On the Born-Infeld Abelian Higgs cosmic strings with symmetric vacuum

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1 On the Born-Infeld Abelian Higgs cosmic strings with symmetric vacuum Dongho Chae Department of Mathematics Sungkyunkwan University Suwon , Korea Abstract We study a system of elliptic equations from the Abelian Born- Infeld system coupled with the Einstein equations under the boundary condition of the symmetric vacuum(nontopological type). When the total string number satisfies 1 N < 1 4πG, where G is the gravitational constant, we construct a family of solutions to the system. The qualitative properties of the solutions are quite different from the solutions with the boundary condition of the broken vacuum symmetry. 1 Introduction We consider the following problem for (u, η): e η (e u 1) u = + 4π n j δ(z z j ), 1 1b (e u 1) (P ) [η + 8πG(e u u)] = 3π G n j δ(z z j ), e u and e η as z, where we denoted z = x 1 + ix C = R, and G > is the gravitational constant. The problem (P ) represent physically the equilibrium configuration of the Abelian Born-Infeld cosmic strings. More precisely, it generates 1

2 a static Einstein equations coupled with the Abelian Born-Infeld electrodynamic equations under the assumption of translation symmetry in one spatial direction(say in x 3 direction). For more details on the derivation of the above system starting from the Lagrangian of the self-gravitating Abelian Born-Infeld theory by applying the Bogoml nyi method combined with the Taubes reduction argument[17], as well as the physical backgrounds, we refer to [3]. Parenthetically, we note that the Born-Infeld field theory, originated in [], has now become a very hot issue of research in the theoretical physics, mainly due to its natural connection to the superstring theory(e.g. [1, 18], and so many articles in the LANL archive). Also in the physics of cosmology there are many studies on the cosmic strings(see e.g.[9, 11, 19, ] and references therein). The Abelian Born-Infeld cosmic string model incorporates the Born-Infeld field theory into the area of cosmic string theories. The boundary condition in (P ), in particular, means that we are assuming symmetric vacuum near infinity. The above system with the different boundary condition, namely with the broken vacuum symmetry boundary condition, e u 1 and e η as z, as well as the similar problem on a compact surface have been studied extensively by Y. Yang[, 3, 1]. Our boundary condition above is different from that of those studies. We remark that this symmetric vacuum boundary condition is not allowed in order to generate finite energy solution of the system in the case of flat(minkowskian) Abelian Born-Infeld theory[3]. On the other hand, our symmetric vacuum boundary condition resembles the nontopological boundary condition in the Chern-Simons Higgs and the related theories studied in [15, 8, 4, 5, 6, 3](The periodic version of the nontopological solutions are studied in [16, 14, 13]). Our aim in this paper is to construct solutions of (P ). Moreover, our construction provides very precise information of the qualitative properties of solutions, including the asymptotes of u, η near infinity. The following is our main Theorem. Theorem 1.1 Suppose {n j } m N {}, and {z j } m R be given. We set N = m n j. Assume 1 N < 1 4πG, (1.1) and b > 1. Then, there exists a constant ε 1 > such that for any ε (, ε 1 ) there exists a family of solutions to (P ), (u 1, u ). Moreover, the solutions we constructed have the following representations: u(z) = ln ρ I ε,δ ε (z) + ε w 1 (ε z ) + ε u 1,ε(εz), (1.) η(z) = ln ρ II ε,δ ε (z) + ε w (ε z ) + ε u,ε(εz) (1.3)

