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Open Access Journal of
eISSN: 2641-9335

Mathematical and Theoretical Physics

Research Article Volume 1 Issue 5

Effect of ion drift on shock waves in an un-magnetized multi-ion plasma

Sijo S,1 Anu V,2 Manesh M,4 G Sreekala,5 Neethu TW,3 Venugopal C6

1Department of Physics, St. Berchman's College, Changanassery, India
2School of Pure and Applied Physics, Mahatma Gandhi University, India
3School of Pure and Applied Physics, Mahatma Gandhi University, India
4School of Pure and Applied Physics, Mahatma Gandhi University, India
5School of Pure and Applied Physics, Mahatma Gandhi University, India
6School of Pure and Applied Physics, Mahatma Gandhi University, India

Correspondence: Venugopal C, School of Pure and Applied Physics, Mahatma Gandhi University, Kottayam, Kerala, India

Received: June 22, 2018 | Published: September 7, 2018

Citation: Sijo S, Anu V, Neethu TW, et al. Effect of ion drift on shock waves in an un-magnetized multi-ion plasma. Open Acc J Math Theor Phy. 2018;1(5):179-184 DOI: 10.15406/oajmtp.2018.01.00031

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Abstract

We investigate the effect of the drift velocity of lighter ions on shock waves in an un-magnetized multi-component plasma. Solar origin hot electrons and lighter ions, positively and negatively charged heavier ions and cometary origin colder electrons form the multi-component plasma. Using the reductive perturbation technique, we have derived the KdVB (Korteweg-deVries-Burgers) equation and its shock wave solution is plotted for different plasma parameters relevant to comet Halley. We find that the strength of the shock profile decreases with increasing drift velocities of the lighter ions and kappa indices of the two components of electrons. On the other hand, the amplitude of the shock increases with increasing kinematic viscosities and densities of the lighter ions. Also, different coefficients of the KdVB equation are strongly affected by the drift velocities of the lighter ions.

Keywords: electrons of solar and cometary origin, kappa distribution functions, lighter ions with drift, kdvb equation, shock waves

Introduction

During the last few decades, multi-component/dusty plasmas have received considerable attention from researchers due to their omnipresence in astrophysical, space and laboratory plasmas. For example, dusty plasmas occur in planetary atmospheres and rings1,2 and comets;3 nearer Earth they occur in the Earth’s mesosphere4 and ionosphere5 and also in laboratory plasmas used for the formation of ‘plasma crystals’6 and fusion devices.7 The addition of dust to plasmas increases their complexity due to the excitation of different eigen modes like Dust Acoustic Waves (DAWs),8 Dust Ion Acoustic Waves (DIAWs),9 Dust Lattice Waves (DLWs),10 etc.

In addition most plasma waves are not linear: hence a study of nonlinearity, along with dispersion and dissipation, becomes important in the investigations of different plasmas. Among nonlinear phenomena, solitons, shocks and double layers are widely seen in different space environments.11 Nonlinearity, along with dispersion, results in the formation of soliton structures with spectacular stability; a plasma supports shock waves if in addition, dissipative effects are also present. The KdVB (Korteweg-deVries-Burgers) equation supports shock waves, with dissipation occurring due to a variety of reasons like wave-particle interactions, turbulence, dust charge, streaming of ions, etc.

A cometary plasma is a true multi-ion plasma composed of species of both solar and cometary origin. Cometary activity begins when it is approaching the Sun: dissociation of water molecules liberates positively charged oxygen and hydrogen and associated photo-electrons; this is in addition to hydrogen and electrons of solar origin.12 Other ions observed in a cometary environment include He+, He2+, H2+, OH+, C+, H2O+, H3O+, CO+ and S+.13,14 Investigations have also revealed the presence of ions of mass >12amu;15 multiply charged ions like O3+ and O7+ and O8+ have been observed16,17 at comet McNaught-Hartley by the spacecrafts Ulysses and Chandra respectively. Thus oxygen ions are an important component in the plasma composition of a comet.

In addition to positively charged ions, negatively charged ions have also been found in three extended mass peaks of 7-19, 22-65 and 85-110amu with energies ranging from 0.03-3.0 keV by the Giotto space-craft at comet Halley. Thus it is now well established that positively and negatively charged ions coexist in different cometary environments.4,18‒20

The presence of different types of in homogeneities makes particle distributions deviate from equilibrium; hence one has to consider non-Maxwellian distributions. Vasyliunas21 first proposed such a distribution while analyzing solar wind data. This distribution is now known as a “kappa” distribution and many space and astrophysical plasmas are best described by this distribution.

The investigation of nonlinear waves in the presence of different drifting components present in plasmas gave a new dimension to nonlinear plasma wave research. Ghosh et al.,22 studied small amplitude dust acoustic solitons in a two component dusty plasma consisting of ions and drifting dust grains, applicable to the F-ring of Saturn. In another study, applicable to the F, G and E rings of Saturn, they investigated dust drift incorporating ion inertial effect along with dust charge variation. They concluded that there existed an instability due to free energy of drift motion of dust grains.23 In yet an another investigation, it was found that the drift velocity of the ions, along with electron inertia, significantly contributes to the formation of double layers and solitary structures in a plasma.24 Chattopadhyay25 investigated the effect of ion temperature on ion acoustic solitary waves in a drifting negative plasma. In 2012, Tribeche et al.,26 studied the effect of drift of dust grains on arbitrary amplitude dust acoustic double layers in a warm dusty plasma with two temperature thermal ions and superthermal electrons. In a study related to ion acoustic double layers in magnetospheric and auroral plasmas, multi-drifting components were considered with nonthermal electrons.27

There are extensive studies on shock waves in the presence of kappa distributed electrons and ions.28‒33 In plasmas with super thermal electrons, it was found that spectral index changes the amplitude of the dust acoustic shock waves significantly.28 In an electron-positron-ion (e-p-i) plasma, applicable to pulsar magnetospheres, the effect of plasma parameters on the strength and steepness of the shock structure was investigated, with electrons and protons being described by kappa distributions.29 In a plasma with a beam, it was noticed that both the amplitude and steepness of the ion-acoustic shock wave accrued, as the spectral index of the superthermal electrons and concentration of impinging positron beam were enhanced. This can also be applied to laboratory beam plasma interaction experiments and space and astrophysical plasmas.30 In an investigation of dust acoustic shock waves in a strongly coupled un-magnetized dusty plasma with kappa described ions, it was found that, the coefficients of KdV-Burger’s equation were significantly modified.31 Later, Kourakis et al.,32 discussed the role of superthermality on the characteristics of electrostatic plasma waves. In a numerical and analytical study of a plasma composed of inertial ions, kappa distributed electrons of two temperatures and negatively charged immobile dust grains, it was seen that the effect of superthermaility significantly modifies the basic features of dust ion acoustic shock waves.33

There are a number of observations showing the existence of nonlinear waves at different comets.34‒39 Cometary missions Giotto and Vega-1 found structures of sub and bow shocks at comet Halley.40 Kennel et al.,41 discussed different plasma waves in the shock interaction regions at comet Giacobini-Zinner. Also, recently nonlinear waves like solitons, etc were found at the same comet.42

Thus for reasons given above, we are interested in studying the effect of streaming lighter ions on shock waves in a five component plasma: a pair of oppositely charged heavier ions, streaming lighter ions and two components of electrons.

Basic equations

As mentioned above we are interested in studying the effect of streaming lighter ions on shock waves in five component plasma. The five components that compose our plasma are a pair of oppositely charged heavier ions, drifting lighter ions and two components of kappa described electrons.

The electrons of both solar and cometary origin are thus described as

n s = n s0 [ 1 e s ψ k B T s ( κ s 3/2) ] κ s +1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBam aaBaaajuaibaGaam4CaaqcfayabaGaeyypa0JaamOBamaaBaaajuai baGaam4CaiaaicdaaKqbagqaamaadmaabaGaaGymaiabgkHiTmaala aabaGaamyzamaaBaaajuaibaGaam4CaaqcfayabaGaeqiYdKhabaGa am4AamaaBaaajuaibaGaamOqaaqcfayabaGaamivamaaBaaajuaiba Gaam4CaaqcfayabaGaaiikaiabeQ7aRnaaBaaajuaibaGaam4Caaqa baqcfaOaeyOeI0IaaG4maiaac+cacaaIYaGaaiykaaaaaiaawUfaca GLDbaadaahaaqabeaacqGHsislcqaH6oWAdaWgaaqcfasaaiaadoha aeqaaKqbakabgUcaRiaaigdacaGGVaGaaGOmaaaaaaa@5AE8@  (1)

where n s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbak aad6gadaWgaaqaaiaadohaaeqaaaaa@3A8D@ and n s0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbak aad6gadaWgaaqaaiaadohacaaIWaaabeaaaaa@3B47@ are respectively the number and equilibrium values of densities of species ‘s’ (s = ce for cometary electrons and s = he for hot solar electrons), e s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwgada Wgaaqcfasaaiaadohaaeqaaaaa@3917@ is the charge, k B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aa8aadaWgaaqcfasaa8qacaWGcbaajuaGpaqabaaaaa@39C8@ is the Boltzmann’s constant, T s , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiva8aadaWgaaqcfasaa8qacaWGZbaajuaGpaqabaGaaiil aaaa@3A92@ the temperature, ψ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5j aacYcaaaa@3964@ the potential. K s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbak aadUeadaWgaaqaaiaadohaaeqaaaaa@3A6A@  is the spectral index of species ‘s’.

