Observation of hole Peregrine soliton in a multicomponent plasma with critical density of negative ions

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, , doi: /jgra.50111, 2013 Observation of hole Peregrine soliton in a multicomponent plasma with critical density of negative ions S. K. Sharma, and H. Bailung Received 18 September 2012; revised 4 December 2012; accepted 4 January 2013; published 28 February [1] The evolution of hole Peregrine soliton (appearing as a deep trough between two crests) from ion-acoustic perturbations excited in a multicomponent plasma with critical density of negative ions has been observed. The observed soliton is described by the rational solution of the cubic nonlinear Schrödinger equation, which can appear as an isolated high peak or a deep hole depending on the phase of the underlying carrier wave relative to the envelope. The measured amplitude of the hole Peregrine soliton (depth from crest to trough) is found to be more than twice the background wave height. The experimental observations are compared with the theoretical results obtained from the solution of the cubic nonlinear Schrödinger equation. The frequency spectrum of the Peregrine soliton is analyzed and found to be triangular in shape. Citation: Sharma, S. K., and H. Bailung (2013), Observation of hole Peregrine soliton in a multicomponent plasma with critical density of negative ions, J. Geophys. Res. Space Physics, 118, , doi: /jgra Introduction [2] It is well known that waves propagating through a plasma medium can transform into a soliton/solitary wave under conditions when the broadening of the wave due to a dispersion effect is balanced by the wave steepening due to nonlinearity. A solitary wave, an important phenomenon in the evolution of nonlinear perturbation, is observed in the laboratory as well as in astrophysical plasma such as in the auroral zone [Pickett et al., 2004], the Earth s magnetosphere [Cattell et al., 2003], the Jovian atmosphere [Maxworthy and Redekopp, 1976], and elsewhere. Theoretical investigations on electrostatic solitary structure formation have also been made in different space multicomponent plasma [Lakhina et al., 2008a, 2008b, 2011]. The evolution of ionacoustic waves into solitons of different types in a multicomponent plasma with negative ions has been extensively studied both experimentally [Ludwig et al., 1984; Nakamura and Tsukabayashi, 1984; Sharma et al., 2008] and theoretically [Das and Tagare, 1975; Watanabe, 1984; Tagare, 1986] using nonlinear model equations, mostly the Korteweg-de Vries (KdV) [Ludwig et al., 1984; Sharma et al., 2008; Das and Tagare, 1975; Washimi and Taniuti, 1966; Ikezi et al., 1970] and modified KdV [Nakamura and Tsukabayashi, 1984; Watanabe, 1984; Tagare, 1986] equations. Another possible soliton mode in multicomponent plasma when the number density of negative ions is equal to a critical value is the envelope soliton, which can be generated from a modulationally unstable ion wave and governed by the cubic 1 Plasma Physics Laboratory, Physical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, Guwahati , India. Corresponding author: S. K. Sharma, Plasma Physics Laboratory, Physical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, Guwahati , India. (sumita_sharma82@ yahoo.com) American Geophysical Union. All Rights Reserved /13/ /jgra nonlinear Schrödinger equation (NLSE) [Saito et al., 1984; Bailung and Nakamura, 1993]. In addition to this, a new type of soliton, known as a Peregrine soliton, has recently been observed in a multicomponent plasma with a critical density of negative ions [Bailung et al., 2011]. Plasma containing negative ions are encountered not only in laboratory discharges but also in space environments such as in the lower ionosphere [Swider, 1988], cometary tails [Chaizy et al., 1991], and in Titan s atmosphere[coates et al., 2007]. [3] The Peregrine soliton is a rational solution of the cubic NLSE first derived by Peregrine [1983]. Unlike the other soliton types, it is localized both in space and time. Another distinctive feature of the Peregrine soliton is that its amplitude is nearly three times the background wave height. A Peregrine soliton is now considered to be a