Research Cite this article: Han X. 2016 The Born–Infeld vortices induced from a generalized Higgs mechanism. Proc. R. Soc. A 472: 20160012. http://dx.doi.org/10.1098/rspa.2016.0012 Received: 5 January 2016 Accepted: 16 March 2016

Subject Areas: mathematical physics Keywords: Born–Infeld electromagnetism, gauge fields, Higgs mechanism, self-dual vortex equations, nonlinear elliptic equations Author for correspondence: Xiaosen Han e-mail: [email protected]

The Born–Infeld vortices induced from a generalized Higgs mechanism Xiaosen Han Institute of Contemporary Mathematics, School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, People’s Republic of China XH, 0000-0001-7665-8421 We construct self-dual Born–Infeld vortices induced from a generalized Higgs mechanism. Two specific models of the theory are of focused interest where the Higgs potential is either of a |φ|4 - or |φ|6 -type. For the |φ|4 -model, we obtain a sharp existence and uniqueness theorem for doubly periodic and planar vortices. For doubly periodic solutions, a necessary and sufficient condition for the existence is explicitly derived in terms of the vortex number, the Born– Infeld parameter, and the size of the periodic lattice domain. For the |φ|6 -model, we show that both topological and non-topological vortices are present. This new phenomenon distinguishes the model from the classical Born–Infeld–Higgs theory studied earlier in the literature. A series of results regarding doubly periodic, topological, and non-topological vortices in the |φ|6 -model are also established.

1. Introduction This work concerns constructing multivortex solutions in the Born–Infeld geometric electromagnetic theory proposed initially to accommodate a finite-energy point charge, modelling the electron, and revived in contemporary theoretical physics owing to its relevance in superstring theory. The complicated structure of the Born–Infeld interaction makes it difficult in general to rigorously construct solutions to the governing equations and an effective approach to gain insight of the solutions has been to explore self-dual reductions, as in the classical gauge field theory. Our study aims at obtaining some rich families of topologically classified static vortex-like solutions in the context of the Born– Infeld theory coupled with a complex Higgs scalar particle whose potential permits self-dual reductions. 2016 The Author(s) Published by the Royal Society. All rights reserved.

In the classical Maxwell theory, electromagnetic interaction over the standard fourdimensional Minkowski space–time is governed by the Lagrangian action density (1.1)

an applied external current density vector and

Fμν = ∂μ Aν − ∂ν Aμ

(1.2)

is the electromagnetic field induced from Aμ . Here and in the following, we observe the summation convention over repeated lower and upper temporal and spatial coordinate indices converted back and forth by the Minkowski metric (gμν ) = (gμν ) = diag{1, −1, −1, −1}. It is well known that the Maxwell theory does not accommodate a finite-energy point-charge which presents an outstanding puzzle for theoretical physicists in modelling particle-like entities such as electrons. More precisely, the electrostatic field generated by a point-charge in the Maxwell theory carries infinite energy. In order to tackle this puzzle and inspired by the transition from the classical mechanics to relativistic mechanics, Born & Infeld [1,2] proposed their celebrated nonlinear theory of electromagnetism with the Lagrangian action density given by 1 2 μν , (1.3) LBI = b 1 − 1 + 2 Fμν F 2b where b > 0 is a suitable parameter, often called the Born–Infeld parameter, characterizing the upper bound of the electromagnetic fields. It is seen that in the large b limit (1.3) recovers the standard Maxwell theory defined by (1.1) with jμ = 0. Over the last two decades or so the Born– Infeld theory (Dirac later adapted the idea of Born–Infeld to come up with an extensible fieldtheoretical model for electron [3], so that in the literature the model is also widely referred to as the Dirac–Born–Infeld model) has been revived because it is shown to appear in supersymmetric field theory and superstring theory [4–9] and has become an actively pursued research topic in modern physics [10–28]. The string theory connection of the Born–Infeld theory mentioned here is referred to the derivation of the Born–Infeld electromagnetic action term in the context of string theory worked out in [29]. In the context of the Abelian Higgs theory, the Maxwell term is modified in the spirit of the Born–Infeld electromagnetism, so that the Lagrangian action density reads 1 2 μν (1.4) + Dμ φDμ φ + W(|φ|2 ), LBIH = b 1 − 1 + 2 Fμν F 2b where Fμν is defined by (1.2), Dμ φ = ∂μ φ − iAμ φ the gauge-covariant derivative, φ a complexvalued scalar field called the Higgs field, and W(·) the Higgs potential which assumes a form, so that there is a spontaneously broken symmetry. The motivation of considering a Higgs field is that it acts as a matter source to generate magnetism as in the Ginzburg–Landau theory of superconductivity [30]. In [31], it is shown that when the potential W(·) in (1.4) takes the form ⎛ ⎞ 2 − 1)2 (|φ| ⎠, (1.5) W(|φ|2 ) = b2 ⎝1 − 1 − b2 there admits a self-dual or BPS (after the pioneering work of Bogomol’nyi [32] and Prasad– Sommerfield [33]) reduction for (1.4) with and without gravitation [31,34], for which cosmic string solutions [35–37] are constructed in [34]. Subsequently, Lin & Yang [38] studied harmonic maps coupled with an Abelian gauge field governed by the Born–Infeld electromagnetic theory with and without gravitation and established the coexistence of vortices and antivortices, and cosmicstrings and antistrings, extending the results in [34,39,40]. Note that there are several interesting studies of BPS structures in the supersymmetric Born–Infeld–Higgs theories. See for example, [41–43]. It is shown that in these studies the highly complicated supersymmetric Born– Infeld–Higgs equations of the BPS types allow a dramatic reduction into those of their classical