The fluid equation of continuity governs the dynamics of the plasma components and is given by

n j t + x .( n j u j )=0;j=i,1and2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcaWGUbWaaSbaaKqbGeaacaWGQbaabeaaaKqbagaacqGH ciITcaWG0baaaiabgUcaRmaalaaabaGaeyOaIylabaGaeyOaIyRaam iEaaaacaGGUaGaaiikaiaac6gadaWgaaqcfasaaiaadQgaaKqbagqa aiaacwhadaWgaaqcfasaaiaadQgaaeqaaKqbakaacMcacqGH9aqpca aIWaGaai4oaiaaywW7caWGQbGaeyypa0JaamyAaiaacYcacaaIXaGa aGjbVlaadggacaWGUbGaamizaiaaysW7caaIYaaaaa@5854@ (2)

The lighter ion (heavier ion (dust)) equations of motion are given below

u i t +( u i . x ) u i = α i β i ψ x +η 2 u i x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcaWG1bWaaSbaaKqbGeaacaWGPbaabeaaaKqbagaacqGH ciITcaWG0baaaiabgUcaRiaacIcacaGG1bWaaSbaaKqbGeaacaWGPb aabeaajuaGcaGGUaWaaSaaaeaacqGHciITaeaacqGHciITcaWG4baa aiaacMcacaGG1bWaaSbaaKqbGeaacaWGPbaabeaajuaGcqGH9aqpcq GHsislcqaHXoqydaWgaaqcfasaaiaadMgaaKqbagqaaiabek7aInaa BaaajuaibaGaamyAaaqabaqcfa4aaSaaaeaacqGHciITcqaHipqEae aacqGHciITcaWG4baaaiabgUcaRiabeE7aOnaalaaabaGaeyOaIy7a aWbaaKqbGeqabaGaaGOmaaaajuaGcaWG1bWaaSbaaKqbGeaacaWGPb aajuaGbeaaaeaacqGHciITcaWG4bWaaWbaaKqbGeqabaGaaGOmaaaa aaaaaa@63AE@  (3)

u i t +( u i . x ) u i = ψ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcaWG1bWaaSbaaKqbGeaacaWGPbaabeaaaKqbagaacqGH ciITcaWG0baaaiabgUcaRiaacIcacaGG1bWaaSbaaKqbGeaacaWGPb aajuaGbeaacaGGUaWaaSaaaeaacqGHciITaeaacqGHciITcaWG4baa aiaacMcacaGG1bWaaSbaaKqbGeaacaWGPbaabeaajuaGcqGH9aqpda WcaaqaaiabgkGi2kabeI8a5bqaaiabgkGi2kaadIhaaaaaaa@5019@  (4)

u 2 t +( u 2 . x ) u 2 = α 2 β 2 ψ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcaWG1bWaaSbaaKqbGeaacaaIYaaajuaGbeaaaeaacqGH ciITcaWG0baaaiabgUcaRiaacIcacaGG1bWaaSbaaKqbGeaacaaIYa aabeaajuaGcaGGUaWaaSaaaeaacqGHciITaeaacqGHciITcaWG4baa aiaacMcacaGG1bWaaSbaaKqbGeaacaaIYaaabeaajuaGcqGH9aqpcq GHsislcqaHXoqydaWgaaqcfasaaiaaikdaaKqbagqaaiabek7aInaa BaaajuaibaGaaGOmaaqcfayabaWaaSaaaeaacqGHciITcqaHipqEae aacqGHciITcaWG4baaaaaa@56E2@  (5)

The above set is completed by the Poisson’s equation, given by

2 ψ x 2 = n 1 (1 z i μ i + μ ce + μ se ) n 2 (1 z 2 μ 2 + μ ce + μ se ) n i + μ ce ( 1 ψ σ ce ( κ ce 3/2) ) κ ce +1/2 + μ se ( 1 ψ σ se ( κ se 3/2) ) κ se +1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaajuaGda WcaaqaaiabgkGi2oaaCaaajuaibeqaaiaaikdaaaqcfaOaeqiYdKha baGaeyOaIyRaamiEamaaCaaajuaibeqaaiaaikdaaaaaaKqbakabg2 da9iaad6gakmaaBaaaleaacaaIXaaabeaajuaGcqGHsislcaGGOaGa aGymaOGaeyOeI0scfaOaamOEaOWaaSbaaSqaaiaadMgaaeqaaKqbak abeY7aTPWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSscfaOaeqiVd0Mc daWgaaWcbaGaam4yaiaadwgaaeqaaOGaey4kaSscfaOaeqiVd0Mcda WgaaWcbaGaam4CaiaadwgaaeqaaKqbakaacMcacaWGUbGcdaWgaaWc baGaaGOmaaqabaqcfaOaeyOeI0IaaiikaiaaigdakiabgkHiTKqbak aadQhakmaaBaaaleaacaaIYaaabeaajuaGcqaH8oqBkmaaBaaaleaa caaIYaaabeaakiabgUcaRKqbakabeY7aTPWaaSbaaSqaaiaadogaca WGLbaabeaakiabgUcaRKqbakabeY7aTPWaaSbaaSqaaiaadohacaWG LbaabeaajuaGcaGGPaGaamOBaOWaaSbaaSqaaiaadMgaaeqaaaqcfa yaaiabgUcaRiabeY7aTnaaBaaajuaibaGaam4yaiaadwgaaKqbagqa amaabmaabaGaaGymaiabgkHiTmaalaaabaGaeqiYdKhabaGaeq4Wdm 3aaSbaaKqbGeaacaWGJbGaamyzaaqabaqcfaOaaiikaiabeQ7aRnaa BaaajuaibaGaam4yaiaadwgaaKqbagqaaiabgkHiTiaaiodacaGGVa GaaGOmaiaacMcaaaaacaGLOaGaayzkaaWaaWbaaeqabaGaeyOeI0Ia eqOUdS2aaSbaaKqbGeaacaWGJbGaamyzaaqabaqcfaOaey4kaSIaaG ymaiaac+cacaaIYaaaaiabgUcaRiabeY7aTnaaBaaajuaibaGaam4C aiaadwgaaKqbagqaamaabmaabaGaaGymaiabgkHiTmaalaaabaGaeq iYdKhabaGaeq4Wdm3aaSbaaKqbGeaacaWGZbGaamyzaaqcfayabaGa aiikaiabeQ7aRnaaBaaajuaibaGaam4CaiaadwgaaeqaaKqbakabgk HiTiaaiodacaGGVaGaaGOmaiaacMcaaaaacaGLOaGaayzkaaWaaWba aeqabaGaeyOeI0IaeqOUdS2aaSbaaKqbGeaacaWGZbGaamyzaaqaba qcfaOaey4kaSIaaGymaiaac+cacaaIYaaaaaaaaa@B241@ (6)

The above equations (2)-(6) are made dimensionless as follows: the densities and speeds of the plasma species are, respectively, normalized by corresponding equilibrium values of densities n s0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBam aaBaaajuaibaGaam4CaiaaicdaaKqbagqaaaaa@3A06@ and ( z 1 k B T 1 m 1 ) 1/2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaWcaaqaaiaadQhadaWgaaqcfasaaiaaigdaaeqaaKqbakaadUga daWgaaqcfasaaiaadkeaaKqbagqaaiaadsfadaWgaaqcfasaaiaaig daaeqaaaqcfayaaiaad2gadaWgaaqcfasaaiaaigdaaeqaaaaaaKqb akaawIcacaGLPaaadaahaaqcfasabeaacaaIXaGaai4laiaaikdaaa qcfaOaaiOlaaaa@45FD@  The space (x) and time (t) variables are normalized by the Debye length ( λ D1 = ( z 1 k B T 1 4π z 1 2 e 2 n 10 ) 1/2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGPaVl aaykW7caaMc8UaaGPaVpaabmaabaGaeq4UdW2aaSbaaKqbGeaacaWG ebGaaGymaaqabaqcfaOaeyypa0ZaaeWaaeaadaWcaaqaaiaadQhada WgaaqcfasaaiaaigdaaeqaaKqbakaadUgadaWgaaqcfasaaiaadkea aKqbagqaaiaadsfadaWgaaqcfasaaiaaigdaaKqbagqaaaqaaiaais dacqaHapaCcaWG6bWaa0baaKqbGeaacaaIXaaabaGaaGOmaaaajuaG caWGLbWaaWbaaKqbGeqabaGaaGOmaaaajuaGcaWGUbWaaSbaaKqbGe aacaaIXaGaaGimaaqabaaaaaqcfaOaayjkaiaawMcaamaaCaaajuai beqaaiaaigdacaGGVaGaaGOmaaaaaKqbakaawIcacaGLPaaaaaa@5B29@  and the inverse of the plasma frequency ( ( ω 1 ) 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGPaVl aaykW7daqadaqaamaabmaabaGaeqyYdC3aaSbaaKqbGeaacaaIXaaa beaaaKqbakaawIcacaGLPaaadaahaaqcfasabeaacqGHsislcaaIXa aaaaqcfaOaayjkaiaawMcaaaaa@4297@ respectively. The kinematic viscosity of the lighter ions is normalized by ω 1 λ D1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyYdC 3aaSbaaKqbGeaacaaIXaaabeaajuaGcqaH7oaBdaqhaaqcfasaaiaa dseacaaIXaaabaGaaGOmaaaaaaa@3E2D@ and potential ψ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbak abeI8a5baa@3A6F@  by k B T e e . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGRbWaaSbaaKqbGeaacaWGcbaajuaGbeaacaWGubWaaSbaaKqb GeaacaWGLbaajuaGbeaaaeaacaWGLbaaaiaac6caaaa@3D64@  Also α i = z i z 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaKqbGeaacaWGPbaabeaajuaGcqGH9aqpdaWcaaqaaiaadQha daWgaaqcfasaaiaadMgaaeqaaaqcfayaaiaadQhadaWgaaqcfasaai aaigdaaeqaaaaajuaGcaGGSaaaaa@4115@ α 2 = z 2 z 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiaadQha daWgaaqcfasaaiaaikdaaKqbagqaaaqaaiaadQhadaWgaaqcfasaai aaigdaaeqaaaaajuaGcaGGSaaaaa@40B1@ β i = m 1 m i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbak abek7aInaaBaaajuaibaGaamyAaaqabaqcfaOaeyypa0ZaaSaaaeaa caWGTbWaaSbaaKqbGeaacaaIXaaajuaGbeaaaeaacaWGTbWaaSbaaK qbGeaacaWGPbaajuaGbeaaaaGaaiilaaaa@431A@ β 2 = m 1 m 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiaad2ga daWgaaqcfasaaiaaigdaaKqbagqaaaqaaiaad2gadaWgaaqcfasaai aaikdaaeqaaaaajuaGcaGGSaaaaa@4099@ μ s = n s0 z 1 n 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaWGZbaajuaGbeaacqGH9aqpdaWcaaqaaiaad6ga daWgaaqcfasaaiaadohacaaIWaaajuaGbeaaaeaacaWG6bWaaSbaaK qbGeaacaaIXaaabeaajuaGcaWGUbWaaSbaaKqbGeaacaaIXaGaaGim aaqcfayabaaaaaaa@4483@ and σ s = T s T 1 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm 3aaSbaaKqbGeaacaWGZbaabeaajuaGcqGH9aqpdaWcaaqaaiaadsfa daWgaaqcfasaaiaadohaaKqbagqaaaqaaiaadsfadaWgaaqcfasaai aaigdaaeqaaaaajuaGcaGG7aaaaa@4110@ where m i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aaBaaabaGaamyAaaqabaGaaiilaaaa@3935@ m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aaBaaajuaibaGaaGymaaqcfayabaaaaa@390E@ and m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aaBaaajuaibaGaaGOmaaqabaaaaa@3881@  and z i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEam aaBaaajuaibaGaamyAaaqcfayabaGaaiilaaaa@39FE@ z 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEam aaBaaajuaibaGaaGymaaqabaaaaa@388D@  and z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEam aaBaaajuaibaGaaGOmaaqabaaaaa@388E@ are respectively the masses and charges of the lighter ions, negatively and positively charged heavier ions respectively. T s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbak aadsfadaWgaaqcfasaaiaadohaaeqaaaaa@3AC1@  and T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbak aadsfakmaaBaaaleaacaaIXaaabeaaaaa@3A6B@  respectively represent the temperatures of the species ‘s’ and negatively charged heavier ions.

