The Born–Infeld vortices induced from a generalized Higgs mechanism

Abstract

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.

In the classical Maxwell theory, electromagnetic interaction over the standard four-dimensional Minkowski space–time is governed by the Lagrangian action density

$${\mathcal{L}}_{\mathrm{Maxwell}}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+A_{\mu }{j}^{\mu },$$ 1.1

where A_μ is the usual gauge potential vector, j^μ an applied external current density vector and

$$F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$$ 1.2

is the electromagnetic field induced from A_μ. 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 (g^μν)=(g_μν)=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

$${\mathcal{L}}_{\mathrm{BI}}=b^2(1-\sqrt{1+\frac{1}{2b^2}{F}_{\mu \nu }{F}^{\mu \nu }}),$$ 1.3

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 j^μ=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 field-theoretical 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

$${\mathcal{L}}_{\mathrm{BIH}}=b^2(1-\sqrt{1+\frac{1}{2b^2}{F}_{\mu \nu }{F}^{\mu \nu }})+D_{\mu }\varphi \overline{D^{\mu }\varphi }+W(|\varphi {|}^2),$$ 1.4

where F_μν is defined by (1.2), D_μϕ=∂_μϕ−iA_μϕ the gauge-covariant derivative, ϕ a complex-valued 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

$$W(|\varphi {|}^2)=b^2(1-\sqrt{1-\frac{{(|\varphi {|}^2-1)}^2}{b^2}}),$$ 1.5

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 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

$${\mathcal{L}}_{\mathrm{AH}}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+D_{\mu }\varphi \overline{D^{\mu }\varphi }-\frac{1}{2}{(|\varphi {|}^2-1)}^2$$ 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 [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

$${\mathcal{L}}_{\mathrm{GBIH}}=b^2(1-\mathcal{F})+w(|\varphi {|}^2)D_{\mu }\varphi \overline{D^{\mu }\varphi }-W(|\varphi {|}^2)$$ 2.1

with

$$\mathcal{F}\equiv \sqrt{1+\frac{G(|\varphi {|}^2)}{2b^2}{F}_{\mu \nu }{F}^{\mu \nu }}$$ 2.2

and

$$W(|\varphi {|}^2)\equiv b^2(1-V(|\varphi {|}^2)),$$ 2.3

where F_μν=∂_μA_ν−∂_νA_μ is the electromagnetic field induced from the gauge field A_μ, D_μϕ=∂_μϕ−iA_μϕ 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

$${\mathrm{\partial }}_{\nu }\left(\frac{G}{\mathcal{F}}{F}^{\nu \mu }\right)=J^{\mu }$$ 2.4

and

$$D_{\mu }(w{D}^{\mu }\varphi )+\frac{\mathrm{\partial }w}{\mathrm{\partial }\overline{\varphi }}{D}_{\mu }\varphi \overline{D^{\mu }\varphi }+\frac{F_{\mu \nu }{F}^{\mu \nu }}{2\mathcal{F}}\frac{\mathrm{\partial }G}{\mathrm{\partial }\overline{\varphi }}-\frac{\mathrm{\partial }W}{\mathrm{\partial }\overline{\varphi }}=0,$$ 2.5

where $J^{\mu }\equiv \mathrm{i}w(|\varphi {|}^2)(\varphi \overline{D^{\mu }\varphi }-\overline{\varphi }{D}^{\mu }\varphi )$ is the current density.

The energy–momentum tensor for the model (2.1) takes the form

$$T_{\mu \nu }=-\frac{1}{\mathcal{F}}{F}_{\mu }^{\beta }{F}_{\beta \nu }+w(|\varphi {|}^2)(\overline{D_{\mu }\varphi }{D}_{\nu }\varphi +\overline{D_{\nu }\varphi }{D}_{\mu }\varphi )-g_{\mu \nu }{\mathcal{L}}_{\mathrm{GBIH}}.$$ 2.6

