Following Ruback, we discuss the evidence for scattering at right angle of two vortices in a head-on collision. The evidence is given in terms of the approximate solutions of the equations of motion. This makes it possible to extend the analysis to the case of a small net repulsive force between the corresponding static vortex configurations.The ordinary differential equations, which result from the ansatz for the approximate solutions, are solved by Taylor series at the origin and by asymptotic series at infinity

Burzlaff, J.

Mc Carthy, P.

English

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A Study of a9Q0 VortexVortex Scattering ProcessJ. Burzlaff”2 and P. Mc Carthy.2AbstractFollowing Ruback, we discuss the evidence for scattering at right angle of two vorticesin a head-on collision. The evidence is given in terms of the approximate solutions of theequations of motion. This makes it possible to extend the analysis to the case of a smallnet repulsive force between the corresponding static vortex configurations.The ordinarydifferential equations, which result from the ansatz for the approximate solutions, aresolved by Taylor series at the origin and by asymptotic series at infinity.DIAS-STP-91-O7PACS numbers: 03.65.Ge, 11.1O.Lm, 11.15.-q‘Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland.2School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland.1 IntroductionOver the years, soliton and soliton-like solutions of nonlinear partial differential equationshave been studied in great detail. One of the most important results of these studieswas the discovery of the unusual behaviour of solitons in a scattering process. In recentyears, mainly based on an idea by Manton [1], results for the scattering of soliton-likeobjects, like magnetic monopoles [2], CF’ skyrmions [3] and cosmic strings or vortices [4]have been obtained. Important numerical work has also been done for example on cosmicstrings or vortices [5] and skyrmions in (2 + 1) dimensions [6] . We consider the work onthe scattering of vortices to be of particular importance because, unlike the other solitonlike objects mentioned, vortices can be produced in the laboratory and with conventionaltechniques [7], it may be possible to study their collisions experimentially.The theoretical predictions for the scattering of soliton-like objects are very exciting.The scattering of slowly moving monopoles, for example, shows an extremely rich structurewhich is partly due to inner degrees of freedom, internal phases. For static vortices theonly degrees of freedom are the positions of the vortices, and any unusual behaviour wouldhence be due to their soliton-like nature. Left-right symmetry in head-on collision wouldonly allow scattering at an angle of 0°, 90° or 180°. For slowly moving vortices at the pointbetween type I and type II superconductivity there is in fact analytic evidence for scatteringat right angle [4]. If the repulsion between the vortices increases and they cannot come veryclose anymore, we would expect to see a switch over to back scattering at a certain valueof the repulsion. There is numerical evidence [8] that for fixed repulsion an increase inthe velocity can bring the vortices close enough together again to produce scattering atright angles. Another parameter that could be changed is the angle between the vortices orcosmic strings, which are the objects one would have in mind in this type of problem. Whenwe turn the strings out of their parallel position we would expect to see not just scatteringbut intercommutation of the strings [5].Of the three parameters, strength of repulsion, velocity and relative angle, in this articlewe only change the first