To derive the KdVB equation, we use the transformations:

ξ= ε 1/2 (xλt)andτ= ε 3/2 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOVdG Naeyypa0JaeqyTdu2aaWbaaKqbGeqabaGaaGymaiaac+cacaaIYaaa aKqbakaacIcacaWG4bGaeyOeI0Iaeq4UdWMaamiDaiaacMcacaaMc8 UaaGPaVlaadggacaWGUbGaamizaiaaykW7cqaHepaDcqGH9aqpcqaH 1oqzdaahaaqcfasabeaacaaIZaGaai4laiaaikdaaaqcfaOaamiDaa aa@53C4@   (7)

Therefore

x = ε 1/2 ξ and t = ε 3/2 τ ε 1/2 λ ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITaeaacqGHciITcaWG4baaaiabg2da9iabew7aLnaaCaaa juaibeqaaiaaigdacaGGVaGaaGOmaaaajuaGdaWcaaqaaiabgkGi2c qaaiabgkGi2kabe67a4baacaaMc8UaaGPaVlaadggacaWGUbGaamiz aiaaykW7daWcaaqaaiabgkGi2cqaaiabgkGi2kaadshaaaGaeyypa0 JaeqyTdu2aaWbaaKqbGeqabaGaaG4maiaac+cacaaIYaaaaKqbaoaa laaabaGaeyOaIylabaGaeyOaIyRaeqiXdqhaaiabgkHiTiabew7aLn aaCaaajuaibeqaaiaaigdacaGGVaGaaGOmaaaajuaGcqaH7oaBdaWc aaqaaiabgkGi2cqaaiabgkGi2kabe67a4baaaaa@6630@    (8)

The different physical quantities in the equations (1-6) can be expressed asymptotically as a power series about their equilibrium values as:

n i,1,2 =1+ε n i,1,2 (1) + ε 2 n i,1,2 (2) +......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBam aaBaaajuaibaGaamyAaiaacYcacaaIXaGaaiilaiaaikdaaeqaaKqb akabg2da9iaaigdacqGHRaWkcqaH1oqzcaWGUbWaa0baaKqbGeaaca WGPbGaaiilaiaaigdacaGGSaGaaGOmaaqaaiaacIcacaaIXaGaaiyk aaaajuaGcqGHRaWkcqaH1oqzdaahaaqcfasabeaacaaIYaaaaKqbak aad6gadaqhaaqcfasaaiaadMgacaGGSaGaaGymaiaacYcacaaIYaaa baGaaiikaiaaikdacaGGPaaaaKqbakabgUcaRiaac6cacaGGUaGaai Olaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaaaa@5AFF@  (9)

u i,x = u 0 +ε u i,x (1) + ε 2 u i,x (2) +......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGPaVl aadwhadaWgaaqcfasaaiaadMgacaGGSaGaamiEaaqabaqcfaOaeyyp a0JaamyDamaaBaaajuaibaGaaGimaaqabaqcfaOaey4kaSIaeqyTdu MaamyDamaaDaaajuaibaGaamyAaiaacYcacaWG4baabaGaaiikaiaa igdacaGGPaaaaKqbakabgUcaRiabew7aLnaaCaaajuaibeqaaiaaik daaaqcfaOaamyDamaaDaaajuaibaGaamyAaiaacYcacaWG4baabaGa aiikaiaaikdacaGGPaaaaKqbakabgUcaRiaac6cacaGGUaGaaiOlai aac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaaaa@5AF7@ (10)

u 1,2 =ε u 1,2 (1) + ε 2 u 1,2 (2) +......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaBaaajuaibaGaaGymaiaacYcacaaIYaaajuaGbeaacqGH9aqpcaaM f8UaaGjbVlabew7aLjaadwhadaqhaaqcfasaaiaaigdacaGGSaGaaG OmaaqaaiaacIcacaaIXaGaaiykaaaajuaGcqGHRaWkcqaH1oqzdaah aaqcfasabeaacaaIYaaaaKqbakaadwhadaqhaaqcfasaaiaaigdaca GGSaGaaGOmaaqaaiaacIcacaaIYaGaaiykaaaajuaGcqGHRaWkcaGG UaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6 caaaa@57B8@  (11)

ψ=ε ψ (1) + ε 2 ψ (2) +......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK Naeyypa0JaeqyTduMaeqiYdK3aaWbaaKqbGeqabaGaaiikaiaaigda caGGPaaaaKqbakabgUcaRiabew7aLnaaCaaajuaibeqaaiaaikdaaa qcfaOaeqiYdK3aaWbaaKqbGeqabaGaaiikaiaaikdacaGGPaaaaKqb akabgUcaRiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6 cacaGGUaGaaiOlaaaa@4FC7@  (12)

η= ε 1/2 η i0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4TdG Maeyypa0JaeqyTdu2aaWbaaKqbGeqabaGaaGymaiaac+cacaaIYaaa aKqbakabeE7aOnaaBaaajuaibaGaamyAaiaaicdaaKqbagqaaaaa@4216@ (13)

Using transformations (8) and equations (9-13) in equations (1-6) and equating different powers of ε, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyTdu Maaiilaaaa@38DB@ we get:

For powers of ε 1/2 : MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyTdu 2aaWbaaKqbGeqabaGaaGymaiaac+cacaaIYaaaaKqbakaacQdaaaa@3BF1@

n i (1) = α i β i l x 2 ( λ u 0 ) 2 ψ (1) , n 1 (1) = ψ (1) λ 2 , n 2 (1) = α 2 β 2 λ 2 ψ (1) , u i (1) = α i β i ( λ u 0 ) ψ (1) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBam aaDaaajuaibaGaamyAaaqaaiaacIcacaaIXaGaaiykaaaajuaGcqGH 9aqpdaWcaaqaaiabeg7aHTWaaSbaaKqbagaajugWaiaadMgaaKqbag qaaiabek7aITWaaSbaaKqbagaajugWaiaadMgaaKqbagqaaiaadYga lmaaDaaajuaGbaqcLbmacaWG4baajuaGbaqcLbmacaaIYaaaaaqcfa yaamaabmaabaGaeq4UdWMaeyOeI0IaamyDamaaBaaabaqcLbmacaaI WaaajuaGbeaaaiaawIcacaGLPaaadaahaaqabeaajugWaiaaikdaaa aaaKqbakabeI8a5TWaaWbaaKqbagqabaqcLbmacaGGOaGaaGymaiaa cMcaaaqcfaOaaGPaVlaacYcacaaMc8UaaGPaVlaad6galmaaDaaaju aGbaqcLbmacaaIXaaajuaGbaqcLbmacaGGOaGaaGymaiaacMcaaaqc faOaeyypa0ZaaSaaaeaacqGHsislcqaHipqElmaaCaaajuaGbeqaaK qzadGaaiikaiaaigdacaGGPaaaaaqcfayaaiabeU7aSnaaCaaabeqa aKqzadGaaGOmaaaaaaqcfaOaaGPaVlaacYcacaaMc8UaaGPaVlaad6 galmaaDaaajuaGbaqcLbmacaaIYaaajuaGbaqcLbmacaGGOaGaaGym aiaacMcaaaqcfaOaeyypa0ZaaSaaaeaacqaHXoqylmaaBaaajuaGba qcLbmacaaIYaaajuaGbeaacqaHYoGydaWgaaqaaKqzadGaaGOmaaqc fayabaaabaGaeq4UdW2aaWbaaeqabaqcLbmacaaIYaaaaaaajuaGcq aHipqElmaaCaaajuaGbeqaaKqzadGaaiikaiaaigdacaGGPaaaaKqb akaaykW7caGGSaGaaGPaVlaaykW7caWG1bWcdaqhaaqcfayaaKqzad GaamyAaaqcfayaaKqzadGaaiikaiaaigdacaGGPaaaaKqbakabg2da 9maalaaabaGaeqySde2cdaWgaaqcfayaaKqzadGaamyAaaqcfayaba GaeqOSdi2cdaWgaaqcfayaaKqzadGaamyAaaqcfayabaaabaWaaeWa aeaacqaH7oaBcqGHsislcaWG1bWaaSbaaeaajugWaiaaicdaaKqbag qaaaGaayjkaiaawMcaaaaacqaHipqElmaaCaaajuaGbeqaaKqzadGa aiikaiaaigdacaGGPaaaaKqbakaaykW7caGGSaaaaa@C2EC@

u 1 (1) = ψ (1) λ , u 2 (1) = α 2 β 2 λ ψ (1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDaS Waa0baaKqbagaajugWaiaaigdaaKqbagaajugWaiaacIcacaaIXaGa aiykaaaajuaGcqGH9aqpdaWcaaqaaiabgkHiTiabeI8a5naaCaaabe qaaKqzadGaaiikaiaaigdacaGGPaaaaaqcfayaaiabeU7aSbaacaaM c8UaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caWG1bWcdaqhaaqcfa yaaKqzadGaaGOmaaqcfayaaKqzadGaaiikaiaaigdacaGGPaaaaKqb akabg2da9maalaaabaGaeqySde2cdaWgaaqcfayaaKqzadGaaGOmaa qcfayabaGaeqOSdi2cdaWgaaqcfayaaKqzadGaaGOmaaqcfayabaaa baGaeq4UdWgaaiabeI8a5TWaaWbaaKqbagqabaqcLbmacaGGOaGaaG ymaiaacMcaaaaaaa@6A94@ (14)