...................................................

where Aμ is the usual gauge potential vector,

jμ

rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160012

LMaxwell = − 14 Fμν Fμν + Aμ jμ ,

2

(1.6)

can be recovered from taking the large b limit in the model (1.4)–(1.5). Thus the Born–Infeld–Higgs theory [31] consisting of (1.4)–(1.5) promises a richer collection of phenomena. For example, it has been shown in [34] that the presence of the Born–Infeld interaction leads to enhanced Bradlow bounds [47,48] for permissible vortex numbers and much relaxed topological obstructions to the existence of cosmic strings with geodesically complete gravitational metrics. Recently, a generalized Born–Infeld–Higgs theory was developed in [49] with an expanded interaction pattern in which the Maxwell term contains a positive Higgs-field-dependent factor. The study of existence of topological defects in such generalized effective field theories [50–54] is of great interest, owing to its various applications in different branches of physics [55–59]. For example, in the classical Abelian Higgs theory context, such an extended interaction dynamics was first explored by Bekenstein [60] in his investigation of the fine-structure constant problem. Later, Bekenstein’s formulation has found applications in cosmology [61,62], in particular, in search of quintessence [63], and in electroweak theory [64] and monopole construction [65]. Our study on the generalized Born–Infeld–Higgs theory [49] in this paper is inspired by those profound applications. In the generalized Born–Infeld–Higgs theory introduced in [49], two prototype BPS models, called the |φ|4 - and |φ|6 -models, respectively, are presented, and the vortex solutions of the governing equations are obtained by numerical methods under radially symmetric assumption. In this work, we first re-derive these governing equation without assuming radial symmetry and then present a complete mathematical existence theory for the vortex solutions of these equations. Besides a series of sharp existence theorems, we also obtain both topological and non-topological solutions in the |φ|6 -model, which is a new feature absent in the standard Born–Infeld–Higgs model [31,34]. The rest of our paper is organized as follows. In §2, we review the generalized Born–Infeld model introduced in [49] and derive the corresponding BPS equations for which we remove the radial restrictions of the field configurations in [49]. In §3, we establish a sharp existence and uniqueness theorem for the |φ|4 -model in both planar case and doubly periodic case, whereas in the former case an explicitly necessary and sufficient condition for existence of vortex solutions spells out in terms of the vortex numbers, Born–Infeld parameter and the size of the domain. Three existence theorems for the |φ|6 -model are established in §4. Section 4a is devoted to the proof of the existence of doubly periodic vortices, for which a necessary condition and a sufficient condition are presented separately. Proofs of the existence of topological and non-topological solutions are carried out in §4b,c by using a monotone iteration and a shooting argument, respectively. In the last section, we make a summary of this work.

2. Generalized Born–Infeld–Higgs model In this section, we review the generalized Born–Infeld–Higgs model proposed in [49] and derive the corresponding BPS equations without radial ansatz for the field configurations. The Lagrangian of the general Born–Infeld–Higgs theory in the (2 + 1)-dimension introduced by [49] reads LGBIH = b2 (1 − F ) + w(|φ|2 )Dμ φDμ φ − W(|φ|2 ) with

(2.1)

F≡

1+

G(|φ|2 ) Fμν Fμν 2b2

(2.2)

...................................................

LAH = − 14 Fμν Fμν + Dμ φDμ φ − 12 (|φ|2 − 1)2

3

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Maxwell theory counterparts, for which the resulted equations are already well understood through the work in [44,45]. It should be noted that the standard well-studied Abelian Higgs theory [44,46] governed by the Lagrangian action density

and W(|φ|2 ) ≡ b2 (1 − V(|φ|2 )),

(2.3)

Dμ (wDμ φ) +

Fμν Fμν ∂G ∂W ∂w Dμ φDμ φ + − = 0, ¯ 2F ∂ φ¯ ∂φ ∂ φ¯

(2.5)

¯ μ φ) is the current density. where Jμ ≡ iw(|φ|2 )(φDμ φ − φD The energy–momentum tensor for the model (2.1) takes the form Tμν = −

1 β F Fβν + w(|φ|2 )(Dμ φDν φ + Dν φDμ φ) − gμν LGBIH . F μ

(2.6)

In the static case and the temporal gauge(A0 = 0), the energy density is E = T00 = b2 (F − 1) + w(|φ|2 )|Dj φ|2 + W(|φ|2 ) = b2 (F − V(|φ|2 )) + w(|φ|2 )|Dj φ|2 ,

(2.7)

here and in what follows, we understand that F=

1+

G(|φ|2 ) 2 F12 . b2

(2.8)

To consider the model (2.1) in a more general setting, we remove the radial ansatz for the field configurations in [49] and carry out a BPS reduction for this model. In the following, we will take a special function V(·) of the form (U(|φ|2 ))2 2 , (2.9) V(|φ| ) = 1 − b2 where the function U(·) is to be determined later. Note the identity |Di φ|2 = |D1 φ ± iD2 φ|2 ± i(D1 φD2 φ − D1 φD2 φ).