In the static case and the temporal gauge(A₀=0), the energy density is

$$\begin{array}{rl}\mathcal{E}& =T_{00}=b^2(\mathcal{F}-1)+w(|\varphi {|}^2)|D_j\varphi {|}^2+W(|\varphi {|}^2)\\ & =b^2(\mathcal{F}-V(|\varphi {|}^2))+w(|\varphi {|}^2)|D_j\varphi {|}^2,\end{array}$$ 2.7

here and in what follows, we understand that

$$\mathcal{F}=\sqrt{1+\frac{G(|\varphi {|}^2)}{b^2}{F}_{12}^2}.$$ 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

$$V(|\varphi {|}^2)=\sqrt{1-\frac{{(U(|\varphi {|}^2))}^2}{b^2}},$$ 2.9

where the function U(⋅) is to be determined later. Note the identity

$$|D_i\varphi {|}^2=|D_1\varphi \pm \mathrm{i}{D}_2\varphi {|}^2\pm \mathrm{i}(D_1\varphi \overline{D_2\varphi }-\overline{D_1\varphi }{D}_2\varphi ).$$ 2.10

Then, the energy density can be rewritten as

$$\begin{array}{rl}\mathcal{E}& =\frac{G}{2\mathcal{F}}{(F_{12}\pm \frac{\mathcal{F}U}{\sqrt{G}})}^2-\frac{G}{2\mathcal{F}}{F}_{12}^2-\frac{\mathcal{F}{U}^2}{2}\mp \sqrt{G}U{F}_{12}\\ & +\frac{b^2}{2\mathcal{F}}{(\mathcal{F}V-1)}^2-\frac{b^2\mathcal{F}}{2}{V}^2-\frac{b^2}{2\mathcal{F}}+b^{2}V\\ & +b^2(\mathcal{F}-V)+w(|\varphi {|}^2)|D_j\varphi {|}^2\\ & =\frac{G}{2\mathcal{F}}{(F_{12}\pm \frac{\mathcal{F}U}{\sqrt{G}})}^2+\frac{b^2}{2\mathcal{F}}{(\mathcal{F}V-1)}^2+w(|\varphi {|}^2)|D_1\varphi \pm \mathrm{i}{D}_2\varphi {|}^2\\ & \mp \sqrt{G}U{F}_{12}\pm \mathrm{i}w(|\varphi {|}^2)[D_1\varphi \overline{D_2\varphi }-\overline{D_1\varphi }{D}_2\varphi ].\end{array}$$ 2.11

To achieve a BPS reduction, later the functions G(⋅),w(⋅),U(⋅) will be chosen suitably such that

$$\frac{\mathrm{d}}{\mathrm{d}t}(\sqrt{G(t)}U(t))=w(t).$$ 2.12

Let

$$H(t)\equiv {\int }_t^{1}w(s)\hspace{0.17em}\mathrm{d}s.$$ 2.13

Hence, the energy density can be reformulated as

$$\begin{array}{rl}\mathcal{E}& =\frac{G}{2\mathcal{F}}{(F_{12}\pm \frac{\mathcal{F}U}{\sqrt{G}})}^2+\frac{b^2}{2\mathcal{F}}{(\mathcal{F}V-1)}^2+w(|\varphi {|}^2)|D_1\varphi \pm \mathrm{i}{D}_2\varphi {|}^2\pm H(0)F_{12}\\ & \mp \mathrm{Im}{\mathrm{\partial }}_j{\epsilon }^{jk}\left(\frac{H(|\varphi {|}^2)-H(0)}{|\varphi {|}^2}\overline{\varphi }{D}_k\varphi \right).\end{array}$$ 2.14

Therefore, the energy admits a lower bound

$$E=\int \mathcal{E}\hspace{0.17em}\mathrm{d}x\ge H(0)|\int F_{12}\hspace{0.17em}\mathrm{d}x|,$$ 2.15

where

$$|\int F_{12}\hspace{0.17em}\mathrm{d}x|\equiv \pm \int F_{12}\hspace{0.17em}\mathrm{d}x.$$ 2.16

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

$$\begin{array}{rl}{F}_{12}\pm \frac{\mathcal{F}U}{\sqrt{G}}& =0,\end{array}$$ 2.17

$$\begin{array}{rl}\mathcal{F}V-1& =0\end{array}$$ 2.18

$$\begin{array}{rl}\text{and}{D}_1\varphi \pm \mathrm{i}{D}_2\varphi & =0.\end{array}$$ 2.19

By the expression of V (⋅) in (2.9), we see that equations (2.17)–(2.19) are equivalent to the following equations

$$F_{12}\pm \frac{U(|\varphi {|}^2)}{\sqrt{G(|\varphi {|}^2)}\sqrt{1-{(U(|\varphi {|}^2))}^2/b^2}}=0$$ 2.20

and

$$D_1\varphi \pm \mathrm{i}{D}_2\varphi =0,$$ 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 first-order 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

$$U(|\varphi {|}^2)=(|\varphi {|}^2-1).$$ 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 $\frac{1}{2}{(|\varphi {|}^2-1)}^2$ as $b\to \mathrm{\infty }$. 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].