Equating terms of powers of ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbak abew7aLbaa@3A28@ and using (14) in Poisson’s equation, we get an expression for the phase velocity of the wave as:

( μ ce σ ce ( κ ce 1/2) ( κ ce 3/2) + μ se σ se ( κ se 1/2) ( κ se 3/2) )( λ 2 (λ u 0 ) 2 ) (1 z 2 μ 2 + μ ce + μ se ) α i β i λ 2 ( 1+(1 z i μ i + μ ce + μ se ) α 2 β 2 ) ( λ u 0 ) 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGda qadaqaamaalaaabaGaeqiVd02aaSbaaKqbGeaacaWGJbGaamyzaaqa baaajuaGbaGaeq4Wdm3aaSbaaKqbGeaacaWGJbGaamyzaaqcfayaba aaamaalaaabaGaaiikaiabeQ7aRnaaBaaajuaibaGaam4yaiaadwga aKqbagqaaiabgkHiTiaaigdacaGGVaGaaGOmaiaacMcaaeaacaGGOa GaeqOUdS2aaSbaaKqbGeaacaWGJbGaamyzaaqabaqcfaOaeyOeI0Ia aG4maiaac+cacaaIYaGaaiykaaaacqGHRaWkdaWcaaqaaiabeY7aTn aaBaaajuaibaGaam4Caiaadwgaaeqaaaqcfayaaiabeo8aZnaaBaaa juaibaGaam4CaiaadwgaaeqaaaaajuaGdaWcaaqaaiaacIcacqaH6o WAdaWgaaqaaKqzadGaam4CaiaadwgaaKqbagqaaiabgkHiTiaaigda caGGVaGaaGOmaiaacMcaaeaacaGGOaGaeqOUdS2cdaWgaaqcfayaaK qzadGaam4CaiaadwgaaKqbagqaaiabgkHiTiaaiodacaGGVaGaaGOm aiaacMcaaaaacaGLOaGaayzkaaGaaiikaiabeU7aSTWaaWbaaKqbag qabaqcLbmacaaIYaaaaKqbakaacIcacqaH7oaBcqGHsislcaWG1bWc daWgaaqcfayaaKqzadGaaGimaaqcfayabaGaaiykaSWaaWbaaKqbag qabaqcLbmacaaIYaaaaKqbakaacMcaaeaacaaMf8UaeyOeI0Iaaiik aiaaigdacqGHsislcaWG6bWcdaWgaaqcfayaaKqzadGaaGOmaaqcfa yabaGaeqiVd02cdaWgaaqcfayaaKqzadGaaGOmaaqcfayabaGaey4k aSIaeqiVd02aaSbaaeaajugWaiaadogacaWGLbaajuaGbeaacqGHRa WkcqaH8oqBdaWgaaqaaKqzadGaam4CaiaadwgaaKqbagqaaiaacMca cqaHXoqylmaaBaaajuaGbaqcLbmacaWGPbaajuaGbeaacqaHYoGylm aaBaaajuaGbaqcLbmacaWGPbaajuaGbeaacqaH7oaBlmaaCaaajuaG beqaaKqzadGaaGOmaaaajuaGcqGHsisldaqadaqaaiaaigdacqGHRa WkcaGGOaGaaGymaiabgkHiTiaadQhalmaaBaaajyaGbaqcLbmacaWG PbaajyaGbeaajuaGcqaH8oqBlmaaBaaajuaGbaqcLbmacaWGPbaaju aGbeaacqGHRaWkcqaH8oqBdaWgaaqaaKqzadGaam4yaiaadwgaaKqb agqaaiabgUcaRiabeY7aTnaaBaaabaqcLbmacaWGZbGaamyzaaqcfa yabaGaaiykaiabeg7aHnaaBaaabaqcLbmacaaIYaaajuaGbeaacqaH YoGydaWgaaqaaKqzadGaaGOmaaqcfayabaaacaGLOaGaayzkaaWaae WaaeaacqaH7oaBcqGHsislcaWG1bWaaSbaaeaajugWaiaaicdaaKqb agqaaaGaayjkaiaawMcaamaaCaaabeqaaKqzadGaaGOmaaaajuaGcq GH9aqpcaaIWaaaaaa@DF04@  (15)

Equating terms of power ε 5/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyTdu 2aaWbaaeqajuaibaGaaGynaiaac+cacaaIYaaaaaaa@3AA9@ and using (14), we get

n i (2) ξ = 2 α i β i ( λ u 0 ) 3 ψ (1) τ + 3 α i 2 β i 2 ( λ u 0 ) 4 ψ (1) ψ (1) ξ + α i β i ( λ u 0 ) 2 ψ (2) ξ α i β i ( λ u 0 ) 3 η i0 2 ψ (1) ξ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGda WcaaqaaiabgkGi2kaad6gadaqhaaqcfasaaiaadMgaaeaacaGGOaGa aGOmaiaacMcaaaaajuaGbaGaeyOaIyRaeqOVdGhaaiabg2da9maala aabaGaaGOmaiabeg7aHnaaBaaajuaibaGaamyAaaqabaqcfaOaeqOS di2aaSbaaKqbafaacaWGPbaabeaaaKqbagaadaqadaqaaiabeU7aSj abgkHiTiaadwhadaWgaaqcfasaaiaaicdaaKqbagqaaaGaayjkaiaa wMcaamaaCaaajuaibeqaaiaaiodaaaaaaKqbaoaalaaabaGaeyOaIy RaeqiYdK3aaWbaaKqbGeqabaGaaiikaiaaigdacaGGPaaaaaqcfaya aiabgkGi2kabes8a0baacqGHRaWkdaWcaaqaaiaaiodacqaHXoqyda qhaaqcfasaaiaadMgaaeaacaaIYaaaaKqbakabek7aInaaDaaajuai baGaamyAaaqaaiaaikdaaaaajuaGbaWaaeWaaeaacqaH7oaBcqGHsi slcaWG1bWaaSbaaKqbGeaacaaIWaaabeaaaKqbakaawIcacaGLPaaa daahaaqcfasabeaacaaI0aaaaaaajuaGcqaHipqEdaahaaqcfasabe aacaGGOaGaaGymaiaacMcaaaqcfa4aaSaaaeaacqGHciITcqaHipqE daahaaqcfasabeaacaGGOaGaaGymaiaacMcaaaaajuaGbaGaeyOaIy RaeqOVdGhaaiabgUcaRmaalaaabaGaeqySde2aaSbaaKqbGeaacaWG PbaajuaGbeaacqaHYoGydaWgaaqcfasaaiaadMgaaeqaaaqcfayaam aabmaabaGaeq4UdWMaeyOeI0IaamyDamaaBaaajuaibaGaaGimaaqc fayabaaacaGLOaGaayzkaaWaaWbaaKqbGeqabaGaaGOmaaaaaaqcfa 4aaSaaaeaacqGHciITcqaHipqEdaahaaqcfasabeaacaGGOaGaaGOm aiaacMcaaaaajuaGbaGaeyOaIyRaeqOVdGhaaaqaaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7cqGHsisldaWcaaqaaiabeg7aHnaa BaaajuaibaGaamyAaaqabaqcfaOaeqOSdi2aaSbaaKqbGeaacaWGPb aabeaaaKqbagaadaqadaqaaiabeU7aSjabgkHiTiaadwhadaWgaaqc fasaaiaaicdaaKqbagqaaaGaayjkaiaawMcaamaaCaaajuaibeqaai aaiodaaaaaaKqbakabeE7aOnaaBaaajuaibaGaamyAaiaaicdaaeqa aKqbaoaalaaabaGaeyOaIy7aaWbaaKqbGeqabaGaaGOmaaaajuaGcq aHipqEdaahaaqcfasabeaacaGGOaGaaGymaiaacMcaaaaajuaGbaGa eyOaIyRaeqOVdG3aaWbaaKqbGeqabaGaaGOmaaaaaaaaaaa@D6C0@  (16) 

  n 1 (2) ξ = 2 λ 3 ψ (1) τ + 3 λ 4 ψ (1) ψ (1) ξ 1 λ 2 ψ (2) ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcaWGUbWaa0baaKqbGeaacaaIXaaabaGaaiikaiaaikda caGGPaaaaaqcfayaaiabgkGi2kabe67a4baacqGH9aqpdaWcaaqaai abgkHiTiaaikdaaeaacqaH7oaBdaahaaqcfasabeaacaaIZaaaaaaa juaGdaWcaaqaaiabgkGi2kabeI8a5naaCaaajuaibeqaaiaacIcaca aIXaGaaiykaaaaaKqbagaacqGHciITcqaHepaDaaGaey4kaSYaaSaa aeaacaaIZaaabaGaeq4UdW2aaWbaaKqbGeqabaGaaGinaaaaaaqcfa OaeqiYdK3aaWbaaKqbGeqabaGaaiikaiaaigdacaGGPaaaaKqbaoaa laaabaGaeyOaIyRaeqiYdK3aaWbaaKqbGeqabaGaaiikaiaaigdaca GGPaaaaaqcfayaaiabgkGi2kabe67a4baacqGHsisldaWcaaqaaiaa igdaaeaacqaH7oaBdaahaaqcfasabeaacaaIYaaaaaaajuaGdaWcaa qaaiabgkGi2kabeI8a5naaCaaajuaibeqaaiaacIcacaaIYaGaaiyk aaaaaKqbagaacqGHciITcqaH+oaEaaaaaa@70B7@ (17)