(2.10)

Then, the energy density can be rewritten as

G FU 2 G 2 F U2 √ F12 ± √ F12 − ∓ GUF12 E= − 2F 2F 2 G +

b2 b2 F 2 b2 (F V − 1)2 − V − + b2 V 2F 2 2F

+ b2 (F − V) + w(|φ|2 )|Dj φ|2

G FU 2 b2 F12 ± √ (F V − 1)2 + w(|φ|2 )|D1 φ ± iD2 φ|2 = + 2F 2F G √ ∓ GUF12 ± iw(|φ|2 )[D1 φD2 φ − D1 φD2 φ].

(2.11)

To achieve a BPS reduction, later the functions G(·), w(·), U(·) will be chosen suitably such that d ( G(t)U(t)) = w(t). dt

(2.12)

...................................................

and

rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160012

where Fμν = ∂μ Aν − ∂ν Aμ is the electromagnetic field induced from the gauge field Aμ , Dμ φ = ∂μ φ − iAμ φ is the gauge-covariant derivative, G(|φ|2 ) and w(|φ|2 ) are positive functions, the generalized potential W(|φ|2 ) is non-negative with 0 < V(|φ|2 ) ≤ 1, b > 0 is the Born–Infeld parameter. The Euler–Lagrange equations of the model (2.1) read

G νμ F = Jμ (2.4) ∂ν F

4

Let H(t) ≡

1

5 w(s) ds.

F12 dx ≡ ± F12 dx.

(2.15)

(2.16)

The above-mentioned lower bound for the energy is achieved only if there hold the following BPS equations

and

FU F12 ± √ = 0, G

(2.17)

FV − 1 = 0

(2.18)

D1 φ ± iD2 φ = 0.

(2.19)

By the expression of V(·) in (2.9), we see that equations (2.17)–(2.19) are equivalent to the following equations U(|φ|2 ) =0 (2.20) F12 ± G(|φ|2 ) 1 − (U(|φ|2 ))2 /b2 and D1 φ ± iD2 φ = 0,

(2.21)

whose radial version was obtained in [49]. With the potential (2.3) and under the condition (2.12), any solution of (2.20)–(2.21) automatically satisfies the original second-order equations (2.4)–(2.5). Therefore, to construct vortex solutions for the model (2.1), in the subsequent sections we only concentrate on the firstorder equations (2.20)–(2.21) with (2.12). Specifically, we will establish a series existence theorems for two typical models when the functions G(·), w(·) and U(·) assume some suitable forms, which satisfy (2.12).

3. Vortices in the |φ|4 -model This section is devoted to the study of the first typical model, for which we establish a sharp existence theorem on the doubly periodic and planar vortices and present the proof. We consider the model (2.1) when the function U(·) takes the form U(|φ|2 ) = (|φ|2 − 1).

(3.1)

From (2.3) and (2.9), we see that the potential W(|φ|2 ) in this case is exactly (1.5), which asymptotically tends to 12 (|φ|2 − 1)2 as b → ∞. In such sense, the model (2.1) with this potential is called |φ|4 -type. To include the symmetric vacuum, φ = 0, one needs to require that b > 1,

(3.2)

which is assumed throughout this section. Indeed, the large b requirement appears originally in [1,2].

...................................................

Therefore, the energy admits a lower bound

E = E dx ≥ H(0) F12 dx

,

(2.14)

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t

Hence, the energy density can be reformulated as

FU 2 b2 G F12 ± √ (F V − 1)2 + w(|φ|2 )|D1 φ ± iD2 φ|2 ± H(0)F12 + E= 2F 2F G 2 jk H(|φ| ) − H(0) ¯ φDk φ . ∓ Im∂j ε |φ|2

where

(2.13)

and w(|φ|2 ) = |φ|2 e|φ| .

2

2

(3.3)

In such setting, we see that H(0) = 1 and the energy lower bound becomes

E ≥ F12 dx

.

(3.4)

Then, the BPS equations (2.20)–(2.21) read as e−|φ| (|φ|2 − 1) 2

F12 ±

1 − (|φ|2 − 1)2 /b2

=0

(3.5)

and D1 φ ± iD2 φ = 0,

(3.6)

whose solutions achieve the lower bound in (3.4). Equation (3.6) implies that the zeros of φ are isolated with integer multiplicities. Denote the zero set of φ by Zφ = {p1 , . . . , pN }, (3.7) where we count the zeros of φ on multiplicities. Our purpose in this section is to establish existence theory for the BPS equations (3.5)–(3.6) in two cases. In the first case, we study equations (3.5)–(3.6) over a doubly periodic domain Ω such that the field configurations subject to the ’t Hooft periodic boundary condition [66,67] under which periodicity is achieved modulo gauge transformations. In the second case, equations (3.5)– (3.6) is studied over the full plane R2 . Finite energy implies the boundary condition for φ on R2 |φ| → 1 as |x| → ∞. (3.8) Our main results for (3.5)–(3.6) read as follows. Theorem 3.1. Consider the BPS equations (3.5)–(3.6) for the configurations (A1 , A2 , φ) with any prescribed zeros of φ given by (3.7). Over a doubly periodic domain Ω, the coupled system (3.5)–(3.6) has a unique solution if and only if N