3 with ρ I δ,ε(z) = εn+ m z z j n j, (1.4) (1 + εz + δ ) a ρ II δ,ε(z) = 4ε aλ 1 (1 + εz + δ ), δ = δ 1 + iδ C, (1.5) where and hereafter we denote a = 8πG. (1.6) In (1.) and (1.3), the function ε δ ε is a continuous function in a neighborhood of, and δ ε as ε. The radial functions w 1, w have the following asymptotic behaviors. w 1 ( z ) = C 1 ln z + O(1), (1.7) w ( z ) = C ln z + O(1) (1.8) as z with the constants C 1, C defined by C 1 = 8[(a + 1)λ 1 + λ ]N!(1 an) a λ ( 1 k=1 N + k) a (1.9) C = 8[(a + 1)λ 1 + λ ]N!(1 an) aλ ( 1 k=1 N 1). a (1.1) The functions u 1,ε, u,ε satisfy sup z R u 1,ε(εz) + u,ε(εz) ln(e + z ) o(1) as ε. (1.11) Remark 1.1: From the second equation of (P ) we obtain [ ] η + 8πG(e u u) + 8πG n j ln z z j =. Hence, η = 8πG(u e u ) 8πG n j ln z z j + h(z), where h(z) is a harmonic function. Under the boundary conditions for u and η, we can assume h(z) c for a constant c. Hence, ( m 16πG e η = c z z j j) n e 8πG(u eu). 3

4 Thus, substituting the asymptotic formula from (1.) [ u(z) = N 4 ] a C 1ε + o(ε ) ln z + O(1) as z and ε we obtain e η(z) = O ( ) z 4 C ε +o(ε ) as z and ε, (1.1) which is consistent with (1.3), and shows extremely weak dependence(via C 1 ) on the total vortex number N of decay properties near infinities of the conformal factor e η. This is in contrast with the case of broken vacuum symmetry boundary condition, e u 1 and e η as z [3], where we have the decay of the conformal factor, e η(z) = O( z 16πGN ) as z, which shows strong dependence of the decay on the total string number N. Remark 1.: We recall the result in Section 1.5 of [3] that the surface M = (R, e η δ jk ) is complete if and only if e R 1 η dx =. According to the representation formula (1.3), this, in turn, is equivalent to (1 + r) 1 C ε +o(ε ) dr =. We observe, however, from (1.1) that C > (< ) if an < (>)1. Thus we conclude that the surface M = (R, e η δ jk ) is complete(incomplete) if an < (>)1. Since we do not know the sign of v,ε, the case an = 1 is inconclusive for the completeness of M. After this work is completed the author find that there is an interesting paper by F. Lin and Y. Yang[1] on the sigma model coupled with the Born-Infeld system and the gravitation, for which the reduced semilinear elliptic system is different from that of (P), and the corresponding mathematical analysis is completely different from ours in the next section. Proof of the Main Theorem In (P ) the second equation added to the first equation times a = 8πG gives (η + ae u ) = aeη (e u 1) 1 1b (e u 1). (.1) 4

5 Replacing the second equation of (P ) by (.1), we obtain the following equivalent system to (P ). u = e η (e u 1) 1 1b (e u 1) + 4π n j δ(z z j ), (.) (η + ae u ) = aeη (e u 1) 1 1b (e u 1). (.3) We consider the following principal part of the system, (.)-(.3). u = λ 1 e η + 4π n j δ(z z j ), (.4) η = aλ 1 e η, (.5) where λ 1 is defined in (1.6). As a family of solution (.5) we have η (z) = ln ρ II δ,ε(z) (.6) with ρ II δ,ε (z) defined in (1.5). In order to solve (.4) we rewrite it as ( ) au a n j ln c z z j = aλ 1 e η, (.7) where c is an arbitrary positive constant. Comparing (.7) with (.5), we find that au a n j ln c z z j = η + h(z) (.8) for a harmonic function, h(z). We choose h(z). Then, substituting η in (.6) into (.8), and solving it for u, and choosing the constant c in as we find that c = εn 8 (aλ 1) 1 a, u (z) = ln ρ I δ,ε(z), (.9) with ρ I δ,ε (z) defined in (1.4). We set gδ,ε(z) I = 1 ( z ε ρi δ,ε, g ε) δ,ε(z) II = 1 ( z ε ρii δ,ε, ε) and define ρ 1 (r), ρ )r) by ρ 1 (r) = r N (1 + r ) a = lim gδ,ε(z), I ρ (r) = ε+ δ 5 8 aλ 1 (1 + r ) = lim gδ,ε(z). II ε+ δ