n 2 (2) ξ = 2 α 2 β 2 λ 3 ψ (1) τ + 3 α 2 2 β 2 2 λ 4 ψ (1) ψ (1) ξ + α 2 β 2 λ 2 ψ (2) ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcaWGUbWaa0baaKqbGeaacaaIYaaabaGaaiikaiaaikda caGGPaaaaaqcfayaaiabgkGi2kabe67a4baacqGH9aqpdaWcaaqaai aaikdacqaHXoqydaWgaaqcfasaaiaaikdaaKqbagqaaiabek7aInaa BaaajuaibaGaaGOmaaqcfayabaaabaGaeq4UdW2aaWbaaeqajuaiba GaaG4maaaaaaqcfa4aaSaaaeaacqGHciITcqaHipqEdaahaaqcfasa beaacaGGOaGaaGymaiaacMcaaaaajuaGbaGaeyOaIyRaeqiXdqhaai abgUcaRmaalaaabaGaaG4maiabeg7aHnaaDaaajuaibaGaaGOmaaqa aiaaikdaaaqcfaOaeqOSdi2aa0baaKqbGeaacaaIYaaabaGaaGOmaa aaaKqbagaacqaH7oaBdaahaaqcfasabeaacaaI0aaaaaaajuaGcqaH ipqEdaahaaqcfasabeaacaGGOaGaaGymaiaacMcaaaqcfa4aaSaaae aacqGHciITcqaHipqEdaahaaqcfasabeaacaGGOaGaaGymaiaacMca aaaajuaGbaGaeyOaIyRaeqOVdGhaaiabgUcaRmaalaaabaGaeqySde 2aaSbaaKqbGeaacaaIYaaabeaajuaGcqaHYoGydaWgaaqcfasaaiaa ikdaaKqbagqaaaqaaiabeU7aSnaaCaaabeqcfasaaiaaikdaaaaaaK qbaoaalaaabaGaeyOaIyRaeqiYdK3aaWbaaKqbGeqabaGaaiikaiaa ikdacaGGPaaaaaqcfayaaiabgkGi2kabe67a4baaaaa@83D5@  (18)

Equating ε 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGPaVl abew7aLnaaCaaajuaibeqaaiaaikdaaaaaaa@3AC2@ terms from Poisson’s equation, taking the derivative and using equations (14-18), we arrive at the KdVB equation as

A ψ (1) τ + ψ (1) ψ (1) ξ +B 3 ψ (1) ξ 3 C 2 ψ (1) ξ 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqam aalaaabaGaeyOaIyRaeqiYdK3aaWbaaKqbGeqabaGaaiikaiaaigda caGGPaaaaaqcfayaaiabgkGi2kabes8a0baacqGHRaWkcqaHipqEda ahaaqcfasabeaacaGGOaGaaGymaiaacMcaaaqcfa4aaSaaaeaacqGH ciITcqaHipqEdaahaaqcfasabeaacaGGOaGaaGymaiaacMcaaaaaju aGbaGaeyOaIyRaeqOVdGhaaiabgUcaRiaadkeadaWcaaqaaiabgkGi 2oaaCaaajuaibeqaaiaaiodaaaqcfaOaeqiYdK3aaWbaaKqbGeqaba GaaiikaiaaigdacaGGPaaaaaqcfayaaiabgkGi2kabe67a4naaCaaa juaibeqaaiaaiodaaaaaaKqbakabgkHiTiaadoeadaWcaaqaaiabgk Gi2oaaCaaajuaibeqaaiaaikdaaaqcfaOaeqiYdK3aaWbaaKqbGeqa baGaaiikaiaaigdacaGGPaaaaaqcfayaaiabgkGi2kabe67a4naaCa aajuaibeqaaiaaikdaaaaaaKqbakabg2da9iaaicdaaaa@6DF2@  (19)

A= A 0 A 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai abg2da9maalaaabaGaamyqamaaBaaajuaibaGaaGimaaqcfayabaaa baGaamyqamaaBaaajuaibaGaaGymaaqcfayabaaaaiaacYcaaaa@3DCB@ B= 1 A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqai abg2da9maalaaabaGaaGymaaqaaiaadgeadaWgaaqcfasaaiaaigda aeqaaaaaaaa@3AEC@ and C= A 2 A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai abg2da9maalaaabaGaamyqamaaBaaajuaibaGaaGOmaaqcfayabaaa baGaamyqamaaBaaajuaibaGaaGymaaqabaaaaaaa@3C91@

Where A 0 = 2 λ 3 + 2 α 2 β 2 λ 3 (1 z i μ i + μ ce + μ se )+ 2 α i β i ( λ u 0 ) 3 (1 z 2 μ 2 + μ ce + μ se ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqam aaBaaajuaibaGaaGimaaqcfayabaGaeyypa0ZaaSaaaeaacaaIYaaa baGaeq4UdW2aaWbaaKqbGeqabaGaaG4maaaaaaqcfaOaey4kaSYaaS aaaeaacaaIYaGaeqySde2aaSbaaKqbGeaacaaIYaaajuaGbeaacqaH YoGydaWgaaqcfasaaiaaikdaaKqbagqaaaqaaiabeU7aSnaaCaaabe qcfasaaiaaiodaaaaaaKqbakaacIcacaaIXaGaeyOeI0IaamOEamaa BaaajuaibaGaamyAaaqcfayabaGaeqiVd02aaSbaaKqbGeaacaWGPb aabeaajuaGcqGHRaWkcqaH8oqBdaWgaaqcfasaaiaadogacaWGLbaa beaajuaGcqGHRaWkcqaH8oqBdaWgaaqcfasaaiaadohacaWGLbaabe aajuaGcaGGPaGaey4kaSYaaSaaaeaacaaIYaGaeqySde2aaSbaaKqb GeaacaWGPbaabeaajuaGcqaHYoGydaWgaaqcfasaaiaadMgaaeqaaa qcfayaamaabmaabaGaeq4UdWMaeyOeI0IaamyDamaaBaaajuaibaGa aGimaaqcfayabaaacaGLOaGaayzkaaWaaWbaaKqbGeqabaGaaG4maa aaaaqcfaOaaiikaiaaigdacqGHsislcaWG6bWaaSbaaKqbGeaacaaI YaaajuaGbeaacqaH8oqBdaWgaaqcfasaaiaaikdaaeqaaKqbakabgU caRiabeY7aTnaaBaaajuaibaGaam4yaiaadwgaaeqaaKqbakabgUca RiabeY7aTnaaBaaajuaibaGaam4CaiaadwgaaeqaaKqbakaacMcaaa a@81BB@

A 1 = 3 λ 4 + 3 α 2 2 β 2 2 λ 4 (1 z i μ i + μ ce + μ se )+ 3 α i 2 β i 2 ( λ u 0 ) 4 (1 z 2 μ 2 + μ ce + μ se ) ( μ ce ( κ ce 1/2)( κ ce +1/2) σ ce 2 ( κ ce 3/2) 2 + μ se ( κ se 1/2)( κ se +1/2) σ se 2 ( κ se 3/2) 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGbbGcdaWgaaWcbaGaaGymaaqabaqcfaOaeyypa0ZaaSaaaeaacqGH sislcaaIZaaabaGaeq4UdW2aaWbaaeqajuaibaGaaGinaaaaaaqcfa Oaey4kaSYaaSaaaeaacaaIZaGaeqySde2aa0baaKqbGeaacaaIYaaa baGaaGOmaaaajuaGcqaHYoGydaqhaaqcfasaaiaaikdaaeaacaaIYa aaaaqcfayaaiabeU7aSnaaCaaajuaibeqaaiaaisdaaaaaaKqbakaa cIcacaaIXaGaeyOeI0IaamOEaOWaaSbaaSqaaiaadMgaaeqaaKqbak abeY7aTPWaaSbaaSqaaiaadMgaaeqaaKqbakabgUcaRiabeY7aTPWa aSbaaSqaaiaadogacaWGLbaabeaajuaGcqGHRaWkcqaH8oqBkmaaBa aaleaacaWGZbGaamyzaaqabaqcfaOaaiykaiabgUcaRmaalaaabaGa aG4maiabeg7aHnaaDaaajuaibaGaamyAaaqaaiaaikdaaaqcfaOaeq OSdi2aa0baaKqbGeaacaWGPbaabaGaaGOmaaaaaKqbagaadaqadaqa aiabeU7aSjabgkHiTiaadwhadaWgaaqcfasaaiaaicdaaKqbagqaaa GaayjkaiaawMcaamaaCaaajuaibeqaaiaaisdaaaaaaKqbakaacIca caaIXaGccqGHsisljuaGcaWG6bGcdaWgaaWcbaGaaGOmaaqabaqcfa OaeqiVd0McdaWgaaWcbaGaaGOmaaqabaGccqGHRaWkjuaGcqaH8oqB kmaaBaaaleaacaWGJbGaamyzaaqabaGccqGHRaWkjuaGcqaH8oqBkm aaBaaaleaacaWGZbGaamyzaaqabaqcfaOaaiykaaqaaiaaywW7caaM e8UaaGzbVlaaysW7caaMe8UaeyOeI0YaaeWaaeaadaWcaaqaaiabeY 7aTnaaBaaajuaibaGaam4yaiaadwgaaeqaaKqbakaacIcacqaH6oWA daWgaaqcfasaaiaadogacaWGLbaabeaajuaGcqGHsislcaaIXaGaai 4laiaaikdacaGGPaGaaiikaiabeQ7aRnaaBaaajuaibaGaam4yaiaa dwgaaeqaaKqbakabgUcaRiaaigdacaGGVaGaaGOmaiaacMcaaeaacq aHdpWCdaqhaaqcfasaaiaadogacaWGLbaabaGaaGOmaaaajuaGcaGG OaGaeqOUdS2aaSbaaKqbGeaacaWGJbGaamyzaaqabaqcfaOaeyOeI0 IaaG4maiaac+cacaaIYaGaaiykamaaCaaajuaibeqaaiaaikdaaaaa aKqbakabgUcaRmaalaaabaGaeqiVd02aaSbaaKqbGeaacaWGZbGaam yzaaqcfayabaGaaiikaiabeQ7aRnaaBaaajuaibaGaam4Caiaadwga aKqbagqaaiabgkHiTiaaigdacaGGVaGaaGOmaiaacMcacaGGOaGaeq OUdS2aaSbaaKqbGeaacaWGZbGaamyzaaqabaqcfaOaey4kaSIaaGym aiaac+cacaaIYaGaaiykaaqaaiabeo8aZnaaDaaajuaibaGaam4Cai aadwgaaeaacaaIYaaaaKqbakaacIcacqaH6oWAdaWgaaqcfasaaiaa dohacaWGLbaajuaGbeaacqGHsislcaaIZaGaai4laiaaikdacaGGPa WaaWbaaKqbGeqabaGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaaaaaa@DC30@