− + 4π N, u +v 2 2 0 1 − (e − 1) /b 1 − 1/b2

(3.19)

which gives the necessary condition stated by (3.9). In what follows, we show that the condition (3.9) is also sufficient to solve (3.12). We achieve this goal by a monotone iteration. To this end, we need to construct suitable super- and subsolutions for (3.20). To get a super-solution for (3.17), we consider the problem 2(eu0 +w − 1) 4π N . +

w = |Ω| 1 − 1/b2

(3.20)

It is seen from [67] that the problem (3.20) has a solution w, satisfying u0 + w < 0, which is also unique if it exists, if and only if (3.9) holds.

...................................................

2e−e (eu − 1) u

u =

7

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We will prove the above theorem by transforming the BPS equations (3.5)–(3.6) into a secondorder nonlinear elliptic problem. To this end, let u = ln |φ|2 and the zeros of φ be given by (3.7). Then, as in [45], equations (3.5)–(3.6) are reduced into the following nonlinear elliptic equation

Then, under the condition (3.9), we have − 1)

1 − 1/b2

+

− 1) 4π N 4π N ≤ , + u +w 2 2 |Ω| |Ω| 0 1 − (e − 1) /b

(3.21)

which says that v ≡ w is a super-solution for (3.17). Next, we construct a subsolution for (3.17) under the condition (3.9). Let us define h(ε) ≡

2(1 − ε) 1 − (1 − ε)/b2

,

(3.22)

which saturates h (ε) < 0

and

h(0) =

2 1 − 1/b2

> h(ε),

∀ ε ∈ (0, 1).

(3.23)

Then, with the condition (3.9), for sufficiently small ε > 0, we have 2(1 − ε)

>

4π N . |Ω|

(3.24)

wε = aε (eu0 +wε − 1) +

4π N |Ω|

(3.25)

aε ≡

1 − (1 − ε)/b2

With (3.24), we infer from [67] that the problem

has a unique solution wε such that u0 + wε < 0 over Ω. Therefore, we conclude from (3.25) that

wε ≥ aε (eu0 +wε −c − 1) +

4π N |Ω|

u +wε −c

2e−e 0 (eu0 +wε −c − 1) 4π N ≥ + u +w |Ω| 1 − (e 0 ε −c − 1)2 /b2

(3.26)

when c > 0 is a suitably large constant. That is to say, for suitably large c, v ≡ wε − c is a subsolution for (3.17) under the condition (3.9). Hence, we get a pair of super- and subsolutions, v¯ and v, of (3.17) over Ω, satisfying v < v¯ everywhere if we take c > 0 suitably large. At this point, we may use a monotone iteration argument to obtain a smooth solution for (3.17). Then the existence of doubly periodic solutions for (3.12) follows. The uniqueness of the solution of (3.17) follows from the fact that the nonlinear function f (t) defined by (3.18) is strictly increasing for t ∈ (0, 1). Now by the above argument and the relations (3.13)–(3.15), we get the existence part stated in theorem 3.1 for the doubly periodic case. The quantized flux and energy formulae, in this case, follow from a direct integration.

(b) Planar solutions To prove theorem 3.1 for the planar case, we study equation (3.12) over R2 . The boundary condition (3.8) now reads u → 0 as |x| → ∞. To construct super- and subsolutions solutions for (3.12) over R2 , we consider the problem

w = λ(ew − 1) + 4π

N

δps ,

λ > 0.

(3.27)

s=1

By [68], we know that (3.27) has a unique solution wλ on R2 for every λ > 0, satisfying wλ < 0 and wλ → 0 as |x| → ∞.

...................................................

w =

8 u +w 2e−e 0 (eu0 +w

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2(eu0 +w

Hence, for λ = λ1 ≡ 2/ 1 − 1/b2 , the problem (3.27) has a solution wλ1 , which satisfies wλ1 < 0 with wλ1 → 0 as |x| → ∞ and

wλ

2e−e 1 (ewλ1 − 1) ≤ + 4π δps 1 − (ewλ1 − 1)2 /b2 s=1 N

(3.28)

in the sense of distribution. Therefore, u¯ ≡ wλ1 is a super-solution for (3.12) over R2 in the sense of distribution. Next, we construct a subsolution for (3.12) on R2 . Noting wλ1 < 0, we may take a number λ2 satisfying λ2 <

2e−1 wλ1

1 − (e

− 1)2 /b2

<

2 1 − 1/b2

= λ1 .