6 We make transforms from (u, η) to (u 1, u ) as follows u(z) = ln ρ I δ,ε(z) + ε w 1 (εz) + ε u 1 (εz) η(z) = ln ρ II δ,ε(z) + ε w (εz) + ε u (εz) (.1) where w 1, w are the radial functions, w j (z) = w j ( z ), j = 1, to be determined below. Then, (.)-(.3) can be transformed into the functional equation, P = (P 1, P ) =, where P 1 (u 1, u, δ, ε) = u 1 gi δ,ε (z)gii (u 1 +u +w 1 +w δ,ε ) (z)eε gii δ,ε (z) e ε (u +w ) ε λ1gii δ,ε (z) + w 1 1 [ε b g I δ,ε (z)eε (u 1 +w 1) 1] ε 1, [ ] P (u 1, u, δ, ε) = u + agδ,ε(z)e I ε (u 1 +w 1 ) agi δ,ε (z)gii (u 1 +u +w 1 +w δ,ε ) (z)eε agii δ,ε (z) e ε (u +w ) ε 1 1 [ε b g I δ,ε (z)eε (u 1 +w 1) 1] aλ 1gII δ,ε (z) ε + w. Now we introduce the functions spaces used in [4]. Let us fix α (, 1) throughout this paper. Following [1], we introduce the Banach spaces X α and Y α as X α = {u L loc(r ) (1 + x R +α ) u(x) dx < } equipped with the norm u X α = R (1 + x +α ) u(x) dx, and Y α = {u W, loc (R ) u X α + u(x) 1 + x 1+ α L (R ) < } equipped with the norm u Y α = u X α + u(x) 1+ x 1+ α L (R. We recall the ) following propositions proved in [4]. Proposition.1 Let Y α be the function space introduced above. Then we have the followings. (i) If v Y α is a harmonic function, then v constant. (ii) There exists a constant C 1 > such that for all v Y α v(x) C 1 v Yα ln(e + x ), x R. 6

7 Proposition. Let α (, 1 ), and let us set where ρ = 8 (1+r ). We have where we denoted ϕ + = Moreover, we have L = + ρ : Y α X α, KerL = Span {ϕ +, ϕ, ϕ }, (.11) r 1 + r cos θ, ϕ = r 1 + r sin θ, ϕ = 1 r 1 + r. ImL = {f X α fϕ R ± = }. (.1) We can check easily that P is a well defined continuous mapping from B ε (Y α ) C R + into (X α ), where B ε = { u 1 Yα + u Yα + δ ε < ε } for sufficiently small ε. In order to have gδ,ε I (z) O(1) as z we impose an <, which is equivalent to (1.1). We now extend continuously P (,,, ε) to ε = by imposing the condition that lim ε P (,,, ε) =. In order to compute the limit lim ε P (,,, ε) we note the fact 1 1b (x 1) λ 1 = 1 1b (x 1) 1 1 b = λ x + O(x ) (.13) as x, where λ is defined in (1.6). Using this fact we obtain and lim P 1(,,, ε) = (λ 1 + λ )ρ 1 ρ + λ 1 ρ w + w 1, ε lim P (,,, ε) = a ρ 1 a(λ 1 + λ )ρ 1 ρ + aλ 1 ρ w + w. ε Hence, the condition lim ε P (,,, ε) = implies the following linear system for w 1, w. w 1 + λ 1 ρ w (λ 1 + λ )ρ 1 ρ =, (.14) w + aλ 1 ρ w a(λ 1 + λ )ρ 1 ρ + a ρ 1 =. (.15) We establish the following lemma about asymptotic behaviors of the solutions w 1, w Y α. 7