prototype of rogue waves mostly observed in the deep ocean and sea as a natural surface wave phenomenon [Shrira and Geogjaev, 2010]. There has been a considerable effort in the last few decades to understand the nature of rogue waves and its possible generic mechanisms [Kharif et al., 2009; Osborne, 2010]. Moreover, experiments reporting the observation of Peregrine soliton, first in fiber optics [Kibler et al., 2010] and soon after in water wave tanks [Chabchoub et al., 2011] and multicomponent plasma [Bailung et al., 2011], have greatly stimulated intense research on rogue waves [Chabchoub et al., 2012a, 2012b]. In recent times, the generation of rogue waves has been studied in different plasma systems relevant to space environments such as active galactic nuclei, plasma magnetospheres [Moslem, 2011; Sabry et al., 2012], Saturn s rings [Moslem et al., 2011b], solar corona, solar wind [Shukla and Moslem, 2012], and Titans atmosphere [El-Labany et al., 2012]. [4] As observed in the laboratory experiments and in numerical simulations [Kibler et al., 2010; Chabchoub et al., 2011; Bailung et al., 2011], the rational solution of NLSE most likely appears as a high peak between two troughs representing a bright (elevated) Peregrine soliton. However, it has been predicted that the solution can appear as an isolated deep trough between two crests representing a hole (dark) Peregrine soliton under conditions depending on the phase 919

2 of the underlying carrier wave [Peregrine, 2005]. Like the bright Peregrine soliton, the amplitude of holes (i.e., depth from crest to trough) may exceed twice the background wave height. It can serve as a prototype of rogue wave holes, which are equally dangerous as rogue waves in deep water [Porubov, 2007]. However, there have been relatively few theoretical and experimental studies on holes. Osborne et al. [2000] described holes as deep troughs occurring before and/or after the largest rogue crests. Porubov [2007] studied the generation and propagation of rogue waves and holes using two-dimensional nonlinear model equations. In an erbium-doped fiber system, He et al. [2012] observed the appearance of dark rogue waves as two (or more) dominant down peaks in its profile. The natural appearance of different shapes of rogue (freak) waves, including holes in the coastal zone of the Baltic Sea, has been recorded and analyzed by Didenkulova and Anderson [2010] and Didenkulova [2011]. In the laboratory, rogue wave holes were recently observed in a water wave tank experiment and described in terms of the rational solution of NLSE [Chabchoub et al., 2012c]. [5] In this work, we present the evolution of hole Peregrine soliton in a multicomponent plasma with negative ions. The frequency spectrum of the observed Peregrine soliton has also been analyzed. The spectral data not only reveals one of the important features of Peregrine soliton but also helps in predicting its growth at the early stages of evolution [Chabchoub et al., 2012b; Akhmediev et al., 2011]. 2. Experimental Setup and Theory 2.1. Experimental Setup [6] The experiment is performed in a double plasma device which is 120 cm in length and 30 cm in diameter [Sharma et al., 2008]. The schematic diagram of the experimental setup is shown in Figure 1. The device consists of two cylindrical magnetic cages, representing source and target section, separated by a stainless steel mesh grid. Each magnetic cage is made up of rectangular magnet bars (~600 G at the surface) with alternate pole orientations to form a cusp field. After achieving a vacuum in the order of torr, argon Figure 1. Schematic diagram of the experimental setup. S, source; T, target; MC, magnetic cage; F, filament, G, grid; V f, filament voltage; V d, discharge voltage; V s, source bias; LP, Langmuir probe; FG, function generator; DSO, digital storage oscilloscope. gas is introduced into the device at partial pressure torr. Plasma is produced in both sections by applying DC discharge between the hot tungsten filaments (five in each section) as the cathode and a magnetic cage as the anode. The cusp field of the magnetic cage helps in minimizing plasma loss at the boundaries, keeping the bulk of the plasma homogeneous and magnetic field free. The discharge voltage andcurrentaremaintainedat50to70vand10to50ma, respectively, in both sections. Typical values of the plasma parameters measured with a planar Langmuir probe of 6 mm diameter are electron density n ~510 8 cm 3 and electron temperature T e ~ 1.0 ev. In order to produce multicomponent plasma, sulfur hexafluoride (SF 6 ) gas is introduced into the Ar discharge, which effectively produces F ions by a dissociative attachment process [Ludwig et al., 1984; Nakamura and Tsukabayashi, 1984]. Some other ion species such as SF 6,SF 5,SF + 3,SF + 4,andSF + 5 are also formed. However, the effects on the propagation of ion-acoustic waves in multicomponent plasma with negative ions are dominated by the lighter ion species. Hence, the multicomponent plasma is considered to be effectively composed of Ar +,F, and electrons [Sharma et al., 2008]. The partial pressure of SF 6 is adjusted in the range of 0 to torr to achieve a critical density of negative ions. This critical condition is confirmed by exciting modified KdV (compressive and rarefactive) solitons, which can only appear when the ratio of the density of negative ions to the positive ions (n /n + = r) is equal to the critical value (~0.1 for the F negative ion and Ar + positive ion mass ratio m /m + = m = 0.476) [Nakamura and Tsukabayashi, 1984]. Ion-acoustic perturbations are excited by applying low frequency (much smaller than ion plasma frequency) sinusoidal signals from a function generator to the source anode. The observed perturbations are detected in the target section using the axially movable Langmuir probe, which is biased above the plasma potential to collect the electron saturation current. The observed signals are then recorded in a digital storage oscilloscope Theory [7] We consider the NLSE derived for ion waves in a multicomponent plasma consisting of Boltzmann electrons, cold positive ions (Ar + ), and negative ions (F )[Saito et al., 1984; Bailung and Nakamura, þ 2 þ q 4 c 2 c ¼ 0: (1) [8] Here, c represents the wave amplitude normalized by the electron temperature (T e ). The normalized time t and space x coordinates in the wave frame are given by, t = a 2 0 o pi t and x = a 0 k D (x V g t) respectively, where a 0 is the smallness parameter representing the initial amplitude of the background carrier wave; x, t, o pi, k D are the distance, time, ion-plasma frequency, and inverse of the electron Debye length, respectively. The group velocity V g of the propagating p wave normalized by the ion-acoustic speed C s (¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kt e =m þ )isgivenbyv g /C s = o 3 (1 r)/[k 3 (1 + r/m)] do/dk. The angular frequency o and the wave number k of the carrier wave are related with the dispersion relation, 920

3 o 2 ¼ k 2 ð1 þ r=mþ= 1 þ k 2 ð1 rþ : (2) [9] The dispersion coefficient p ¼ 3o 5 =2k 4 ½ð1 rþ= ð1 þ r=mþš 2 is always negative, whereas the nonlinear coefficient q is a complicated function of wave number k, and negative ion density ratio r [Saito et al., 1984; Bailung and Nakamura, 1993]. The value of q has been calculated numerically and it has been found that there exists a critical value of wave number (k c ) above (below) which the nonlinear coefficient q becomes negative (positive). At k = k c, q vanishes. In the absence of negative ions, the critical wave number is found to be 1.47, i.e., for k > 1.47, ion waves may become modulationally unstable (pq > 0 satisfying the Lighthill condition) [Shimizu and Ichikawa, 1972]. However, at such a large wave number region, ion waves experience heavy Landau damping owing to low phase velocity and, therefore, modulational instability cannot be observed. With the addition of negative ions, the critical wave number decreases drastically and the region for q < 0 extends to a smaller wave number region. At the critical negative ion density (r = 0.1 for m = 0.476), k c becomes zero, q remains negative for any wave number and, therefore, modulational instability of ion waves occur for all k. Bailung and Nakamura [1993] observed the modulational instability of ion-acoustic waves in a multicomponent plasma with negative ions in a laboratory. [10] The rational solution possessed by the NLSE (1) is given by [Moslem, 