and A 2 = α i β i η i0 ( λ u 0 ) 3 (1 z 2 μ 2 + μ ce + μ se ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqam aaBaaajuaibaGaaGOmaaqcfayabaGaeyypa0ZaaSaaaeaacqaHXoqy daWgaaqcfauaaiaadMgaaeqaaKqbakabek7aInaaBaaajuaibaGaam yAaaqabaqcfaOaeq4TdG2aaSbaaKqbGeaacaWGPbGaaGimaaqabaaa juaGbaWaaeWaaeaacqaH7oaBcqGHsislcaWG1bWaaSbaaKqbGeaaca aIWaaajuaGbeaaaiaawIcacaGLPaaadaahaaqcfasabeaacaaIZaaa aaaajuaGcaGGOaGaaGymaiabgkHiTiaadQhadaWgaaqcfasaaiaaik daaKqbagqaaiabeY7aTnaaBaaajuaibaGaaGOmaaqabaqcfaOaey4k aSIaeqiVd02aaSbaaKqbGeaacaWGJbGaamyzaaqabaqcfaOaey4kaS IaeqiVd02aaSbaaKqbGeaacaWGZbGaamyzaaqcfayabaGaaiykaaaa @60FC@    (20)

Solution of KdVB equation

To find the solution to (19), we use the transformation χ=f( ξVτ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm Maeyypa0JaamOzamaabmaabaGaeqOVdGNaeyOeI0IaamOvaiabes8a 0bGaayjkaiaawMcaaaaa@4105@  of the co-moving frame with speed V. A convenient method to solve the KdVB equation is the “tanh method”.43,44 Using boundary conditions ψ (1) , ψ (1) χ , 2 ψ (1) χ 2 , 3 ψ (1) χ 3 0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aaWbaaKqbGeqabaGaaiikaiaaigdacaGGPaaaaKqbakaacYcadaWc aaqaaiabgkGi2kabeI8a5naaCaaajuaibeqaaiaacIcacaaIXaGaai ykaaaaaKqbagaacqGHciITcqaHhpWyaaGaaiilamaalaaabaGaeyOa Iy7aaWbaaKqbGeqabaGaaGOmaaaajuaGcqaHipqEdaahaaqcfasabe aacaGGOaGaaGymaiaacMcaaaaajuaGbaGaeyOaIyRaeq4Xdm2aaWba aKqbGeqabaGaaGOmaaaaaaqcfaOaaiilamaalaaabaGaeyOaIy7aaW baaKqbGeqabaGaaG4maaaajuaGcqaHipqEdaahaaqcfasabeaacaGG OaGaaGymaiaacMcaaaaajuaGbaGaeyOaIyRaeq4Xdm2aaWbaaKqbGe qabaGaaG4maaaaaaqcfaOaeyOKH4QaaGimaiaacYcaaaa@630E@ as χ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm MaeyOKH4QaeyOhIukaaa@3B99@  for a localized solution, we can write (19) as

AV ψ (1) τ + ψ (1) ψ (1) ξ +B 3 ψ (1) ξ 3 C 2 ψ (1) ξ 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaamyqaiaadAfadaWcaaqaaiabgkGi2kabeI8a5naaCaaajuaibeqa aiaacIcacaaIXaGaaiykaaaaaKqbagaacqGHciITcqaHepaDaaGaey 4kaSIaeqiYdK3aaWbaaKqbGeqabaGaaiikaiaaigdacaGGPaaaaKqb aoaalaaabaGaeyOaIyRaeqiYdK3aaWbaaKqbGeqabaGaaiikaiaaig dacaGGPaaaaaqcfayaaiabgkGi2kabe67a4baacqGHRaWkcaWGcbWa aSaaaeaacqGHciITdaahaaqcfasabeaacaaIZaaaaKqbakabeI8a5n aaCaaajuaibeqaaiaacIcacaaIXaGaaiykaaaaaKqbagaacqGHciIT cqaH+oaEdaahaaqcfasabeaacaaIZaaaaaaajuaGcqGHsislcaWGdb WaaSaaaeaacqGHciITdaahaaqcfasabeaacaaIYaaaaKqbakabeI8a 5naaCaaajuaibeqaaiaacIcacaaIXaGaaiykaaaaaKqbagaacqGHci ITcqaH+oaEdaahaaqcfasabeaacaaIYaaaaaaajuaGcqGH9aqpcaaI Waaaaa@6FBA@  (21)

Using the transformation ψ (1) ( α )= i=0 n a i α i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aaWbaaKqbGeqabaGaaiikaiaaigdacaGGPaaaaKqbaoaabmaabaGa eqySdegacaGLOaGaayzkaaGaeyypa0ZaaabmaeaacaWGHbWaaSbaaK qbGeaacaWGPbaabeaaaKqbafaacaWGPbGaeyypa0JaaGimaaqcfasa aiaad6gaaKqbakabggHiLdGaeqySde2aaWbaaKqbGeqabaGaamyAaa aajuaGcaaMc8Uaaiilaaaa@4DD0@   α=tanhχ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde Maeyypa0JaciiDaiaacggacaGGUbGaaiiAaiabeE8aJjaaykW7caGG Saaaaa@40D8@ we can arrive at the solution as

ψ (1) =AV+8B k 2 + C 2 25B 12 k 2 tanh 2 [ k( ξVτ ) ] 12 5 kC[ 1tanh[ k( ξVτ ) ] ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aaWbaaKqbGeqabaGaaiikaiaaigdacaGGPaaaaKqbakabg2da9iaa dgeacaWGwbGaey4kaSIaaGioaiaadkeacaWGRbWaaWbaaKqbGeqaba GaaGOmaaaajuaGcqGHRaWkdaWcaaqaaiaadoeadaahaaqcfauabeaa caaIYaaaaaqcfayaaiaaikdacaaI1aGaamOqaaaacqGHsislcaaIXa GaaGOmaiaadUgadaahaaqcfasabeaacaaIYaaaaKqbakGacshacaGG HbGaaiOBaiaacIgadaahaaqcfasabeaacaaIYaaaaKqbaoaadmaaba Gaam4AamaabmaabaGaeqOVdGNaeyOeI0IaamOvaiabes8a0bGaayjk aiaawMcaaaGaay5waiaaw2faaiabgkHiTmaalaaabaGaaGymaiaaik daaeaacaaI1aaaaiaadUgacaWGdbWaamWaaeaacaaIXaGaeyOeI0Ia ciiDaiaacggacaGGUbGaaiiAamaadmaabaGaam4AamaabmaabaGaeq OVdGNaeyOeI0IaamOvaiabes8a0bGaayjkaiaawMcaaaGaay5waiaa w2faaaGaay5waiaaw2faaaaa@7293@

The speed of the co-moving frame is related to the coefficients of the KdVB equation as V= 100 B 2 k 2 C 2 +60kBC 25AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvai abg2da9maalaaabaGaaGymaiaaicdacaaIWaGaamOqamaaCaaajuai beqaaiaaikdaaaqcfaOaam4AamaaCaaajuaibeqaaiaaikdaaaqcfa OaeyOeI0Iaam4qamaaCaaajuaibeqaaiaaikdaaaqcfaOaey4kaSIa aGOnaiaaicdacaWGRbGaamOqaiaadoeaaeaacaaIYaGaaGynaiaadg eacaWGcbaaaaaa@4AC1@ where k= ±C 10B , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aai abg2da9maalaaabaGaeyySaeRaam4qaaqaaiaaigdacaaIWaGaamOq aaaacaaMc8Uaaiilaaaa@3FB7@  obtained from the boundary conditions.

Results

We have derived equations applicable to any multi-ion/dusty plasma environment; the figures, are, however, plotted using parameters relevant to comet Halley. The majority lighter ion density was set at 4.95cm-3 , and their temperature at T i =8× 10 4 K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srVq0Jbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGubWaaSbaaKqbGeaacaWGPbaabeaajuaGcqGH9aqpcaaI4aGaey41 aqRaaGymaiaaicdadaahaaqcfasabeaacaaI0aaaaKqbakaadUeaaa a@429D@ and solar electron temperature at12 T he =2× 10 5 K. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srVq0Jbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGubWaaSbaaKqbGeaacaWGObGaamyzaaqcfayabaGaeyypa0JaaGOm aiabgEna0kaaigdacaaIWaWaaWbaaKqbGeqabaGaaGynaaaajuaGca WGlbGaaiOlaaaa@4433@ The negatively charged heavier ions (oxygen) was detected at an peak energy of 1eV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWI8iIocaaIXaGaaiyzaiaacAfaaaa@3A4B@ with densities18 1c m 3 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0le9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyizIm QaaGymaiaadogacaWGTbWaaWbaaKqbGeqabaGaeyOeI0IaaG4maaaa juaGcaGGUaaaaa@3E08@ The negatively and positively charged heavier ion densities are therefore n 1 =0.05c m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0le9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBam aaBaaajuaybaGaaGymaaqabaqcfaOaeyypa0JaaGimaiaac6cacaaI WaGaaGynaiaadogacaWGTbWaaWbaaKqbGfqabaGaeyOeI0IaaG4maa aaaaa@412E@ and n 2 =0.5c m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBam aaBaaajuaybaGaaGOmaaqcfayabaGaeyypa0JaaGimaiaac6cacaaI 1aGaam4yaiaad2gadaahaaqcfawabeaacqGHsislcaaIZaaaaaaa@4075@ respectively at a temperature12,18