one. We review the evidence for scattering at right angle of slowly-moving vortices at values of the parameters for which the net force between static vorticesis zero. The ansatz used leads to ordinary differential equations which we solve by Taylorseries at the origin and by asymptotic series at infinity. When we turn on a small repulsionbetween the vortices we find that they still scatter at right angle, which, of course, isessential if we want to see 90° scattering experimentially.2 The Approximate SolutionThe Ginzburg-Landau model of a superconductor in a magnetic field in direction z is givenby the Lagrangian density£ = (D)(D --- (1)is the complex Riggs field, and D = 0— iAc and F1, = — OA in terms of thegauge potentials A,,u, v = 0, 1,2. The metric is g = diag(+1, —1, —1). The Euler-Lagrangeequations areDD + — 1) = 0,1+ - (D] =0. (2)For all A the Euler-Lagrange equations have static, finite energy n-vortex solutions ofthe form [9]cb(r,9) = e J’S)’ A0 = 0,A1(r, ) = —ExJa(r), i,j = 1,2, (3)wherer(rf’)’ — n2f(a — 1)2 — r2Af(f2 — 1) = 0,(4)andf(0) = a(0) = 0, roT)== 1. (5)In the special case A = 1, it can be shown [10] that the solutions actually satisfy thefirst-order Bogomolnyi equations [11], and f and a satisfyrf’—n(l—a)f= a1+f2_ 1=0. (6)In this case, there exists also a 2n-parameter family of static n-vortex solutions describingvortices located at arbitrary positions [10]. The reason for its existence is the fact that forA = 1 the net force between static vortices is zero.Now the scattering of slowly-moving vortices during the time from shortly before toshortly after the collision is studied. We setç(t, ) = ) + (t, ),A1(t,) = A;(f) + A(t,),A0(t,) = 0, (7)and assume that the solution is not too different from the configurations (3) of vorticessitting on top of each other. In fact, (, A1) is taken to be the solution (3) for A = 1 andn = 2, and(t, ) = s) + t),A(t,) = a1() + tB(f) (8)where A = 1 + ., 0 < \ << 1, t e (—s, ), E << 1,and ( + Ap, A1 + a) satisfies the staticequations of motion linearized in A. Under these assumptions it is possible to linearize theequations of motion (2) in , A, and A0. This leads to the equations- 2iA’D - + (I 12_1 + + + 2 -1) =0,+ A I 2 +[ñj - + - (ñj] =0,aa0A + [O0-ao] = 0, (9)2where D, = — iA and F:, = — 9jA.The solution (3) to the equations (2) are analytic in A [12], and can be expanded =+ A: + Aa1. Hence, (Ap,Aa) is a solution of the inhomogeneous system ofequations (9). The homogeneous system is the one which had to be solved in the caseA = 1. In this case, Ruback[4], extending work by Weinberg[13], found the solutions= 2(c + i3)e6f’(r),B1 + iB2 = — i)(rff” — rf’2 + ff’), (10)and= 2f(r)k(r),B1 + iB2 e[rk’(r) + 2k], (11)and= 2if(r)k(r),B1 + iB2 = e[rk’(r) + 2k], (12)where k(r) satisfies the equation+ (f2 + )k =0. (13)The solutions (10) lead to fields (‘ + + tB) which are of the form) + t) = e(+ d),A() + tB() = A(+ ) + OiX, (14)wherex = t( — )(cos9 — asin9) (15)and d = 2t(c,,3), to first order in t. Hence, these solutions describe overall translations.In the next section, we discuss (13) in detail and show that it has solutions with asymptotic behaviour k ce,c 0, at infinity and k “ cr2 +c2r,cj. 0, at the origin,and that all other non-zero solutions have asymptotic behaviour k c3e_’ +c4er,c 0,at infinity. Without loss of generality we can set c = 1. That c1 = 0 cannot hold can beseen as follows: Asymptotic behaviour e_r at infinity and c2 at the origin would implythe existence of a point r = r0 for which k(ro) > 0,k’(ro) = 0 and k”(ro) 0. This is notconsistent with (13). Therefore, k is strictly monotonic decreasing from infinity to zero,as r increases, and there exists a point r= p > 0 such that k(p) = 