2011] cðx; t Þ ¼ p 2 ffiffi q 41þ ð itþ 1 þ 4t 2 þ 2x 2 =p 1 (3) expðitþ: (4) discharge conditions have been studied using the rational solution of NLSE (1) in our previous experiment [Bailung et al., 2011]. An example of fully grown bright Peregrine soliton observed at a distance of 12 cm from the separation grid (point of excitation) is shown in Figure 2a along with the applied signal. The top trace represents the applied signal with f c = 400 khz, f m = 27 khz, carrier excitation voltage V c = 6 V and f = 0. The amplitude of the observed perturbation is represented in terms of dn/n, where dn is the perturbed electron density. The measured amplitude of the soliton is found to be ~0.22 and the group velocity is ~ cm.s 1. The soliton amplitude at this position is nearly 2.50 times the average carrier wave height. The time series data of the observed signal at 12 cm is compared with the theoretical bright Peregrine soliton obtained from equation (5) and is shown in Figure 2b. The parameters used in the numerical calculations are f =0, o = o pi, o pi /2p = 691 khz, k = k D, k D = 33.3 cm 1, x = 12 cm, C s = cm.s 1. These values correspond to the experimentally measured values. The value of a 0 = is δn/n (0.05/div) (a) Applied signal [11] Depending on the phase of the underlying carrier wave relative to the envelope, solution (4) can describe a bright (elevated) and hole (dark) Peregrine soliton. The development of the initial wave packet with the incorporation of carrier-envelope phase f is then given by [Chabchoub et al., 2012c] ðx; t Þ ¼ Refcðx; tþexp½ikx ð ot þ fþšg; (5) where c(x,t) is the dimensional form of equation (4). The carrier-envelope phase f = 0 and p in equation (5) corresponds to the development of bright and hole Peregrine solitons, respectively. 3. Experimental Results and Discussions [12] In order to excite ion-acoustic Peregrine soliton, a slowly amplitude-modulated continuous sinusoidal signal from a function generator is applied to the source anode. The carrier frequency (f c ) and the modulation frequency (f m ) vary in the range of 350 to 400 khz and 25 to 31 khz, respectively. The phase of the carrier relative to the envelope can also be controlled, which is the decisive factor for the formation of bright and hole Peregrine soliton. When the carrier-envelope phase is zero, i.e., the carrier maxima coincides with the envelope maxima, the perturbation evolves into a bright (elevated) Peregrine soliton. The formation of such a Peregrine soliton and its characteristics under similar Normalized amplitude (0.05/div) Time (20 μsec/div) (b) Theory Experiment Time (5.0 μsec/div) Figure 2. (a) Observed bright Peregrine soliton recorded at 12 cm from the separation grid. The top trace represents the applied signal with f c = 400 khz, f m = 27 khz, carrier excitation voltage V c = 6 V, and f = 0. (b) Comparison of the time series data of the observed signal (solid line) at 12 cm with the theoretical bright Peregrine soliton (dashed line) obtained from the rational solution of the NLSE [equation (5)]. The parameters used in the theoretical calculations are f =0, o = o pi, o pi /2p = 691 khz, k = k D, k D = 33.3 cm 1, V g = cm.s

4 selected to fit the observed signal. The measured dispersion coefficient p for o = under the present experimental condition is 0.5. This value of p agrees with that obtained by using equations (2) and (3). Nearly equal values of p were measured in a previous experiment under similar conditions [Bailung and Nakamura, 1993]. The value of nonlinear coefficient (q/4) is obtained from its expression given by Saito et al. [1984] and is found to be 0.4. The value of q measured by Bailung and Nakamura [1993], under similar conditions, is also found to be negative. This confirms the Lighthill condition pq > 0 for modulational instability to occur. A good agreement between the experimental signal and the theoretical result is observed in Figure 2b. We also analyzed a slightly modified rational solution of the NLSE derived by Moslem et al. [2011a, 2011b] and Sabry et al. [2012] for describing rogue waves for either positive or negative dispersion coefficient. It is found that the time series data of the bright Peregrine soliton, obtained by