Figure 1 depicts the effect of the drift velocities of the lighter ions on the shock profile. The parameters used for the figure are: n i0 =4.95 cm -3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGUbWaaSbaaKqbGeaacaWGPbGaaGimaaqcfayabaGaeyypa0JaaGin aiaac6cacaaI5aGaaGynaiGacogacaGGTbWaaWbaaKqbGeqabaGaai ylaiaacodaaaqcfaOaaiilaaaa@4479@ n 10 =0.05 cm -3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBam aaBaaajuaibaGaaGymaiaaicdaaKqbagqaaiabg2da9iaaicdacaGG UaGaaGimaiaaiwdaciGGJbGaaiyBamaaCaaajuaibeqaaiaac2caca GGZaaaaKqbakaacYcaaaa@4284@ n 20 =0.5 cm -3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBam aaBaaajuaibaGaaGOmaiaaicdaaKqbagqaaiabg2da9iaaicdacaGG UaGaaGynaiGacogacaGGTbWaaWbaaKqbGeqabaGaaiylaiaacodaaa qcfaOaaGPaVlaacYcaaaa@4356@ T ce =2× 10 4 K, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGubWaaSbaaKqbGeaacaWGJbGaamyzaaqabaqcfaOaeyypa0JaaGOm aiabgEna0kaaigdacaaIWaWaaWbaaKqbGeqabaGaaGinaaaajuaGca WGlbGaaiilaaaa@442B@ T he =2× 10 5 K, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srVq0Jbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGubWaaSbaaKqbGeaacaWGObGaamyzaaqcfayabaGaeyypa0JaaGOm aiabgEna0kaaigdacaaIWaWaaWbaaKqbGeqabaGaaGynaaaajuaGca WGlbGaaiilaaaa@4431@ T 1 =1.16× 10 4 K, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGubWaaSbaaKqbGeaacaaIXaaabeaajuaGcqGH9aqpcaaIXaGaaiOl aiaaigdacaaI2aGaey41aqRaaGymaiaaicdadaahaaqcfasabeaaca aI0aaaaKqbakaadUeacaGGSaaaaa@4540@ z 1 =2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEam aaBaaajuaibaGaaGymaaqabaqcfaOaeyypa0JaaGPaVlaaikdacaGG Saaaaa@3D18@ z 2 =4, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG6bWaaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcaaMc8UaaGin aiaacYcaaaa@3ED0@ B 0 =8× 10 5 G, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGcbWaaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpcaaI 4aGaey41aqRaaGymaiaaicdadaahaaqcfasabeaacqGHsislcaaI1a aaaKqbakaaykW7caWGhbGaaiilaaaa@43E7@ κ ce = κ he =3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH6oWAdaWgaaqcfasaaiaadogacaWGLbaajuaGbeaacqGH 9aqpcqaH6oWAdaWgaaqcfasaaiaadIgacaWGLbaajuaGbeaacqGH9a qpcaaIZaaaaa@4234@ and η i0 =0.5. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4TdG 2aaSbaaKqbGeaacaWGPbGaaGimaaqabaqcfaOaeyypa0JaaGimaiaa c6cacaaI1aGaaGPaVlaac6cacaaMc8oaaa@41AE@ The upper plot (blue in color) depicts the shock profile without a drift velocity for the lighter ions, as discussed by Manesh et al.,45 the shock is produced by the effects of heavier positive and negative ions. The profile in the middle (green in color) is for a drift velocity of lighter ions u 0 =0.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG1bWaaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpcaaIWaGaaiOl aiaaikdaaaa@3DF8@ and lower (red in color) is for u 0 =0.4. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG1bWaaSbaaKqbGeaacaaIWaaabeaajuaGcqGH9aqpcaaIWaGaaiOl aiaaisdacaGGUaaaaa@3EAC@ From the plots it is seen that as drift velocities of the lighter ions increase, the amplitude of the shock profile decreases.

Figure 1 Plot of shock profiles for different drift velocities of the lighter ions.

The shock profiles, for different kappa indices of both solar and cometary electrons, are depicted in Figure 2. The lower plot is for κ ce = κ he =3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH6oWAdaWgaaqcfasaaiaadogacaWGLbaajuaGbeaacqGH 9aqpcqaH6oWAdaWgaaqcfasaaiaadIgacaWGLbaajuaGbeaacqGH9a qpcaaIZaaaaa@4234@ for u 0 =0.3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG1bWaaSbaaKqbGeaacaaIWaaabeaajuaGcqGH9aqpcaaIWaGaaiOl aiaaiodaaaa@3DF9@ (continuous blue line) and u 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG1bWaaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpcaaIWaGaaGjb Vdaa@3E17@ (blue dashed line); the middle plot is for κ ce = κ he =5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca aMe8UaeqOUdS2aaSbaaKqbGeaacaWGJbGaamyzaaqcfayabaGaeyyp a0JaeqOUdS2aaSbaaKqbGeaacaWGObGaamyzaaqabaqcfaOaeyypa0 JaaGynaaaa@4558@ for u 0 =0.3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG1bWaaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpcaaIWaGaaiOl aiaaiodaaaa@3DF9@ (continuous green line) and u 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG1bWaaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpcaaIWaGaaGjb Vdaa@3E17@ (green dashed line) while the upper plot is for κ ce = κ he =10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca aMe8UaeqOUdS2aaSbaaKqbGeaacaWGJbGaamyzaaqabaqcfaOaeyyp a0JaeqOUdS2aaSbaaKqbGeaacaWGObGaamyzaaqabaqcfaOaeyypa0 JaaGymaiaaicdaaaa@460E@ for u 0 =0.3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG1bWaaSbaaKqbGeaacaaIWaaabeaajuaGcqGH9aqpcaaIWaGaaiOl aiaaiodaaaa@3DF9@ (continuous red line) and u 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG1bWaaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpcaaIWaGaaGjb Vdaa@3E17@ (dashed red line). The other parameters are the same as in figure 1. We find that the strengths of the shock profiles decrease as the kappa values increase, irrespective of the drift velocities of the lighter ions. Lower values of kappa indices indicate the presence of more superthermal particles in the plasma; these superthermal particles thus support larger shock profiles. Also, as the plasma approaches a Maxwellian distribution, the effect of the drift velocity diminishes. In addition, the strength of the shock profile decreases for non-zero values of drift velocity as compared to the profile for a zero drift velocity of the lighter ions; in agreement with Figure 1.

Figure 2 Plot of shock profiles for different kappa indices with and without drift for the lighter ions.

Next, Figure 3 shows the variation of shock profiles in the plasma for different values of kinematic viscosity with drift velocity of lighter ions included ( u 0 =0.1). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca GGOaGaamyDamaaBaaajuaibaGaaGimaaqabaqcfaOaeyypa0JaaGim aiaac6cacaaIXaGaaiykaiaac6caaaa@4002@ The upper curve (blue in color) is for kinematic viscosity η i0 =0.1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcq aH3oaAdaWgaaqcfasaaiaadMgacaaIWaaabeaajuaGcqGH9aqpcaaI WaGaaiOlaiaaigdacaGGSaaaaa@4047@ the middle curve (green in color) is for η i0 =0.3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcq aH3oaAdaWgaaqcfasaaiaadMgacaaIWaaabeaajuaGcqGH9aqpcaaI WaGaaiOlaiaaiodaaaa@3F99@ and the lower curve (red in color) is for η i0 =0.5. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcq aH3oaAdaWgaaqcfasaaiaadMgacaaIWaaabeaajuaGcqGH9aqpcaaI WaGaaiOlaiaaiwdacaGGUaaaaa@404D@ The other parameters are the same as in Figure 1. From the figure it is seen that the size of the shock profile increases with increasing values of kinematic viscosities of the lighter ions.

Figure 3 Plot of Shock profiles for different values of kinematic viscosity with a drift velocity of lighter ions.

Figure 4 is a plot of shock profiles for different densities and drift velocities of the lighter ions. The upper plot (blue in color) is for n i0 =3, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGUbWaaSbaaKqbGeaacaWGPbGaaGimaaqabaqcfaOaeyypa0JaaG4m aiaacYcaaaa@3E24@ middle one (green in color) for n i0 =4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGUbWaaSbaaKqbGeaacaWGPbGaaGimaaqcfayabaGaeyypa0JaaGin aaaa@3D75@ and lower plot (red in color) is for n i0 =5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WGUbWaaSbaaKqbGeaacaWGPbGaaGimaaqabaqcfaOaeyypa0JaaGyn aaaa@3D76@ with drift velocities of lighter ions as u 0 =0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG1bWaaSbaaKqbGeaacaaIWaaabeaajuaGcqGH9aqpcaaIWaGaaiOl aiaaiwdaaaa@3DFB@ and u 0 =0.1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGca WG1bWaaSbaaKqbGeaacaaIWaaabeaajuaGcqGH9aqpcaaIWaGaaiOl aiaaigdaaaa@3DF7@ (denoted respectively by continuous and dashed lines). The other parameters are the same as in figure 1. From the plots, it is evident that the amplitude of the shock profile increases as the density of lighter ions increases.

Figure 4 Plot of shock profiles for different densities and drift velocities of lighter ions.

Figure 5 depicts values of coefficient A as a function of the drift velocities of the lighter ions. The other parameters for the plot remain the same as in Figure 1. From the plot, it is seen that value of coefficient increases exponentially as the drift velocity increases.

Figure 5 Plot of coefficient A for different drift velocities of lighter ions.

Finally, Figure 6 illustrates the values of the coefficient B as a function of the drift velocities of the lighter ions. The other parameters for the plot remain the same as in Figure 1.It is obvious from the plot that the variation of B, with the drift velocity of the lighter ions, is almost linear.

Figure 6 Plot of coefficient B for different drift velocities of lighter ions.

Figure 1 and Figure 2 given above reveal that the shock amplitudes decrease with increasing streaming velocities of the lighter ions. For the given plasma system some of the plasma instabilities, whose growth rates are a function of the streaming velocity, that can be excited are the two stream instability, the ion acoustic instability, etc. Thus the decrease in shock amplitude could be due to a diversion of the streaming energy to excite these instabilities. At this point it may be noted that the existence of both these waves were noticed in numerical simulation studies of shock dynamical behavior.46 A superthermal distribution tends to a Maxwellian distribution when the spectral index κ>. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH6oWApaGaeyOeI0IaeyOpa4JaeyOhIuQaaiOlaaaa@3C7D@ thus damping the wave; a larger streaming velocity is now required to drive the instability. Hence the shock amplitude decreases when the spectral index κ>, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH6oWApaGaeyOeI0IaeyOpa4JaeyOhIuQaaiilaaaa@3C7B@ Such a decrease in the amplitude of the shock wave with spectral index was also found in a study of ion acoustic shock waves in a dissipative plasma with superthermal electrons and positrons.47 Figure 3 reveals that the amplitude of the shock wave increases with increasing lighter ion kinematic viscosity. This is due to the increased dissipation in the system and has been reported earlier in an electron-positron-ion plasma.48 Keeping the ion kinematic viscosity a constant and increasing the lighter ion density again increases the dissipation and hence explains the results of figure. Finally, the equations describing the system can be made applicable to any multi-ion environment where both polarities of heavier ions are present along with superthermal electrons and drift velocity of lighter ions.

