 0)|, respectively, have six, four, and two zeros symmetrically located on the ξ axis. Therefore, the solutions |3 (ξ , 0)|, |2 (ξ , 0)|, and |1 (ξ , 0)| have ﬁve, three, and one local maxima, respectively. In general, the number of zeros and local maxima for the jth order rational solution are, respectively, (2j) and (2 j − 1 ). Many laboratory experiments conﬁrmed all of the above characteristics of super RW solutions, for instance, Chabchoub et al. [65] found that the experimental and theoretical values of the ampliﬁcation in the water-wave tank for 2 are very close to excellent. Furthermore, Pathak et al. [18] noticed that the energy of the super RWs concen-

Fig. 6. The fundamental RW proﬁle is plotted against ξ for different values of the temperature degeneracy T where k = 3.

trates to a smaller localized area with amplitude ampliﬁcation up to 5 times of the background carrier wave in a multicomponent plasma with negative ions. Also, they compared the experimental results with second-order RWs of the NLSE and found that there was full consensus among them. The dependence of the fundamental RW proﬁle on the electron temperature degeneracy T is depicted in Fig. 6. It is obvious that both the amplitude and the width of the RW decrease with the

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increase of the electron temperature T. We speculate that this behavior could be explained as follows: the high electron temperature reduces the nonlinearity of the system which reducing the freak wave amplitude. 5. Summary The nonlinear propagation of modulated ion-acoustic wavepackets propagating in an electron–ion plasma is investigated taking into account the effect of adiabatically trapped electrons. While the ions were considered as inertial species. A ﬂuid model was proposed and the basic equations are reduced to a nonlinear Schrödinger equation using a reductive perturbation method, i.e., derivative expansion technique. Depending on the criteria for modulational instability, i.e., the product of the coeﬃcients of the dispersion and the nonlinear interaction terms (PQ), the regions of stability and instability have been determined precisely for various regimes. The effect of the temperature T of the degenerate trapped electrons on the modulational instability (MI) regions is reported. It is found that the region of the MI of the ion-acoustic waves becomes wider with the increase of T. Moreover, the analysis showed that the MI region becomes dominant for large values of the carrier wavenumber (small carrier wavelengths). Within the modulational unstable region, i.e. for PQ > 0, it is possible for a random perturbation of the amplitude to grow and therefore leads to the creation of breathers waves. In addition, the inﬂuence of the degeneracy temperature T on the nonlinear bright structures, i.e. the Akhmediev breather (AB), Kuznetsov–Ma breather (KM) soliton, rogue waves (RWs), and higher-order RWs is examined. Our results reveal that the higher values of the degeneracy temperature T reduces the nonlinearity of the system, and therefore the amplitude of the breathers become smaller. The present investigation should be helpful in understanding the propagation of the nonlinear bright structures such as the rogue waves and their family in the fast ignition scenario for inertial conﬁnement fusion, white dwarf stars, and in short pulsed petawatt laser technology [36]. Moreover, for dense plasma situations, the effect of external magnetic ﬁeld on the MI and ionacoustic breathers waves may also be problem of great importance, but beyond the scope of our present work and could be investigated in the future. Acknowledgments This work is dedicated to the memory of Professor Mouloud Tribeche. References [1] Wazwaz AM, Beijing U. Partial differential equations and solitary waves theory. Higher Education Press; 2009. [2] Wazwaz AM. Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn 2015;83:591. [3] Ruderman MS, Talipova T, Pelinovsky E. Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions. J Plasma Phys 2008;74:639. [4] Ruderman MS. Freak waves in laboratory and space plasmas. Eur Phys J Spec Top 2010;185:57. [5] Bouzit O, Gougam LA, Tribeche M. Solitons and freak waves in a mixed nonextensive high energy-tail electron distribution. Phys Plasmas 2014;21:062101. [6] Lü X. Madelung ﬂuid description on a generalized mixed nonlinear Schrödinger equation. Nonlinear Dyn 2015;81:239. [7] Misra AP, Samanta S. Double-layer shocks in a magnetized quantum plasma. Phys Rev E 2010;82:037401. [8] Treumann RA, Baumjohann W. Advanced space plasma physics. London: Imperial College Press; 1997. p. 264. [9] Washimi H, Taniuti T. Propagation of ion-acoustic solitary waves of small amplitude. Phys Rev Lett 1966;17:966. [10] El-Labany SK, Moslem WM, El-Bedwehy NA, Sabry R, El-Razek HNA. Rogue wave in Titan’s atmosphere. Astrophys Space Sci 2012;338:3. [11] Lü X, Ma WX, Yu J, Lin F, Khalique CM. Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model. Nonlinear Dyn 2015;82:1211.

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