using this modified solution, also agrees with the experimentally observed signal. [13] In order to excite the hole Peregrine soliton, we carefully set the phase of the carrier relative to the envelope equal to p, so that the carrier minima coincides with the envelope maxima. The temporal evolution of the observed perturbation due to an initial signal of f c =400kHz,f m =27kHz andf = p is shown in Figure 3a from x = 6 to 10 cm and in Figure 3b from x = 12 to 15 cm. Here, x represents the distance from the separation grid. The top signal in Figure 3a is the applied amplitude modulated wave. As the perturbation propagates, it undergoes self-modulation and eventually develops into an isolated deep hole between two crests (at ~12 cm). Moving further away (>12 cm from the separation grid), it undergoes decay. The evolution mechanism of this hole Peregrine soliton is similar to that observed for a bright Peregrine soliton case [Bailung et al., 2011], except that the initial carrier-envelope phase which results in a deep hole in this case instead of a high peak during the evolution process. The measured group velocity of the observed perturbation is found to be the same as that of the bright Peregrine soliton (as shown in Figure 2a). At position x = 12 cm, the measured depth of the hole (from crest to trough) is dn/n ~ 0.20, which is nearly 2.12 times the average carrier wave height. [14] A direct comparison of the observed hole Peregrine soliton, with the rational solution of the NLSE with f = p, is shown in Figure 4. The theoretical curve represented by the dashed line is obtained from equation (5) using the parameters f = p, a 0 = 0.035, x = 12 cm, V g = cm.s 1. Other essential parameters used in the numerical calculations are the same as that used for the evaluation of the theoretical curve in Figure 2b. The size, shape, and position of the observed soliton agree very well with the theoretical prediction based on experimental conditions. The carrier signal on either side of the soliton also matches with the theoretical curve. The time series data of the hole Peregrine soliton obtained by using the solution derived by Moslem et al. [2011a, 2011b] also agrees with the observed hole Peregrine soliton in this experiment. [15] We analyzed the frequency spectrum of the observed Peregrine soliton by obtaining the fast Fourier transform of the experimental time series data. A typical frequency spectrum of a full-grown Peregrine soliton observed at 12 cm is shown in Figure 5. The carrier and modulation frequencies Normalized amplitude (0.20/div) Normalized amplitude (0.10/div) (a) (b) Time (50 μsec/div) Time (50 μsec/div) Applied signal x = 6 cm 8 cm 10 cm x = 12 cm 13 cm 14 cm 15 cm Figure 3. Perturbations recorded at different distances (x) from the separation grid (a) x = 6 to 10 cm. Top trace is the applied signal with carrier and modulation frequencies at 400 and 27 khz, respectively, and V c = 6 V (b) x =12 to 15 cm. Normalized amplitude (0.05/div) Time (5.0 μsec/div) Theory Experiment Figure 4. Comparison of the hole Peregrine soliton observed at 12 cm (solid line) with the theoretical result (dashed line) obtained from equation (5) with carrier-envelope phase f = p. The parameters used in the theoretical calculations are a 0 = 0.035, o = o pi, o pi /2p = 691 khz, k =0.575 k D, k D = 33.3 cm 1, V g = cm.s

5 Power (arb. scale) Frequency (khz) Figure 5. Frequency spectrum of the observed Peregrine soliton at 12 cm from the separation grid. Carrier and modulation frequencies of the applied signal are 400 and 27 khz, respectively. The highest peak located at 400 khz indicates the background carrier wave. in this case are 400 and 27 khz, respectively. The shape of the spectra (particularly the portion corresponding to the modulated part of the signal) was found to be triangular as expected from theory. The highest peak (located at 400 khz) clearly shows the background carrier wave frequency. Additional peaks with smaller magnitudes appearing to the left (376, 312, and 260 khz) and right (427 and 457 khz) side of the central peak (400 khz) represent sideband frequencies. Both the bright and hole Peregrine solitons with the same set of carrier and modulation frequencies exhibit similar frequency spectra. The triangular spectrum of the Peregrine soliton has already been confirmed in previous experiments in fiber optic and water waves [Kibler et al., 2010; Chabchoub et al., 2011]. 