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Head-on collisions of electrostatic solitons in nonthermal plasmas Frank Verheest1,2 , Manfred A Hellberg2 and Willy A. Hereman3 1 Sterrenkundig Observatorium, Universiteit Gent, Belgium School of Chemistry and Physics, University of KwaZulu-Natal, Durban, SA Applied Mathematics and Statistics, Colorado School of Mines, Golden CO, USA 2 3 Phys. Rev. E 86, 036402 (2012) Introduction Basic formalism Generic case Critical case Summary Outline 1 Introduction 2 Basic formalism 3 Generic case 4 Critical case 5 Summary ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Motivation Recently, many papers appeared in a plasma physics context about head-on collisions of two electrostatic solitons, which intrinsically is a very interesting problem Before going on, we remind ourselves that KdV equations possess exact solutions for interactions between N solitons, with full nonlinearity during overtaking t x Although rarely mentioned, this is valid only for solitons propagating in direction that underlies basic derivation of parent equation ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Head-on collisions In contrast, head-on collisions between two electrostatic solitons can only be dealt with by approximate methods, which limit the range of validity but might offer interesting insight Framework is based on Poincaré-Lighthill-Kuo formalism of strained coordinates (which yields here KdV/mKdV families of equations plus phase/time shifts that occur during interaction) and was adapted for solitary waves on shallow water KdV equation is typical paradigm which fixes our intuitive ideas of what solitary waves look like, having nonlinear steepening balanced by dispersion, on slow timescale compared to linear propagation velocity ∂ϕ ∂ϕ ∂3ϕ + Aϕ +B =0 ∂τ ∂ξ ∂ξ 3 However, when plasma composition is critical in sense that A = 0, another expansion and/or stretching scheme is needed, leading to a modified KdV equation with cubic nonlinearity ∂ϕ ∂3ϕ ∂ϕ + C ϕ2 +D =0 ∂τ ∂ξ ∂ξ 3 ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Remarks on models and procedures used in literature Almost all papers in recent plasma literature are theoretical and use PLK formalism, without worrying about or explaining inherent limitations or use of certain assumptions Even though most plasma models investigated allow for criticality (A = 0), none of this is discussed, so that only generic half of problem is treated This is like deducing KdV equation by tacitly assuming that A 6= 0, and forgetting about possible mKdV discussion Original derivation of KdV equation was, of course, for water waves, which can only occur as humps (so that A 6= 0 here), but plasma is much richer medium Besides this plethora of theoretical papers, looking very similar and seldom giving any figures, one experimental paper stands out: Harvey et al., Phys. Rev. E 81, 057401 (2010) Harvey et al. discuss experimental observations of interaction of two counter-propagating solitons of equal amplitude in a monolayer strongly coupled dusty plasma Their results include propagation delays after interaction, compared to a single soliton, and also that combined amplitudes during interaction are less than simple linear superposition ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Experimental results Data from experiment [Harvey et al., Phys. Rev. E 81, 057401 (2010)] t 6 4 2 -4 2 -2 4 x t -2 x -4 Results schematically show delay in transmission after interaction, but do not indicate height, which is found to be less than linear superposition ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Outline 1 Introduction 2 Basic formalism 3 Generic case 4 Critical case 5 Summary ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Model and basic equations Although recent literature encompasses quite a variety of plasma compositions and electrostatic waves, treatment is very similar and leads to KdV equations plus time delays, only coefficients change Hence, for clarity of exposition, consider plasma consisting of cold ions and nonthermal (“Cairns") electrons, which is simplest two-species plasma allowing for criticality Normalized continuity and momentum equations for ions are coupled to Poisson’s equation ∂n ∂ + (nu) = 0 ∂t ∂x ∂u ∂u ∂ϕ +u + =0 ∂t ∂x ∂x ∂2ϕ + n − (1 − βϕ + βϕ2 ) exp(ϕ) = 0 ∂x2 where only parameter is β, measuring degree of nonthermality For β = 0 Maxwellian electrons are recovered, and for β > 4/7 original phase space distribution develops wings which might lead to bump-on-tail or two-beam unstable cases ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Stretching and expansions Stretching of independent variables reflects propagation in opposite directions ξ = ε(x − ca t) + ε2 P(ξ, η, τ ) + ... η = ε(x + ca t) + ε2 Q(ξ, η, τ ) + ... τ = ε3 t Both space coordinates have to use same linear acoustic phase velocity in medium, here 1 ca = (1 − β)− 2 with proper choice of normalization This is coupled to expansions for densities, velocities and electrostatic potential n = 1 + εn1 + ε2 n2 + ε3 n3 + ε4 n4 + ... u = εu1 + ε2 u2 + ε3 u3 + ε4 u4 + ... ϕ = εϕ1 + ε2 ϕ2 + ε3 ϕ3 + ε4 ϕ4 + ... Steps in expansion are left general, so as to deal with generic and critical compositions in one coherent treatment ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Lower order results and bifurcation To lowest order perturbations obey ∂n1 ∂u1 ∂u1 ∂n1 1 − + + =0 ∂ξ ∂ξ ∂η 1 − β ∂η 1 ∂u1 ∂ϕ1 ∂ϕ1 ∂u1 √ − + + =0 ∂ξ ∂ξ ∂η 1 − β ∂η √ n1 − (1 − β)ϕ1 = 0 and lead to separability at linear level n1 = (1 − β) ϕ1ξ + ϕ1η p u1 = 1 − β ϕ1ξ − ϕ1η ϕ1 = ϕ1ξ + ϕ1η Next order leads to bifurcation (2 − 6β + 3β 2 )ϕ21ξ = 0 & (2 − 6β + 3β 2 )ϕ21η = 0 so that in generic case ϕ1ξ = ϕ1η = 0 or at critical parameters β = βc = (3 − ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 √ 3)/3 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Outline 1 Introduction 2 Basic formalism 3 Generic case 4 Critical case 5 Summary ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Coupled Korteweg-de Vries equations and phase shifts After much complicated/tedious algebra one arrives at KdV equations and phase shifts ∂ϕ2ξ ∂ϕ2ξ ∂ 3 ϕ2ξ + A ϕ2ξ +B =0 ∂τ ∂ξ ∂ξ 3 & ∂P = S ϕ2η ∂η ∂ϕ2η ∂ϕ2η ∂ 3 ϕ2η − A ϕ2η −B =0 ∂τ ∂η ∂η 3 & ∂Q = S ϕ2ξ ∂ξ where A= 2 − 6β + 3β 2 2(1 − β)3/2 & B= 1 2(1 − β)3/2 One-soliton solutions for each KdV equation are 3vξ sech2 κξ (ξ − vξ τ ) A 3vη = sech2 [κη (η + vη τ )] A ϕ2ξ = & ϕ2η & where κξ = (1 − β)3/4 r vξ 2 ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 6Sκη {tanh [κη (η + vη τ )] + 1} A(1 − β)3/2 6Sκξ Q= tanh κξ (ξ − vξ τ ) − 1 A(1 − β)3/2 P= & κη = (1 − β)3/4 r vη 2 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Behaviour of KdV coefficients Expressions for coefficients in KdV equations are A= 2 − 6β + 3β 2 2(1 − β)3/2 & B= 1 2(1 − β)3/2 Changes of A and B (full and dashed curves) as β is increased, showing how A goes from positive to negative values at βc and that B remains positive and increases monotonically from 1/2 A,B 2.0 1.5 1.0 0.5 Β 0.1 0.2 0.3 0.4 0.5 0.6 -0.5 -1.0 Case A = 1 and B = 1/2 is for Maxwellian electrons (β = 0) ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Illustrations for Maxwellian and Cairns electrons Only same sign polarities (positive or negative) are possible for both waves Left figure: values for Maxwellian (β = 0) electrons, so that A = 1 (potential and density humps) and B = 1/2 (propagation delays due to interaction) Right figure: values for Cairns (β = 0.5) electrons, so that A < 0 (potential and density dips) but B > 1/2 > 0 (propagation delays due to interaction) ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Visualization of propagation delays Comparison of analytical result with schematic corresponding to Harvey et al. experiments t 6 4 2 -4 2 -2 4 x -2 -4 Left figure: top view showing propagation delay for slower positive soliton at β = 0.25, where stronger soliton has been omitted for graphical clarity Right figure: repetition of earlier schematic of Harvey et al. results ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Comparison of amplitudes Compare exact solution for overtaking solitons to PLK approach for head-on collisions t x Overtaking soliton amplitudes are depressed during nonlinear interaction, whereas PLK formalism leads to simple linear superposition However, this linear superposition is also at variance with experiments by Harvey et al. and points to weak point of PLK approach Caution is needed here, but at present there seems to be no alternative theory to deal with head-on collisions... ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Outline 1 Introduction 2 Basic formalism 3 Generic case 4 Critical case 5 Summary ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Modified Korteweg-de Vries equations and phase shifts Algebra in critical case is even more involved but leads to mKdV equations plus phase shifts ∂ϕ1ξ ∂ϕ1ξ ∂ 3 ϕ1ξ + 0.353 ϕ21ξ + 1.140 =0 ∂τ ∂ξ ∂ξ 3 & ∂P = 0.449 ϕ21η ∂η ∂ϕ1η ∂ϕ1η ∂ 3 ϕ1η − 0.353 ϕ21η − 1.140 =0 ∂τ ∂η ∂η 3 & ∂Q = 0.449 ϕ21ξ ∂ξ Solutions can now be of either sign and thus allow for collision between counterstreaming negative and positive polarity solitons √ √ ϕ1ξ = ± 4.125 vξ sech 0.937 vξ (ξ − vξ τ ) √ √ ϕ1η = ± 4.125 vη sech 0.937 vη (η + vη τ ) and P = 8.162 Q = 8.162 √ √ √ tanh 0.937 vη (η + vη τ ) + 1 √ tanh 0.937 vξ (ξ − vξ τ ) − 1 vη vξ ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Illustrations for combination of positive and negative solitons Criticality can lead to two counter-propagating positive or negative solitons, but graphs are similar to examples produced for generic case Unusual is combination of a positive and a negative polarity soliton Combining solitons of different polarity in two-soliton model for single KdV equation is apparently not possible! ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Outline 1 Introduction 2 Basic formalism 3 Generic case 4 Critical case 5 Summary ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas Introduction Basic formalism Generic case Critical case Summary Summary In generic case KdV equations govern collision of left- and right-propagating solitons, with corresponding phase shifts At critical plasma composition modified KdV equations are needed, with corresponding phase shifts, a case not addressed before For A ≷ 0 and B > 1/2 (no critical compositions): polarities are positive/negative, but phase shifts are equivalent to delays compared to single-soliton trajectory Criticality needs A = 0, but coefficient of nonlinear term in mKdV equations is positive, avoiding hypothetical case of supercriticality which has occasionally surfaced in literature... Comparison with recent experimental observations of two counter-propagating solitons of equal amplitude in a monolayer strongly coupled dusty plasma indicates qualitative agreement regarding delays occurring after interaction However, amplitude of overlapping solitons during collision was less than sum of initial soliton amplitudes, which cannot correctly be dealt with by available PLK formalism Survey of literature indicates that analytical treatment of more complicated plasma models is fully analogous, without qualitative changes ITCPS: Advanced Plasma Concepts, Faro, 24 September 2012 Head-on collisions of solitons in nonthermal plasmas