1/(2 I t ). For thesolution (11) we haveI= f2(l + 4tkcos29 + 4t2k) f2( —2 t k)2. (16)The zeros of the Higgs field, which give the locations where the magnetic field penetratesthe superconductor, are therefore at r= p, 9 = and 9 = for t > 0, and at r = p, 9 = 0and 9 = ir for t < 0. This solution describes 90° scattering.3Futhermore, for the solution (11) the energy density reads(r,9) (f2 — 1)2 + 4f2(i— a)2 + 8(I)2 + 16akt()2(a — 1) cos(2)+ 2t(k’f + (l — a))2 + 2tf(k’ + )2(l + 4ktcos(29) + (2kt))— cos(28)tf2(k’+ )[(a — 1) + 2kat cos(29) — tk cos(2&) + 2akt]— 4t2k’f(k’ + )sin2(2&) +f2kt cos(26)(2 cos(28)fkt + f2 1)+(f2 + 4ktf2 cos(29) + (2kft)2 — 1)2. (17)By investigating the terms, one finds that the energy for a function k is finite if k hasasymptotic behaviour k e at infinity, or infinite if k has asymptotic behaviour kat infinity. In Fig.1, the energy density (17) has been plotted for t = 1,t = 0 and t =respectively. The plot shows how the scattering process proceeds: As the two vorticesapproach each other, their energy densities form a cylinder-like structure from which twovortices emerge at right angle for t > 0.The arguments we gave for the solution (11) can easily be repeated to also show that thesolution (12) describes a 900 scattering process. We have therefore four linearly independent,gauge inequivalent approximate solutions whose superposition describes 90° scattering plustranslation. Since the parameter space for two static vortices is four- dimensional we do notexpect more solutions for slowly-moving vortices. No special initial conditions are thereforerequired in an experiment. Head-on collision of slowly-moving vortices should always leadto 90° scattering. An important point made in this section is that all the above argumentsholdforA=1+A> 1,A<< 1.Finally, we have to address the problem that the approximate solution for t e (—E,E)(which we used to discriminate against 0° and 180° scattering in favour of 90° scattering)is not a scattering solution. However, we can take the configuration for t = 0 as initialdata for a solution for t E which we know exists [14J. For t E (—E,E),E <<1, the linearization which leads to eqs.(9) should be justified. Therefore, the solutionswe discussed should be approximations for t E (—e,e) to the scattering solution for t E(—,c), although we have not rigously proven this. In fact, we are not aware of anyrigorous proof which establishes that any of the configurations discussed in the literatureare approximate solutions.3 Series SolutionsIn this section, we solve the equations (6) and (13) near the origin and use the equationsto find asymptotic expansiond near infinity. The technique used has been developed forthe Euler-Lagrange equations (4) [15]. First, we study the equations for small r. Theequation (6) can be rewritten asf(r) — 2r f a(s)f(s)dSa(r) = r2 - lfsf2(3)ds (18)4A Green’s function for the linear equation r(rk’)’ 4k isg(r,p) = H(r-p)(- 42)which leads to the integral equationk(r) + cr± jT(- p (19)Given the behaviour of f and a at the origin [9], where f 7012 and a r2, we can proveby induction that f, a and k are of the formf(r) = fr2,a(r) =k(r) = kr2, k0 = 0, (20)This leads to the following recurrence relations for n 21= 1 — 1= fnlfn26n, l+n2+1,7i ,fl2=1k— 4(n2— 1)(21)‘1 ,fl2,T131To prove the convergence of the Taylor series we show by induction thatI f (n± 1)2I a I 1)2’! k I (n± 1)2 (22)hold for sufficiently large n and M 1. The estimate for f, we need for the proof, is of theformI I + 1)2 (n— n1 + 1)2. (1 + .r)2(n — x + 1)24M’1 1 1 1 2n+13(n+2)2(’ (23)Using similar estimates for a,1 and twice for k,1 we can complete the induction proof forthe inequalities (22). These inequalities imply