(3.29)

Then, for λ = λ2 , the problem (3.27) admits a unique solution wλ2 , saturating wλ2 < 0 with wλ2 → 0 as |x| → ∞ and

wλ2 = λ2 (ewλ2 − 1) + 4π

N

δps

s=1

≥

2e−1 (ewλ2 − 1) 1 − (ewλ1 − 1)2 /b2

+ 4π

N

δps

s=1

wλ

2e−e 2 (ewλ2 − 1) ≥ + 4π δps 1 − (ewλ2 − 1)2 /b2 s=1 N

(3.30)

in the sense of distribution. In the last inequality of (3.30), we use the fact wλ2 < wλ1 < 0 ensured by (3.29) and the maximum principle. Then, we conclude from (3.30) that u = wλ2 is a subsolution for (3.12) over R2 in the sense of distribution. To carry out an iteration, we need to remove the source terms in (3.12) by introducing the background function u0 = −

N

ln(1 + |x − ps |−2 ),

(3.31)

s=1

which satisfies

u0 = 4π

N

δps − g with g ≡

s=1

N s=1

4 . (1 + |x − ps |2 )2

(3.32)

The substitution u = u0 + v reduces (3.12) into u +v

2e−e 0 (eu0 +v − 1) + g.

v = 1 − (eu0 +v − 1)2 /b2

(3.33)

From the above argument, we see that v¯ = −u0 + wλ1 and v = −u0 + wλ2 , saturating v < v¯ on R2 , are a pair of super- and subsolutions of (3.33), respectively. Then, we can apply a monotone iteration procedure to construct a smooth solution v for (3.33) with v → 0 as |x| → ∞. As the doubly periodic case, the uniqueness of the solution of (3.33) follows from the strict monotonicity of the function f (t) defined by (3.18). Next, we prove the decay estimates for the planar solution.

...................................................

N

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2

wλ1 = (ewλ1 − 1) + 4π δps 1 − 1/b2 s=1

9

Let R0 > max1≤s≤N {|ps |} + 1. When |x| > R0 , we have

10

2e−e (eu − 1) u

.

(3.34)

Noting u → 0 as |x| → ∞, and linearizing (3.34) near u = 0 at infinity, we have u = h(x)u with h(x) → 2e−1 as |x| → ∞. It is standard to prove that, for any ε ∈ (0, 1), there exist positive constants C(ε) and R(ε) such that √ |u(x)| ≤ C(ε)e−(1−ε) Lp -estimates,

2e−1 |x|

as |x| > R(ε).

u ∈ W 2,p ,

we have Then, by (3.35) and elliptic ∂j u → 0 as |x| → ∞(j = 1, 2). When |x| > R0 , we obtain u

(3.35)

∀ p ≥ 1 when |x| > R0 , which yields

u

2eu−e (2 − eu )∂j u 2eu−e (eu − 1)2 ∂j u

(∂j u) = + . (1 − (eu − 1)2 /b2 )3/2 1 − (eu − 1)2 /b2

(3.36)

Again, we have (∂j u) = h(x)(∂j u) with h(x) → 2e−1 as |x| → ∞. Hence, the same decay estimate as (3.35) also holds for |∇u|. In view of the relations (3.13)–(3.15) and the above argument, we get the existence part and the decay estimates stated in theorem 3.1 for the planar case. The quantized flux and energy formula for the planar case follow from these decay estimates and integration.

4. Vortices in the |φ|6 -model Here, we concentrate on the second typical model, for which a series of existence theorems are developed. We consider the model (2.1) when the function U(·) assumes the form (4.1) U(|φ|2 ) = |φ|2 (|φ|2 − 1). It follows from (2.3) and (2.9) that the potential W(|φ|2 ) for this case takes the form [49] ⎛ ⎞ 2 (|φ|2 − 1)2 |φ| ⎠, W(|φ|2 ) = b2 ⎝1 − 1 − b2

(4.2)

which asymptotically tends to 12 |φ|2 (|φ|2 − 1)2 as b → ∞. That is why we call the model with this potential |φ|6 -type. As in §3, to accommodate the symmetric vacuum, φ = 0, we also need the requirement (3.2), which will be observed throughout this section. To make (2.12) hold in this case, we take the functions G(·) and w(·) given in [49] G(|φ|2 ) =

(3 + |φ|2 )2 , 9|φ|2

2 w(|φ|2 ) = (1 + |φ|2 ). 3

Then, within this setting, we see that H(0) = 1 and the energy lower bound (2.15) reads

E ≥ F12 dx

.

(4.3)

(4.4)

Hence, with (4.1) and (4.3), the BPS equations (2.20)–(2.21) become F12 ±

3|φ|2 (|φ|2 − 1) =0 (3 + |φ|2 ) 1 − |φ|2 (|φ|2 − 1)2 /b2

(4.5)

and D1 φ ± iD2 φ = 0,

(4.6)

whose solutions attain the lower bound in (4.4). The main purpose of this section is to develop existence theories for the BPS equations (4.5)– (4.6) over a doubly periodic domain Ω and the full plane R2 , respectively.

...................................................