8 Lemma.1 Let C 1, C be the numbers introduced in (1.9), (1.1) respectively. Then, there exist radial solutions w 1 ( z)), w ( z ) of (.14)-(.15), belong to Y α, and satisfy the asymptotic formula in (1.7) and (1.8) respectively. Proof: From (.14) a (.15) we obtain (aw 1 w aρ 1 ) =. We seek w 1, w with aw 1 w aρ 1 Y α. Then, it follows that aw 1 w aρ 1 =constant by ([1], Proposition 1.1). We choose this constant=. Then, ρ w = aρ w 1 aρ 1 ρ. Substituting this into (.14) we obtain the following reduced system for w 1, w. w 1 + aλ 1 ρ w 1 = [(a + 1)λ 1 + λ ]ρ 1 ρ, (.16) w = aw 1 aρ 1. (.17) Let us set f(r) = [(a + 1)λ 1 + λ ]ρ 1 ρ. Then, it is found in [1, 4] that the ordinary differential equation(with respect to r), (.16) has a solution w 1 (r) Y α given by with { r w 1 (r) = ϕ (r) ( ) 1 + r (1 r) φ f (r) := 1 r r φ f (s) φ f (1) ds + φ f(1)r (1 s) 1 r r } ϕ (t)tf(t)dt, (.18) where φ f (1) and w 1 (1) are defined as limits of φ f (r) and w 1 (r) as r 1. From the formula (.18) we find that r w 1 (r) = ϕ (r) as r, where ( ) 1 + s I(s) ds + (bounded function of r) (.19) 1 s s I(s) = [(a + 1)λ 1 + λ ] s ϕ (t)tρ 1 (t)ρ (t)dt. Since ϕ (r) 1 as r, (1.7) follows if we show I = I( ) = [(a + 1)λ 1 + λ ] = 8[(a + 1)λ 1 + λ ]N!(1 an) a λ ( 1 k=1 N + k) = C 1. a ϕ (r)rρ 1 (r)ρ (r)dr 8

9 Indeed, substituting r = t in the integrand of I, we have I = 4[(a + 1)λ 1 + λ ] aλ 1 = 4[(a + 1)λ 1 + λ ] aλ 1 = 4[(a + 1)λ 1 + λ ] aλ 1 (1 t)t N dt (1 + t) 3+ a [ t N ] t N+1 dt dt (1 + t) 3+ a (1 + t) 3+ a [ ] N! ( + k) (N + 1)! ( + k) a a k= N k=1 N = 4[(a + 1)λ [ ] 1 + λ ]N! aλ ( 1 k=1 N + k) a + 1 N (N + 1) a = 8[(a + 1)λ 1 + λ ]N!(1 an) a λ ( 1 k=1 N + k) = C 1. (.) a The formula (1.8), on the other hand, follows from (.18), observing C = ac 1, since ρ 1 (r) = O(1) as r. This completes the proof of Lemma.1. Now we compute the linearized operator of P. By direct computation we have gδ,ε I lim (z) = 4 ε δ 1 a ρ gδ,ε I 1ϕ +, lim (z) = 4 ε δ a ρ 1ϕ, δ= δ= gδ,ε II lim (z) = 4ρ ϕ +, ε δ 1 δ= gδ,ε II lim (z) ε δ = 4ρ ϕ. δ= Let us set P u 1,u,δ(,,, ) = A. Then, using the above preliminary computations and.13, we obtain and A 1 [v 1, v, β] = v 1 + λ 1 ρ v +4 [(1 + 1a ] )(λ 1 + λ )ρ 1 ρ λ 1 ρ w (ϕ + β 1 + ϕ β ), A [v 1, v, β] = v + aλ 1 ρ v +4 [(1 + a)(λ 1 + λ )ρ 1 ρ aλ 1 ρ w ] (ϕ + β 1 + ϕ β ) 4 [ρ 1 (ϕ + β 1 + ϕ β )]. For the linearized operator A[ ] we will establish the following key lemma. 9