With (3.1) if one takes G(|ϕ|²)≡1, w(|ϕ|²)≡1, which satisfies (2.12) automatically, then the Born–Infeld–Higgs model (1.4) with potential (1.5) proposed in [31] is recovered, which can be viewed as a special case of model (2.1). For the corresponding BPS equations, existence theory was established in [34].

Next, we consider more general form of G(⋅) and w(⋅). To ensure (2.12), the functions G(⋅) and w(⋅) can be chosen as [49]

$$G(|\varphi {|}^2)={\mathrm{e}}^{2|\varphi {|}^2}\mathrm{and}w(|\varphi {|}^2)=|\varphi {|}^2{\mathrm{e}}^{|\varphi {|}^2}.$$ 3.3

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

$$E\ge |\int F_{12}\hspace{0.17em}\mathrm{d}x|.$$ 3.4

Then, the BPS equations (2.20)–(2.21) read as

$$F_{12}\pm \frac{{\mathrm{e}}^{-|\varphi {|}^2}(|\varphi {|}^2-1)}{\sqrt{1-{(|\varphi {|}^2-1)}^2/b^2}}=0$$ 3.5

and

$$D_1\varphi \pm \mathrm{i}{D}_2\varphi =0,$$ 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

$$Z_{\varphi }=\{p_1,\dots ,p_N\},$$ 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 ${\mathbb{R}}^2$. Finite energy implies the boundary condition for ϕ on ${\mathbb{R}}^2$

$$|\varphi |\to 1\text{as}\hspace{0.17em}|x|\to \mathrm{\infty }.$$ 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

$$N<\frac{|\mathit{\Omega }|}{2\pi \sqrt{1-1/b^2}}.$$ 3.9

Over the full plane ${\mathbb{R}}^2,$ there exists a unique solution for the coupled system (3.5)–(3.6), which satisfies the decay estimates at infinity,

$$|D_1\varphi |+|D_2\varphi |,F_{12},|\varphi {|}^2-1=\mathrm{O}({\mathrm{e}}^{-(1-\epsilon )\sqrt{2{\mathrm{e}}^{-1}}|x|}),$$ 3.10

where ε∈(0,1) is arbitrarily small.

In both cases, the solutions carry minimum energy, and the magnetic flux Φ and energy E are of the quantized values given by

$$\mathit{\Phi }=\int F_{12}\hspace{0.17em}\mathrm{d}x=\pm 2\pi N\mathrm{and}E=\int \mathcal{E}\hspace{0.17em}\mathrm{d}x=2\pi N.$$ 3.11

Remark 3.2 —

It is interesting to see that the necessary and sufficient condition (3.9) for the existence of the doubly periodic solutions of equations (3.5)–(3.6) coincides with that found in [34] for the compact case. However, the planar solution of (3.5)–(3.6) decays at a different rate compared with that in [34].

We will prove the above theorem by transforming the BPS equations (3.5)–(3.6) into a second-order nonlinear elliptic problem. To this end, let $u=\ln |\varphi {|}^2$ 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

$$\mathrm{\Delta }u=\frac{2{\mathrm{e}}^{-{\mathrm{e}}^u}({\mathrm{e}}^u-1)}{\sqrt{1-{({\mathrm{e}}^u-1)}^2/b^2}}+4\pi \sum _{s=1}^N{\delta }_{p_s}.$$ 3.12

Note the relations [45]

$$\begin{array}{rl}\varphi & =\exp (\frac{1}{2}u(x)+\mathrm{i}\sum _{s=1}^N\arg (x-p_s)),\end{array}$$ 3.13

$$\begin{array}{rl}{A}_1(x)& =-\mathrm{Re}\{\mathrm{i}\overline{\mathrm{\partial }}\ln \varphi (x)\},A_2(x)=-\mathrm{Im}\{\mathrm{i}\overline{\mathrm{\partial }}\ln \varphi (x)\}\end{array}$$ 3.14

$$\begin{array}{rl}\text{and}|\varphi {|}^2& ={\mathrm{e}}^u,|D_j\varphi {|}^2=\frac{1}{2}{\mathrm{e}}^u|\mathrm{\nabla }u{|}^2.\end{array}$$ 3.15

Hence, to prove theorem 3.1, it is sufficient to establish existence and uniqueness results for the problem (3.12) over a doubly periodic domain Ω and over the full plane ${\mathbb{R}}^2$, respectively.