1 − (eu − 1)2 /b2

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u =

In view of the structure of the potential (4.2), on R2 finite energy implies that there are two kinds of boundary conditions for φ at space infinity,

|φ| → 0

(4.8)

and as |x| → ∞,

which are called topological and non-topological, respectively, as that for the Chern–Simons model in [45]. Then, on R2 , both topological and non-topological vortices may arise in this |φ|6 -type Born–Infeld–Higgs model, which is a new phenomenon absent in the classical Born– Infeld–Higgs model [31,34]. Our main results for (4.5)–(4.6) read as follows. The first result is concerned with the existence of doubly periodic solutions of (4.5)–(4.6). Theorem 4.1. Consider the BPS equations (4.5)–(4.6) for the configurations (A1 , A2 , φ) over a doubly periodic domain Ω with any prescribed zeros of φ given by (3.7). (Necessary condition) If there is a solution of (4.5)–(4.6), then there must hold N

N + 2 is a constant depending on α. Furthermore, for the above solution, the magnetic flux and energy read as Φ= F12 dx = 2π (N + β), E = E dx = 2π (N + β). R2

R2

(4.12)

(4.13)

Remark 4.4. Unlike the |φ|4 -model studied in the previous section, for the |φ|6 -model, we have no uniqueness of the vortex solutions. In fact, such phenomenon also appears in the self-dual Abelian Chern–Simons–Higgs model [45,69,70].

...................................................

(4.7)

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|φ| → 1 as |x| → ∞

11

(4.14)

Note the topological and non-topological boundary conditions (4.7) and (4.8) in the new variable become u → 0 as

|x| → ∞

(4.15)

and u → −∞

as

|x| → ∞,

(4.16)

respectively. In §4a–c, we prove the above three existence theorems separately by studying the problem (4.14) over a doubly periodic domain Ω and over the full plane with (4.15) and (4.16).

(a) Doubly periodic solutions Here, we aim to prove theorem 4.1. For this purpose, we consider equation (4.14) over a doubly periodic domain Ω. By maximum principle, we see that any solutions of (4.14) are negative on Ω. We first derive the necessary condition (4.9) for the existence of solutions of (4.14). Let u0 be defined by (3.16). Let u = u0 + v. We have

v =

4π N 6eu0 +v (eu0 +v − 1) . + u +v u +v 2 2 |Ω| 1 − e 0 (e 0 − 1) /b

(3 + eu0 +v )

(4.17)

If v is a solution of (4.17), we have u0 + v < 0. It is easy to see that the number λ0 defined below satisfies t(1 − t) > 0. (4.18) λ0 ≡ max 0≤t≤1 (3 + t) 1 − t(t − 1)2 /b2 Then integrating equation (4.17) over Ω gives 0=

Ω

6eu0 +v (eu0 +v − 1) dx + 4π N (3 + eu0 +v ) 1 − eu0 +v (eu0 +v − 1)2 /b2

> −6λ0 |Ω| + 4π N, which yields the necessity of condition (4.9) for solving (4.14) over Ω. To construct a solution for (4.14) over Ω, in what follows, we need to construct suitable superand subsolutions. First, we observe that u¯ ≡ 0 is a super-solution for (4.14) in the sense of distribution. Next, we aim to get a suitable subsolution for (4.14). Consider the problem 3 δps .

w = ew (ew − 1) + 4π 2 N

(4.19)

s=1

By [45,69,71,72], we know that (4.19) admits a solution w with w < 0 if 3 16π N > 2 |Ω|

(4.20)

...................................................

6eu (eu − 1) + 4π δps . (3 + eu ) 1 − eu (eu − 1)2 /b2 s=1 N

u =

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To prove the above existence theorems, as in §3, we transform equations (4.5)–(4.6) equivalently into a nonlinear elliptic problem. Denote the zeros of φ given by (3.7). Let u = ln |φ|2 . Then, equations (4.5)–(4.6) are reduced into the nonlinear elliptic equation

and |Ω| is suitably large. Then, we see that on Ω

6ew (ew − 1) + 4π δps w w w 2 2 (e + 3) 1 − e (e − 1) /b s=1 N

≥

(4.21)

holds in the sense of distribution. Hence, u ≡ w is a subsolution of equation (4.14) over Ω in the sense of distribution. ¯ are a pair of sub- and superAs a consequence, v ≡ −u0 + w and v¯ ≡ −u0 , satisfying v < v, solutions of (4.17). Then, we can use a standard monotone iteration procedure to get a smooth solution v of (4.17) satisfying u0 + v < 0 over Ω. In view of the relations (3.13)–(3.15) and the above argument, we get the existence part of theorem 4.1. By integration, we get the quantized flux and energy stated in theorem 4.1.

(b) Topological solutions Here, we carry out the proof of theorem 4.2. To this end, we need to study equation (4.14) over R2 with boundary condition (4.15). It follows from maximum principle that any solutions of (4.14) over R2 are negative. Let u0 be the background function defined by (3.31) and the function g defined by (3.32). Setting u = u0 + v, we reduce equation (4.14) as

v =

6eu0 +v (eu0 +v − 1) +g (3 + eu0 +v ) 1 − eu0 +v (eu0 +v − 1)2 /b2

(4.22)

and the boundary condition (4.15) becomes v → 0 as

|x| → ∞.