that the series (20) converge and solve (6)5and (13) for r < i//i7. Analyticity of the solutions discussed in section 2 then implies thatthey are included in the set of series solutions (20).For large r, f and a have the following asymptotic behaviour [9]:f = 1 +f1(r)e + F2(r),a = 1 + ai(r)e_r + A2(r), (24)where fi(r) and ai(r) are polynomially bounded, and F2(r) and A2(r) approach zero fasterthan rmer for any power of m. This implies that, to leading order, k satisfiesr2k” + rk’ — (,2 + 4)k = 0. (25)To this order,k(r) =c1H’(ir) +c2H(ir), (26)where are Hankel functions. Their asymptotic behaviour [16] and the finite energycondition require c2 = 0. Therefore,k(r) = ki(r)e_r + 1(2(r),(27)where ki(r) is polynoniially bounded and K2(r) approachs zero faster than rme_ for anypower of m.We now prove by induction thatf(r) = f(r)e =: F(r),a(r) = a(r)e =: ErAri(r) =:k(r) = k(r)e =: EKn(r), (28)where and k are polynomially bounded. Equations (6) imply for f = 1 + F anda=1+’rAF” + F’ — F = F2 + 4A2 + + 4A2F (29)andF+F—F = [ f12,+rl ,r1+ 4 á1a2ö2+——ç 1 —rr-I-1 ,TL2,rL31=:c(r)eT, (30)6ThereforeF(r) E j p[H(2)(ip)H(l)(ir) — (31)Substituting for a,(p) and calculating the integral leads to terms of the form f(r)e_nr,where f is polynomially bounded. Solutions of the homogeneous equation have the wrongasymptotic behaviour and cannot be added to (31).To prove that a and k are of the form (28) we use-icA(r)=p(p)edp, (32)where= —f — fn1f2n,z+n2• (33)fl1,T2=lOne also findsK(r) = j p[H2(ip)H’(ir) — (34)where= 2 + (35)flj,21 T1,Tl2,fl3=1To prove convergence of the series, we assume there exist numbers M and R such thatrirsupf>RIrf(r)e 2 < (n + 1)2’supr>ra(r)e 2 < (n + 1)2’ljn3Ztpr>Rjrk(r)e2I < (n+ 1)2’ (36)for large enough n. Taking R large enough, we can bound HS’2(ir)t by e and derivethe estimateSUpr>RITfN(T)e_N2I < v2_13upr>RIrQN(r)eI. (37)The induction hypothesis leads toVW’ 11 1SZLpr>RJrfN(r)eNn/21 <—(ni + 1)2 (V — n1 + 1)2N—rz1—2+ 2 (n1 + 1)2 (n2 + 1)2 N — n1 — n2 + 1)21(38)The inequality in (23) then completes the induction proof for f,(r). Similar arguments forã(r) and k(r) establish all inequalities (36). Thus, the series (28) converge for r > R andr > 2 log M.74 References1. N.S.Manton, Phys. Lett. 11OB (1982) 54.2. M.F. Atiyah and N.J. Hitchin, The Geometry and Dynamics of Magnetic Monopoles,Princeton Univ. Press 1988, and references therein.3. R.S. Ward, Phys. Lett. 158B (1985) 424.4. P.J. Ruback, Nuc. Phys. B296 (1988) 669.5. P. Laguna and R.A. Matzner, Phys. Rev. D 41 (1989) 1751, and references therein.6. M. Peyrard, B. Piette and W.J. Zakrzewski, University of Durham, Preprints DTP90/37, DTP-90/39, and references therein.7. R.P. Huebener, Magnetic Flux Structures in Superconductors, Springer 1979.8. R.A. Matzner, Comp. Phys. 2 (1988) 51.9. B. Plohr, Princeton University Doctoral Dissertation, October 1980; J. Math. Phys.(1981) 2184.10. A. Jaffe and C. Taubes, Vortices and Monopoles, Birkhäuser 1980.11. E.B. Bogomolnyi, Yad. Fiz. 24. (1976) 861 [Soy. J. Nuci. Phys. 24 (1976) 449].12. S.-N. Chow and J.K. Hale, Methods of Bifurcation Theory, Springer 1982.13. E.J. Weinberg, Phys. Rev. D j (1979) 3008.14. J.Burzlaff and V. Moncrief, J. Math. Phys. . (1985) 1368.15. W. Rühl, University of Kaisersiantern, unpublished report, 1977.16. M. Abromowitz and l.A. Stegun (ed.) Handbook of Mathematical Functions, Dover1965.Figure CaptionFigure 1: The energy density (17) for t = , t = 0 and t =8EjE.8j.4t=1/2t055—5//;;5/;;/_Figure1(i)Figure1(ii)(Tr!)I:31niTx

A Study of a 90° Vortex-Vortex Scattering Process

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A Study of a 90° Vortex-Vortex Scattering Process

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