(a). Doubly periodic solutions

Here, we prove theorem 3.1 for the doubly periodic case. For this purpose, we need to study equation (3.12) over a doubly periodic domain Ω.

The maximum principle implies that the solutions of (3.12) on Ω are negative. Let u₀ be the unique solution of

$$\mathrm{\Delta }{u}_0=-\frac{4\pi N}{|\mathit{\Omega }|}+4\pi \sum _{s=1}^N{\delta }_{p_s},{\int }_{\mathit{\Omega }}{u}_0\hspace{0.17em}\mathrm{d}x=0.$$ 3.16

The substitution of u=u₀+v in (3.12) leads to

$$\mathrm{\Delta }v=\frac{2{\mathrm{e}}^{-{\mathrm{e}}^{u_0+v}}({\mathrm{e}}^{u_0+v}-1)}{\sqrt{1-{({\mathrm{e}}^{u_0+v}-1)}^2/b^2}}+\frac{4\pi N}{|\mathit{\Omega }|}.$$ 3.17

We first derive a necessary condition for solving (3.12).

Note the function

$$f(t)=\frac{2{\mathrm{e}}^{-t}(t-1)}{\sqrt{1-{(t-1)}^2/b^2}}$$ 3.18

is strictly increasing for 0<t≤1. Letting v be a solution of (3.17), we have u₀+v<0. Then integrating (3.17) over Ω gives us

$$0={\int }_{\mathit{\Omega }}\frac{2{\mathrm{e}}^{-{\mathrm{e}}^{u_0+v}}({\mathrm{e}}^{u_0+v}-1)}{\sqrt{1-{({\mathrm{e}}^{u_0+v}-1)}^2/b^2}}\hspace{0.17em}\mathrm{d}x+4\pi N>-\frac{2|\mathit{\Omega }|}{\sqrt{1-1/b^2}}+4\pi N,$$ 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

$$\mathrm{\Delta }w=\frac{2({\mathrm{e}}^{u_0+w}-1)}{\sqrt{1-1/b^2}}+\frac{4\pi N}{|\mathit{\Omega }|}.$$ 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.

Then, under the condition (3.9), we have

$$\mathrm{\Delta }w=\frac{2({\mathrm{e}}^{u_0+w}-1)}{\sqrt{1-1/b^2}}+\frac{4\pi N}{|\mathit{\Omega }|}\le \frac{2{\mathrm{e}}^{-{\mathrm{e}}^{u_0+w}}({\mathrm{e}}^{u_0+w}-1)}{\sqrt{1-{({\mathrm{e}}^{u_0+w}-1)}^2/b^2}}+\frac{4\pi N}{|\mathit{\Omega }|},$$ 3.21

which says that $\overline{v}\equiv w$ is a super-solution for (3.17).

Next, we construct a subsolution for (3.17) under the condition (3.9). Let us define

$$h(\epsilon )\equiv \frac{2(1-\epsilon )}{\sqrt{1-(1-\epsilon )/b^2}},$$ 3.22

which saturates

$$h^{\prime }(\epsilon )<0\text{and}h(0)=\frac{2}{\sqrt{1-1/b^2}}>h(\epsilon ),\mathrm{\forall }\hspace{0.17em}\epsilon \in (0,1).$$ 3.23

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

$$a_{\epsilon }\equiv \frac{2(1-\epsilon )}{\sqrt{1-(1-\epsilon )/b^2}}>\frac{4\pi N}{|\mathit{\Omega }|}.$$ 3.24

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

$$\mathrm{\Delta }{w}_{\epsilon }=a_{\epsilon }({\mathrm{e}}^{u_0+w_{\epsilon }}-1)+\frac{4\pi N}{|\mathit{\Omega }|}$$ 3.25

has a unique solution w_ε such that u₀+w_ε<0 over Ω.