Conclusion

The KdVB equation has been derived in an un-magnetized plasma composed of kappa function described electrons (of both cometary and solar origin), positively and negatively charged heavier ions and lighter ions with a streaming velocity. The shock profiles have been studied for various parameters like kinematic viscosities of the lighter ions, drift velocities and kappa values. It is seen that the drift velocity has a strong influence on the shock wave: its amplitude decreases as the drift velocities of the lighter ions increases. Also, the amplitude of the shock profiles increases with a decrease in the kappa indices of the electrons and increases with an increase in the densities of the lighter ions. That is, increasing the number of superthermal electrons and lighter ions supports high amplitude shock waves in the plasma. In other words, increasing the number density of the dissipative plasma species increases the amplitude of the shock waves, irrespective of their polarity. Also, different coefficients of the KdVB equation are strongly affected by the drift velocities of the lighter ions.

Acknowledgements

We thank the reviewer for the very useful comments which have added immensely to the scientific content of the paper.

Conflict of interest

The author declares that there is no conflict of interest.

References

  1. Goertz CK. Dusty Plasmas in the Solar System. Rev Geophys. 1989;27(2):271–292.
  2. Northrop TG. Dusty Plasmas. Phys Scripta. 1992;45(5):475–490  
  3. Egrand C, Duprant J, Dartois E, et al. Variations in cometary dust composition from Giotto to Rosetta, clues to their formation mechanisms. MNRAS. 2016;462(Suppl 1):323–S330.
  4. Havnes O, Troim J, Blix T, et al. First detection of charged particles in the Earth’s mesosphere. J Geophys Res. 1996;101(A5):10839–10847.
  5. Dubinskii A Popel SI. Formation and evolution of dusty plasmas in the ionosphere. JETP Letters. 2012;96(1):21–26
  6. Thomas H, Morfill GE, Demmel V, et al. Plasma crystal: coulomb crystallization in a dusty plasma. Phys Rev Lett. 1994;73(5);652–655.  
  7. Winter J. Dust in Fusion Devices. Plas Phys Contr Fus. 2004;12B:583
  8. Rao NN, Shukla PK, Yu, et al. Dust–acoustic waves in dusty plasmas. Planet Space Sci. 1990;38(4):543–546.
  9. Shukla PK, Silin VP. Dust ion–acoustic wave. Physica Scr. 1992;45(5):508.
  10. Melandso M. Lattice Waves in dust plasma crystals. Phys Plasmas. 1996;3(11):3890–3901.
  11. Shukla PK, Mamun AA. Solitons, shocks and vortices in dusty plasmas. New J Phys. 2003;5(1):17–37.
  12. Brinca AL, Tsurutani BT. Unusual characteristics of the electromagnetic waves excited by cometary newborn ions with large perpendicular energies. Exploration of Halley’s Comet. 1987;187:311–319.
  13. Balsiger H, Altwegg K, Bühler F, et al. Ion composition and dynamics at comet Halley. Nature. 1986;321:330–334.
  14. Rubin M, Hansen KC, Gombosi TI, et al. Ion composition and chemistry in the coma of Comet 1P/Halley—A comparison between Giotto's Ion Mass Spectrometer and our ion–chemical network. Icarus. 2009;199(2):505–519.
  15. Ipavich F, Galvin A, Gloeckler G, et al. Comet Giacobini–Zinner: In Situ observations of energetic heavy ions. Science. 1986;232(4748):366–369.
  16. Neugebauer M, Gloeckler G, Gosling JT, et al. Encounter of the Ulysses spacecraft with the ion tail of comet McNaught. Astrophy J. 2007;667(2):1262–1266.
  17. Kharchenko V, Rigazio M, Dalgarno A, et al. Charge abundances of the solar wind ions inferred from cometary X–ray spectra. Astrophys J Lett. 2003;585:73–L75.
  18. Chaizy P, Reme H, Sauvaud, JA, et al. Negative ions in the coma of comet Halley. Nature . 1991;349:393–396.
  19. Ellis TA, Neff JS. Numerical simulation of the emission and motion of neutral and charged dust from P/Halley. Icarus. 1991;91(2):280–296.
  20. Chow VW, Mendis DA, Rosenberg M. Role of grain size and particle velocity distribution in secondary electron emission in space plasmas. J Geophys Res. 1993;98(A11):19065–19076.
  21. Vasyliunas VM. A survey of low–energy electrons in the evening sector of the magnetosphere with OGO 1 and OGO 3. J Geophys Res. 1968;73(9):2839–2884.
  22. Ghosh S, Sarkar S, Khan M. Effect of finite ion inertia and dust drift on small amplitude dust acoustic soliton. Planet. Space Sci. 2000;48:609–614.
  23. Ghosh S, Chaudhuri TK, Sarkar S, et al. Small amplitude nonlinear dust acoustic wave propagation in Saturn's F, G and E rings. Astrophys Space Sci. 2001;278(4):465–477.
  24. Paul SN, Chattopadhyay S, Bhattacharya SK, et al. On the study of ion–acoustic solitary waves and double–layers in a drift multi–component plasma with electron–inertia. Pramana. 2003;60(6):1217–1233.
  25. Chattopadhyay S. Effect of ionic temperatures on ion–acoustic solitary waves in a drift negative–ion plasma with single temperature electron. Fizika A –J Exper Theor Phys–Zagreb. 2010;19:31–46.
  26. Tribeche M, Younsi S, Zerguini TH. Arbitrary amplitude dust–acoustic double–layers in a warm dusty plasma with suprathermal electrons, two–temperature thermal ions, and drifting dust grains. Astrophys Space Sci. 2012;339(2):243–247.
  27. Ghosh B, Paul SN, Das C, et al. Electrostatic double layers in a multi–component drifting plasma having nonthermal electrons. Braz J Phys. 2013;43(1–2):28–33.
  28. Kundu SK, Ghosh DK, Chatterjee P, et al. Shock waves in a dusty plasma with positive and negative dust, where electrons are superthermally distributed. Bulg J Phys. 2011;38(2011):409–419.
  29. Shah A, Saeed R. Nonlinear Korteweg–de Vries–Burger equation for ion–acoustic shock waves in the presence of kappa distributed electrons and positrons. Plasma Phys Control Fus. 2011;53(9):095006
  30. Shah A, Mahmood S, Haque Q. Ion acoustic shock waves in presence of superthermal electrons and interaction of classical positron beam. Phys Plasmas, 2012;19:032302
  31. Pakzad HR. Dust acoustic shock waves in strongly coupled dusty plasmas with kappa–distributed ions. Indian J Phys. 2012;86(8):743–747.
  32. Kourakis I, Sultana S, Hellberg MA. Dynamical characteristics of solitary waves, shocks and envelope modes in kappa–distributed non–thermal plasmas: an overview. Plasma Phys Control Fus. 2012:54(12):124001–124007
  33. Alam MS, Masud MM, Mamun AA. Effects of bi–kappa distributed electrons on dust–ion–acoustic shock waves in dusty superthermal plasmas. Chinese Phys. B 2013;22:115202.
  34. Hyder CL, Brandt JC, Roosen RG. Tail structures far from the head of Comet Kohoutek I. Icarus. 1974;23:601–610.
  35. Ershkovich AI. Solar wind interaction with the tail of comet Kohoutek. Planet. Space Sci. 1976;24(3):287–291.
  36. Ershkovich AI, Heller AB. Helical waves in type–1 comet tails. Astrophys. Space Sci. 1977;48:365–377.
  37. Hada T, Kennel CF, Buti B. Stationary nonlinear Alfvén waves and solitons. J Geophys Res. 1989;94:65–77.
  38. Tsurutani BT, Smith EJ, Matsumoto H, et al. Highly nonlinear magnetic pulses at comet Giacobini–Zinner. Geophys. Res. Lett.1990;17(6):757–760.
  39. Kotsarenko NY, Koshevaya SV, Stewart GA, et al. Electrostatic spatially limited solitons in a magnetised dusty plasma. Planet Space Sci. 1998;46(4):429–433.
  40. Coates AJ. Heavy ion effects on cometary shocks. Adv Space Res. 1995;15(8–9):403–413.
  41. Kennel CF, Coroniti FV, Scarf FL, et al. Plasma waves in the shock interaction regions at comet Giacobini–Zinner. Geophys. Res Lett.1986;13(9):921–924.
  42. Voelzke MR, Izaguirre LS. Morphological analysis of the tail structures of comet P/Halley 1910 II. Planet Space Sci. 2012;65(1):104–108.
  43. Malfliet W. Solitary wave solutions of nonlinear wave equations. American J Phys. 1992;60:650–654.
  44. Malfliet W. The tanh method:a tool for solving certain classes of nonlinear evolution and wave equations. J Comput Appl Math. 2004;164–165:529–541.
  45. Manesh M, Neethu TW, Neethu J, et al. Korteweg–deVries–Burgers (KdVB equation) equation in a five component plasma cometary plasma with kappa described electrons and ions. J Theor Appl Phys. 2015;10(4):280–296.
  46. Shimada N, Hoshino M. Effect of strong thermalization on shock dynamical behavior. J Geophys Res. 2005;110(2):A02105
  47. Pakzad HR. Ion Acoustic shock waves in dissipative plasma with superthermal electrons and positrons. Astrophys Sp Sci. 2011;331(1):169–174
  48. Ghosh DB, Chatterjee P, Mandal PJ, et al. Nonplanar ion–acoustic shocks in electron–positron–ion plasmas:effect of superthermal electrons. Pramana–J Phys. 2013;81(3):491–501
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