10 Lemma. The operator A : Y α R X α defined above is onto. Moreover, the kernel of A is given by KerA = Span{(1, ); ( ϕ ± a, ϕ ±); ( ϕ a, ϕ )} {(, )}. (.1) Thus, if we decompose Y α R = U α KerA, where we set U α = (KerA), then A is an isomorphism from U α onto X α. In order to prove the above lemma we need the following: Proposition.3 Let w Y α solve (.14)-(.15), then I ± = [(1 + a)(λ R 1 + λ )ρ 1 ρ aλ 1 ρ w ] ϕ +dx (ρ R 1 ϕ + )ϕ ± dx. (.) Proof: Integrating by part, we obtain { } I ± = [(a + 1)(λ1 + λ R )ρ 1 ρ aλ 1 w ρ ]ϕ ± ρ 1 ϕ ± ϕ ± dx = [((a + 1)λ R 1 + (a + 1)λ )ρ 1 ρ aλ 1 w ρ ]ϕ ±dx, (.3) where we used (.1 ϕ ± = aλ 1 ρ ϕ ±. (Note that L = + aλ 1 ρ.) Below we list useful formulas, which can be checked by elementary computations. ϕ ±ρ = 1 { } cos 16 L ρ θ sin, (.4) θ Also, from (.15), we have ϕ ± = aλ { 1 cos 8 r ρ θ sin θ }, (.5) ρ = aλ 1 (r 1)ρ, (.6) Lw = a(λ 1 + λ )ρ 1 ρ a ρ 1. (.7) Using (.4)-(.7), and integrating by parts, we transform the integral suc- 1

11 cessively as follows. I ± = [(a + 1)λ R 1 + (a + 1)λ ]ρ 1 ρ ϕ ±dx a π { } cos w (L ρ ) θ 16 sin dθrdr θ π { aλ1 [(a + 1)λ 1 + (a + 1)λ ] = r ρ 1 ρ aλ } { } 1 cos 8 16 (Lw )ρ θ sin dθrdr θ { aλ1 [(a + 1)λ 1 + (a + 1)λ ] = π r ρ 1 ρ aλ } [a(λ 1 + λ )ρ 1 ρ a ρ 1 ]ρ rdr { } aλ1 [(a + 1)λ 1 + (a + 1)λ ] = π r ρ 1 ρ a λ 1 (λ 1 + λ ) ρ 1 ρ + a λ ρ 1 ρ rdr { aλ1 [(a + 1)λ 1 + (a + 1)λ ] = π r ρ 1 ρ a λ 1 (λ 1 + λ ) ρ 1 ρ 8 16 } + a3 λ 1 16 (r 1)ρ 1 ρ rdr = λ 1a[λ 1 (a + 1) + λ ]π 16 = 4[λ 1(a + 1) + λ ]π aλ 1 = [λ 1(a + 1) + λ ]π aλ 1 = [λ 1(a + 1) + λ ]π aλ 1 [ [(a + 1)r a]ρ 1 ρ rdr [(a + 1)r a]r N+1 dr (Setting r = t) (1 + r ) a +4 ] at N dt (1 + t) a +4 [ (a + 1)t N+1 (1 + t) a +4 (a + 1)(N + 1)! 3 k= N = [λ 1(a + 1) + λ ]πn! aλ 3 ( [(a 1 k= N + k) + 1)(N + 1) a a = [λ 1(a + 1) + λ ]π(3a + )N N! aλ 1 3 k= N This completes the proof of Proposition.3.. ( a We are now ready to prove Lemma 3.1. ] ( + k) an! ( + k) a a 3 k=3 N ( )] a + N + k) >. (.8) Proof of Lemma.: Given (f 1, f ) X α, we want first to show that there exists (v, η) Y α, β = (β 1, β ) R such that which can be rewritten as A(v 1, v, β) = (f 1, f ), (.9) v 1 + λ 1 ρ v +4 [(1 + 1a ] )(λ 1 + λ )ρ 1 ρ λ 1 ρ w (ϕ + β 1 + ϕ β ) = f 1, (.3) 11