Therefore, we conclude from (3.25) that

$$\begin{array}{rl}\mathrm{\Delta }{w}_{\epsilon }& \ge a_{\epsilon }({\mathrm{e}}^{u_0+w_{\epsilon }-c}-1)+\frac{4\pi N}{|\mathit{\Omega }|}\\ & \ge \frac{2{\mathrm{e}}^{-{\mathrm{e}}^{u_0+w_{\epsilon }-c}}({\mathrm{e}}^{u_0+w_{\epsilon }-c}-1)}{\sqrt{1-{({\mathrm{e}}^{u_0+w_{\epsilon }-c}-1)}^2/b^2}}+\frac{4\pi N}{|\mathit{\Omega }|}\end{array}$$ 3.26

when c>0 is a suitably large constant. That is to say, for suitably large c, $\underset{\_}{v}\equiv w_{\epsilon }-c$ is a subsolution for (3.17) under the condition (3.9).

Hence, we get a pair of super- and subsolutions, $\overline{v}$ and $\underset{\_}{v}$, of (3.17) over Ω, satisfying $\underset{\_}{v}<\overline{v}$ 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 ${\mathbb{R}}^2$. The boundary condition (3.8) now reads u→0 as $|x|\to \mathrm{\infty }$.

To construct super- and subsolutions solutions for (3.12) over ${\mathbb{R}}^2$, we consider the problem

$$\mathrm{\Delta }w=\mathrm{\lambda }({\mathrm{e}}^w-1)+4\pi \sum _{s=1}^N{\delta }_{p_s},\mathrm{\lambda }>0.$$ 3.27

By [68], we know that (3.27) has a unique solution w_λ on ${\mathbb{R}}^2$ for every λ>0, satisfying w_λ<0 and w_λ→0 as $|x|\to \mathrm{\infty }$.

Hence, for $\mathrm{\lambda }={\mathrm{\lambda }}_1\equiv 2/\sqrt{1-1/b^2},$ the problem (3.27) has a solution w_λ1, which satisfies w_λ1<0 with w_λ1→0 as $|x|\to \mathrm{\infty }$ and

$$\begin{array}{rl}\mathrm{\Delta }{w}_{{\mathrm{\lambda }}_1}& =\frac{2}{\sqrt{1-1/b^2}}({\mathrm{e}}^{w_{{\mathrm{\lambda }}_1}}-1)+4\pi \sum _{s=1}^N{\delta }_{p_s}\\ & \le \frac{2{\mathrm{e}}^{-{\mathrm{e}}^{w_{{\mathrm{\lambda }}_1}}}({\mathrm{e}}^{w_{{\mathrm{\lambda }}_1}}-1)}{\sqrt{1-{({\mathrm{e}}^{w_{{\mathrm{\lambda }}_1}}-1)}^2/b^2}}+4\pi \sum _{s=1}^N{\delta }_{p_s}\end{array}$$ 3.28

in the sense of distribution. Therefore, $\overline{u}\equiv w_{{\mathrm{\lambda }}_1}$ is a super-solution for (3.12) over ${\mathbb{R}}^2$ in the sense of distribution.

Next, we construct a subsolution for (3.12) on ${\mathbb{R}}^2$.

Noting w_λ1<0, we may take a number λ₂ satisfying

$${\mathrm{\lambda }}_2<\frac{2{\mathrm{e}}^{-1}}{\sqrt{1-{({\mathrm{e}}^{w_{{\mathrm{\lambda }}_1}}-1)}^2/b^2}}<\frac{2}{\sqrt{1-1/b^2}}={\mathrm{\lambda }}_1.$$ 3.29

Then, for λ=λ₂, the problem (3.27) admits a unique solution w_λ2, saturating w_λ2<0 with w_λ2→0 as $|x|\to \mathrm{\infty }$ and

$$\begin{array}{rl}\mathrm{\Delta }{w}_{{\mathrm{\lambda }}_2}& ={\mathrm{\lambda }}_2({\mathrm{e}}^{w_{{\mathrm{\lambda }}_2}}-1)+4\pi \sum _{s=1}^N{\delta }_{p_s}\\ & \ge \frac{2{\mathrm{e}}^{-1}({\mathrm{e}}^{w_{{\mathrm{\lambda }}_2}}-1)}{\sqrt{1-{({\mathrm{e}}^{w_{{\mathrm{\lambda }}_1}}-1)}^2/b^2}}+4\pi \sum _{s=1}^N{\delta }_{p_s}\\ & \ge \frac{2{\mathrm{e}}^{-{\mathrm{e}}^{w_{{\mathrm{\lambda }}_2}}}({\mathrm{e}}^{w_{{\mathrm{\lambda }}_2}}-1)}{\sqrt{1-{({\mathrm{e}}^{w_{{\mathrm{\lambda }}_2}}-1)}^2/b^2}}+4\pi \sum _{s=1}^N{\delta }_{p_s}\end{array}$$ 3.30

in the sense of distribution. In the last inequality of (3.30), we use the fact w_λ2<w_λ1<0 ensured by (3.29) and the maximum principle.