4. Conclusion [16] The evolution of ion-acoustic bright and hole Peregrine soliton in a multicomponent plasma with negative ions has been studied using the rational solution of the cubic NLSE. The dispersion coefficient in the NLSE is negative for all values of the wave number k. The nonlinear coefficient, on the other hand, becomes negative for any value of k only when the density of negative ions (r) is equal to a critical value (r c = 0.1). Under such conditions, i.e., r = r c and pq > 0 with p < 0 and q < 0, a modulated sinusoidal signal with frequency f < f pi becomes unstable and selfmodulation of the wave occurs, which eventually leads to the formation of an ion-acoustic Peregrine soliton. In the presence of negative ions, the phase velocity of ion waves increases with amplitude, which imposes the self-modulation of the wave. The condition for the generation of bright and hole Peregrine soliton out of an initial amplitude-modulated signal depends on the relative phase between the carrier and the envelope. When the carrier minima (maxima) coincide with the envelope maxima, hole (bright) Peregrine soliton is found to emerge. The observed solitons are compared with the rational solution of the NLSE. The comparison shows a good agreement between experimental observation and theoretical results. The amplitudes of the hole and bright Peregrine solitons observed in the experiment were found to be 2.50 and 2.12 times, respectively, the average carrier wave height. The spectrum of the observed Peregrine soliton is analyzed by obtaining its fast Fourier transform. The measurement confirms the triangular-shaped spectrum of the Peregrine soliton as expected from theory. The present experiment not only reveals a new type of ion-acoustic soliton mode in a multicomponent plasma but also brings a novel media plasma into the picture where these rogue wave prototypes can be excited and studied in a laboratory environment. Moreover, laboratory observations of rogue waves would significantly contribute to the ongoing study of this phenomenon in different astrophysical plasma environments. [17] Acknowledgments. The authors thank Dr. Y. Nakamura for useful discussion and guidance. One of the authors SKS thanks the Department of Science and Technology, Government of India, for supporting the work under INSPIRE Faculty Scheme. References Akhmediev, N., A. Ankiewicz, J. M. Soto-Crespo, and J. M. Dudley (2011), Rogue wave early warning through spectral measurements?, Phys. Lett. A, 375, Bailung, H., and Y. Nakamura (1993), Observation of modulational instability in a multicomponent plasma with negative ions, J. Plasma Phys., 50, Bailung, H., S. K. Sharma, and Y. Nakamura (2011), Observation of Peregrine solitons in a multicomponent plasma with negative ions, Phys. Rev. Lett. 107, Cattell, C., et al. (2003), Large amplitude solitary wave in and near the Earth s magnetosphere, magnetopause and bow shock: Polar and Cluster observations, Nonlinear Processes Geophys., 10, Chabchoub, A., N. P. Hoffmann, and N. Akhmediev (2011), Rogue wave observation in a water wave tank, Phys. Rev. Lett., 106, Chabchoub, A., N. P. Hoffmann, M. Onorato, and N. Akhmediev (2012a), Super rogue waves: observation of a higher-order breather in water waves, Phys. Rev. X, 2, Chabchoub, A., S. Neumann, N. P. Hoffmann, and N. Akhmediev (2012b), Spectral properties of the Peregrine soliton observed in a water wave tank, J. Geophys. Res., 117, C00J Chabchoub, A., N. P. Hoffmann, N. Akhmediev (2012c), Observation of rogue wave holes in a water wave tank, J. Geophys. Res., 117, C00J Chaizy, P. H., et al. (1991), Negative ions in the coma of comet Halley, Nature, 34, Coates, A. J., et al. (2007), Discovery of heavy negative ions in Titan s atmosphere, Geophys. Res. Lett., 34, L Das, G. C., and S. G. Tagare (1975), Propagation of ion-acoustic waves in a multicomponent plasma, Plasma Phys., 17, Didenkulova, I. (2011), Shapes of freak waves in the coastal zone of the Baltic sea (Tallinn Bay), Boreal Environ. Res., 