(4.23)

We directly see that u¯ = 0 is a super-solution of (4.14) with (4.15) in the sense to distribution. Next, we seek a subsolution for (4.14) with (4.15). From [45,73], we know that on R2 the problem 3 δps

w = ew (ew − 1) + 4π 2 N

(4.24)

s=1

has a solution w, which satisfies w → 0 at infinity and w < 0 on R2 . Then, for this w on R2 , there holds 3 δps

w = ew (ew − 1) + 4π 2 N

s=1

6ew (ew − 1) + 4π δps (ew + 3) 1 − ew (ew − 1)2 /b2 s=1 N

≥

(4.25)

in the sense of distribution. In other words, u ≡ w is a subsolution for (4.14) with (4.15) in the sense of distribution. Therefore, we get a pair of super- and subsolutions, v¯ ≡ −u0 and v ≡ −u0 + w for equation (4.22) with (4.23). As previously, we may follow a monotone iteration argument to obtain a smooth solution v, satisfying u0 + v < 0, for (4.22) with (4.23). Accordingly, with the help of the relations (3.13)–(3.15), we get the existence part of theorem 4.2. The decay estimates, quantized flux and energy formula in theorem 4.2 follow from a similar argument as that in §3b. Then the proof of theorem 4.2 is complete.

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δps

s=1

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3

w = ew (ew − 1) + 4π 2

13 N

(c) Non-topological solutions

14

6eu (eu − 1) + 4π Nδ(x), (3 + eu ) 1 − eu (eu − 1)2 /b2

(4.26)

with the boundary condition (4.16). Write equation (4.26) and the boundary condition (4.16) in the radial variable r = |x| as 1 6eu (eu − 1) , urr + ur = u r (3 + e ) 1 − eu (eu − 1)2 /b2

r > 0,

(4.27)

and lim

r→0

u(r) = 2N, ln r

lim u(r) = −∞.

r→∞

(4.28)

To solve this problem, we employ a shooting argument initiated in [70,74] and used in [75]. Let t = ln r. We transform the problem (4.27)–(4.28) into utt =

6e2t eu (eu − 1) , (3 + eu ) 1 − eu (eu − 1)2 /b2

t∈R

(4.29)

and lim ut = 2N,

t→−∞

lim u = −∞.

(4.30)

t→∞

To start the shooting procedure, we need to consider the following initial-value problem utt =

6e2t eu (eu − 1) , (3 + eu ) 1 − eu (eu − 1)2 /b2

t∈R

(4.31)

and u(t0 ) = −α,

ut (t0 ) = 0,

(4.32)

where the parameters α > 0, t0 ∈ R. By the standard ODE theory, we see that the problem (4.31)–(4.32) admits a unique global solution for any α > 0, t0 ∈ R. In the sequel, we will find a solution of (4.27)–(4.28) by solving (4.31)–(4.32) for some suitable parameters α > 0 and t0 . Our first purpose is to achieve the boundary condition at −∞ in (4.30). By integration of (4.31), we obtain ut (t) =

t t0

6e2s eu(s) (eu(s) − 1) ds. (3 + eu(s) ) 1 − eu(s) (eu(s) − 1)2 /b2

(4.33)

Noting u(t0 ) = −α < 0, we see from (4.33) that u(t) < −α for all t < t0 . Then the limit γ (t0 , α) ≡ lim ut (t) t→−∞

(4.34)

exists and the convergence is uniform with respect to α and t0 . Hence, γ (t0 , α) is a continuous function of α and t0 .

...................................................

u =

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Here, we aim to prove the existence of non-topological vortices for the BPS equations (4.5)–(4.6) given by theorem 4.3. Without loss of generality, assuming the point p is the origin, we need to study equation (4.14) with all pi being the origin, that is

It follows from equation (4.31) that

15 2t u

utt > −2a0 e e ,

Setting v = u + 2t in (4.36) yields

1 1 − (4/27)/b2

.

(4.36)

vtt > −2a0 ev .

(4.37)

Note that vt = ut + 2 and ut > 0 for t < t0 ensured by (4.33). Multiplying (4.37) by vt and by integration, we see that for t < t0 4 − (vt )2 > 4a0 (ev(t) − e2t0 −α ),

(4.38)

0 < (vt )2 < 4(1 + a0 )e2t0 −α .

(4.39)

which gives

Then, we obtain

0 < ut < 2( 1 + a0 e2t0 −α − 1) ≡ K0

for t < t0 ,

(4.40)

− α − K0 (t0 − t) < u(t) < −α

t < t0 .

(4.41)

which implies for

It is straightforward to see that the function es (es − 1) is decreasing on (−∞, − ln 2). Let α ≥ ln 2. We see from (4.41) that 6e2t eu (eu − 1) for t < t0 , (4.42) utt < (3 + e−α ) integrating which over (−∞, t0 ) concludes t0 6 e2t e−α−K0 (t0 −t) (e−α−K0 (t0 −t) − 1) dt γ (t0 , α) > − (3 + e−α ) −∞

1 e−α 6e2t0 −α − = (3 + e−α ) K0 + 2 2(K0 + 1) > =

3e2t0 −α (3 + e−α )(K0 + 2) 3e2t0 −α . 1 + a0 e2t0 −α

2(3 + e−α )

(4.43)

Therefore, it follows from (4.43) that, for given α, there exists some t0 such that γ (t0 , α) > 2N.

(4.44)

γ (t0 , α) ≤ 2( 1 + a0 e2t0 −α − 1).