12 and Let us set v + aλ 1 ρ v + 4 [(1 + a)(λ 1 + λ )ρ 1 ρ aλ 1 ρ w ] (ϕ + β 1 + ϕ β ) 4 [ρ 1 (ϕ + β 1 + ϕ β )] = f. (.31) β 1 = 1 f 4I + R ϕ + dx, β = 1 f 4I R ϕ dx, (.3) where I ± > is defined in (.). We introduce f by f = f β 1 ϕ + β ϕ. (.33) Using the fact π ϕ + ϕ dθ =, we find easily R f ϕ ± dx =. (.34) Hence, by (.1) there exists v Y α such that v + aλ 1 ρ v = f. Thus we have found (v, β 1, β ) Y α R satisfying (.31). Given such (v, β 1, β ), in order to construct v 1 Y α satisfying (.3), we consider the following equation, obtained by (.3) a (.31), (av 1 v + 4ρ 1 ϕ + β 1 + 4ρ 1 ϕ β ) = af 1 f. (.35) For any harmonic function h 1 (x) the function v 1 (x) = 1 ln( x y )(af πa R 1 (y) f (y))dy + 1 a (v 4ρ 1 ϕ + β 1 4ρ 1 ϕ β ) + h 1 (x) (.36) satisfies (.3). The requirement v 1 Y α implies h 1 (x)(x) Constant thanks to Proposition.1(i).. We have just finished the proof that A : Y α R X α is onto. Now it is easy to check that the restricted operator(denoted by the same symbol), A : U α X α is one to one. We omit the details. This completes the proof of the lemma. We are now ready to prove our main theorem. Proof of Theorem 1.1: Lemma. shows that P (v,ξ,β) (,,, ) : U α X α X α is an isomorphism for α (, 1 ). Then, the standard implicit function theorem(see e.g. [4]), applied to the functional P : U α ( ε, ε ) X α X α, implies that there exists a constant ε 1 (, ε ) and a continuous function ε ψε := (v1,ε, v,ε, δε) from (, ε 1 ) into a neighborhood of in U α such that P (u 1,ε, u,ε, δε, ε) = (, ), for all ε (, ε 1 ). 1

13 This completes the proof of Theorem 1.1. The representation of solutions u 1, u, and the explicit form of ρ I ε,δ (z), ε ρii ε,δ (z),, together with the asymptotic behaviors of w 1, w described in Lemma.1, the fact that u 1,ε, u,ε Y α, ε combined with Proposition.1, implies that the solutions satisfy the boundary condition in (P ). Now, from Proposition.1 we obtain that for each j = 1,, u j,ε(x) C u j,ε Yα (ln + x + 1) C ψ ε Uα (ln + x + 1). (.37) This implies then u j,ε(εx) C ψ ε Uα (ln + εx + 1) C ψ ε Uα (ln + x + 1). From the continuity of the function ε ψ ε from (, ε ) into U α and the fact ψ = we have ψ ε Uα as ε. (.38) The proof of (1.11) follows from (.37) combined with (.38). This completes the proof of Theorem 1.1 Acknowledgements This work was supported by Korea Research Foundation Grant KRF-- 15-CS3. References [1] S. Baraket and F. Pacard, Construction of singular limits for a semilinear elliptic equation in dimension, Calc. Var. and P.D.E.,6, (1998), pp [] M. Born and L. Infeld, Foundation of the new field theory, Proc. Roy. Soc. A, 144, (1934), pp [3] H. Chan, C. C. Fu, and C. S. Lin, Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation Comm. Math. Phys. 31 (), no., pp [4] D. Chae and O. Yu Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys. 15, (), pp [5] D. Chae and O. Yu Imanuvilov, Non-topological solitons in a generalized self-dual Chern-Simons theory, Calc. Var. P.D.E., 16,(3), pp [6] D. Chae and O. Yu Imanuvilov, Non-topological multivortex solutions to the self-dual Maxwell-Chern-Simons-Higgs systems, J. Funct. Anal., 196, (), pp

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