16(suppl. A), Didenkulova, I., and C. Anderson (2010), Freak waves of different types in the coastal zone of the Baltic sea, Nat. Hazards Earth Syst. Sci., 10, El-Labany, S. K., W. M. Moslem, N. A. El-Bedwehy, R. Sabry, and H. N. Abd El-Razek (2012), Rogue wave in Titan s atmosphere, Astrophys. Space Sci., 338, 3 8. He, J., S. Xu, and K. Porezian (2012), New types of rogue wave in an erbidium-doped fibre system, J. Phys. Soc. Jpn., 81, Ikezi, H., R. J. Taylor, and D. R. Baker (1970), Formation and interaction of ion-acoustic solitons, Phys. Rev. Lett., 25, Kharif, C., E. Pelinovsky, and A. Slunyaev (2009), Rogue Waves in the Ocean, pp , Springer, Heidelberg. Kibler, B., et al. (2010), The Peregrine soliton in nonlinear fiber optics, Nat. Phys., 6, Lakhina, G. S., A. P. Kakad, S. V. Singh, and F. Verheest (2008a), Ion and electron acoustic soliton in two-electron temperature space plasmas, Phys. Plasmas, 15, Lakhina, G. S., S. V. Singh, A. P. Kakad, F. Verheest, and R. Bharuthram (2008b), Study of nonlinear ion- and electron-acoustic waves in multicomponent space plasmas, Nonlinear Processes Geophys., 15, Lakhina, G. S., S. V. Singh, A. P. Kakad (2011), Ion- and electron-acoustic solitons and double layers in multi-component space plasmas, Adv. Space Res., 47, Ludwig, G. O., J. L. Ferreira, and Y. Nakamura (1984), Observation of ionacoustic rarefaction solitons in a multicomponent plasma with negative ions, Phys. Rev. Lett., 52,

6 Maxworthy, T., and L. G. Redekopp (1976), New theory of the Great Red Spot from solitary waves in the Jovian atmosphere, Nature, 260, Moslem, W. M. (2011), Langmuir rogue waves in electron-positron plasmas, Phys. Plasmas, 18, Moslem, W. M., P. K. Shukla, and B. Eliasson (2011a), Surface plasma rogue waves, Europhys. Lett., 96, Moslem, W. M., R. Sabry, S. K. El-Labany, and P. K. Shukla (2011b), Dust-acoustic rogue waves in a nonextensive plasma, Phy. Rev. E., 84, Nakamura, Y., and I. Tsukabayashi (1984), Observation of modified Korteweg-de Vries solitons in a multicomponent plasma with negative ions, Phys. Rev. Lett., 52, Osborne, A. R. (2010), Nonlinear Ocean Waves and the Inverse Scattering Transform, Int. Geophys. Ser., vol. 97, Elsivier, Amsterdam. Osborne, A., M. Onorato, and M. Serio (2000), The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains, Phys. Lett. A, 275, Peregrine, D. H. (1983), Water waves, nonlinear Schrödinger equation and their solutions, J. Aust. Math. Soc. Ser. B Appl. Math., 25, Peregrine, D. H. (2005), Where can theory contribute to practical concerns about rogue waves? Paper presented at Workshop on Rogue Waves, International Centre for Mathematical Sciences, Edinburgh. Pickett, J. S., et al. (2004), Solitary waves observed in the auroral zone: the Cluster multi-spacecraft perspective, Nonlinear Processes Geophys., 11, Porubov, A. V. (2007), On formation of rogue waves and holes in ocean, Rend. Sem. Mat. Univ. Pol. Torino., 65, Sabry, R., W. M. Moslem, and P. K. Shukla (2012), Amplitude modulation of hydromagnetic waves and associated rogue waves in magnetoplasmas, Phys. Rev. E., 86, Saito, M., S. Watanabe, and H. Tanaka (1984), Modulational instability of ion wave in plasma with negative ions, J. Phys. Soc. Jpn., 53, Sharma, S. K., K. Devi, N. C. Adhikary, and H. Bailung (2008), Transition of ion-acoustic perturbations in multicomponent plasma with negative ions, Phys. Plasmas, 15, Shimizu, K., Y. H. Ichikawa (1972), Automodulation of ion oscillation modes in plasma, J. Phys. Soc. Jpn., 33, Shrira, V. I., and V. V. Geogjaev (2010), What makes the Peregrine soliton so special as a prototype of freak waves?, J. Eng. Math., 67, Shukla, P. K., and W. M. Moslem (2012), Alfvénic rogue waves, Phys. Lett. A, 376, Swider, W. (1988), Electron loss and the determination of electron concentrations in the D-region, Pure Appl. Geophys., 127, Tagare, S. G. (1986), Effect of ion temperature on ion-acoustic soliton in a two ion warm plasma with adiabatic positive and negative ions and isothermal electrons, J. Plasma Phys., 36, Washimi, H., and T. Taniuti (1966), Propagation of ion-acoustic solitary waves of small amplitude, Phys. Rev. Lett., 17, Watanabe, S. (1984), Ion acoustic soliton in plasma with negative ion, J. Phys. Soc. Jpn., 53,