(4.45)

From (4.40), we may obtain

Hence, for given α, there exists a suitable t0 , so that γ (t0 , α) < 2N.

(4.46)

Consequently, noting (4.44) and (4.46), for any α ≥ ln 2, we may find a suitable t0 = t0 (α) such that (4.47) γ (t0 , α) = 2N. Therefore, we realize the boundary condition at −∞ in (4.30). In what follows, we aim to achieve the boundary condition at ∞ in (4.30).

...................................................

a0 ≡

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where

(4.35)

− 2β ≡ lim ut =

∞

t→∞

t0

6e2s eu(s) (eu(s) − 1) ds < 0 is finite. (3 + eu(s) ) 1 − eu(s) (eu(s) − 1)2 /b2

(4.49)

Then, (4.48) implies β > 1. Hence, the boundary condition at ∞ in (4.30) is established. Next, we derive a more precise estimate for β. Multiplying (4.31) by ut and integrating over (−∞, +∞), and using the fact β > 1, we have ∞ 6e2t eu(t) (eu(t) − 1)ut (t) dt 2(β 2 − N2 ) = −∞ (3 + eu(t) ) 1 − eu(t) (eu(t) − 1)2 /b2 ∞ d u(t) 6es (es − 1) = e2t ds dt s dt −∞ (3 + e ) 1 − es (es − 1)2 /b2 −∞

∞ u(t)

6es (es − 1)

2t =e ds s s s 2 2

−∞ (3 + e ) 1 − e (e − 1) /b −2 = −2 where ˜ H(u) ≡

∞ −∞

∞

−∞

u(t)

−∞

− 1) ds dt (3 + es ) 1 − es (es − 1)2 /b2 ∞ 6e2t eu(t) (eu(t) − 1) ˜ dt + 2 e2t H(u(t)) dt, −∞ (3 + eu(t) ) 1 − eu(t) (eu(t) − 1)2 /b2 e2t

6es (es

−∞

6eu (eu − 1) − (3 + eu ) 1 − eu (eu − 1)2 /b2

u −∞

6es (es − 1) ds. (3 + es ) 1 − es (es − 1)2 /b2

(4.50)

(4.51)

Noting b > 1 and u < 0, it is direct to check that ˜ dH(u) 3e2u (8b2 + 3[eu − 1]2 [e2u − 1]) >0 = 2 u du b (e + 3)2 (1 − eu (eu − 1)2 /b2 )3/2

(4.52)

˜ ˜ and H(−∞) = 0, which yields H(u) > 0. Then, we conclude from (4.50) that 2(β 2 − N2 ) > 4(β + N),

(4.53)

which gives β > N + 2. Hence by the above argument and the relations (3.13)–(3.15), we get the existence and the asymptotic estimates of the solutions stated in theorem 4.3. The flux and energy formula follow from integration and these asymptotic estimates. Then, we complete the proof of theorem 4.3.

5. Summary Although the Born–Infeld theory enables a point charge at rest to carry finite energy, the nonlinearity it introduces makes it difficult to construct exact solutions to the governing equations, and so far, analytic insight can only be gained through exploring the self-dual or BPS reductions of the problem, which is also a well-known common feature in classical gauge field theory. This study enriches our knowledge on the exact solutions of the Born–Infeld theory with following main results. (i) Along the framework of [49], we derived without the radial ansatz the BSP vortex equations of the Born–Infeld theory in presence of a complex scalar field following a generalized Higgs mechanism of the Bekenstein-type.

...................................................

t0

Therefore, the limit

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Noting u < 0, we see from (4.33) that ut < 0 for all t > t0 . Then, we have u < −α for all t ∈ R \ {t0 }, which implies the integral ∞ e2s+u(s) ds is convergent. (4.48)

Data accessibility. This work does not have any experimental data. Competing interests. I declare I have no competing interests. Funding. This work was supported by National Natural Science Foundation of China under grants nos. 11201118 and 11471100, and the Key Foundation for Henan colleges under grant no. 15A110013.

Acknowledgements. I thank the anonymous referees for their useful suggestions and valuable comments on this work.

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Note that our existence theorem concerning non-topological vortices for the |φ|6 -model is only established when all the vortices concentrate at one point. It will be an interesting future problem to develop an existence theory for the non-topological vortices when the vortices are arbitrarily distributed. We will deal with this problem in a forthcoming work.

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(ii) We established a series of existence theorems for the generalized BPS vortex equations [49]. Two typical models with the |φ|4 - and |φ|6 -type Higgs potentials are rigorously analysed. For the |φ|4 -model, a sharp existence and uniqueness theorem is obtained for both doubly periodic and planar cases. In the former case, a necessary and sufficient condition for the existence of vortex solutions is explicitly deduced in terms of the vortex number, the Born–Infeld parameter, and the size of the lattice domain. For the existence of doubly periodic vortices in the |φ|6 -model, necessary and sufficient conditions in terms of the vortex number, the Born–Infeld parameter, and the size of the lattice domain are separately stated. Owing to the specific structure of the Higgs potential, both topological and non-topological vortices are constructed for the |φ|6 -model, which is a novel phenomenon distinguishing the |φ|6 -model from that found in the classical Born–Infeld–Higgs model [31,34].

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