Then, we conclude from (3.30) that $\underset{\_}{u}=w_{{\mathrm{\lambda }}_2}$ is a subsolution for (3.12) over ${\mathbb{R}}^2$ 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

$$u_0=-\sum _{s=1}^N\ln (1+|x-p_s{|}^{-2}),$$ 3.31

which satisfies

$$\mathrm{\Delta }{u}_0=4\pi \sum _{s=1}^N{\delta }_{p_s}-g\text{with}\hspace{0.17em}g\equiv \sum _{s=1}^N\frac{4}{{(1+|x-p_s{|}^2)}^2}.$$ 3.32

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

$$\mathrm{\Delta }v=\frac{2{\mathrm{e}}^{-{\mathrm{e}}^{u_0+v}}({\mathrm{e}}^{u_0+v}-1)}{\sqrt{1-{({\mathrm{e}}^{u_0+v}-1)}^2/b^2}}+g.$$ 3.33

From the above argument, we see that $\overline{v}=-u_0+w_{{\mathrm{\lambda }}_1}$ and $\underset{\_}{v}=-u_0+w_{{\mathrm{\lambda }}_2}$, saturating $\underset{\_}{v}<\overline{v}$ on ${\mathbb{R}}^2$, 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 $|x|\to \mathrm{\infty }$.

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.

Let $R_0>\underset{1\le s\le N}{max}\{|p_s|\}+1$. When |x|>R₀, we have

$$\mathrm{\Delta }u=\frac{2{\mathrm{e}}^{-{\mathrm{e}}^u}({\mathrm{e}}^u-1)}{\sqrt{1-{({\mathrm{e}}^u-1)}^2/b^2}}.$$ 3.34

Noting u→0 as $|x|\to \mathrm{\infty }$, and linearizing (3.34) near u=0 at infinity, we have Δu=h(x)u with h(x)→2e⁻¹ as $|x|\to \mathrm{\infty }$. It is standard to prove that, for any ε∈(0,1), there exist positive constants C(ε) and R(ε) such that

$$|u(x)|\le C(\epsilon ){\mathrm{e}}^{-(1-\epsilon )\sqrt{2{\mathrm{e}}^{-1}}|x|}\text{as}\hspace{0.17em}|x|>R(\epsilon ).$$ 3.35

Then, by (3.35) and elliptic L^p-estimates, we have u∈W^2,p, ∀ p≥1 when |x|>R₀, which yields ∂_ju→0 as $|x|\to \mathrm{\infty }(j=1,2)$. When |x|>R₀, we obtain

$$\mathrm{\Delta }({\mathrm{\partial }}_{j}u)=\frac{2{\mathrm{e}}^{u-{\mathrm{e}}^u}(2-{\mathrm{e}}^u){\mathrm{\partial }}_{j}u}{\sqrt{1-{({\mathrm{e}}^u-1)}^2/b^2}}+\frac{2{\mathrm{e}}^{u-{\mathrm{e}}^u}{({\mathrm{e}}^u-1)}^2{\mathrm{\partial }}_{j}u}{{(1-{({\mathrm{e}}^u-1)}^2/b^2)}^{3/2}}.$$ 3.36

Again, we have Δ(∂_ju)=h(x)(∂_ju) with h(x)→2e⁻¹ as $|x|\to \mathrm{\infty }$. 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

$$U(|\varphi {|}^2)=\sqrt{|\varphi {|}^2}(|\varphi {|}^2-1).$$ 4.1

It follows from (2.3) and (2.9) that the potential W(|ϕ|²) for this case takes the form [49]

$$W(|\varphi {|}^2)=b^2(1-\sqrt{1-\frac{|\varphi {|}^2{(|\varphi {|}^2-1)}^2}{b^2}}),$$ 4.2

which asymptotically tends to $\frac{1}{2}|\varphi {|}^2{(|\varphi {|}^2-1)}^2$ as $b\to \mathrm{\infty }$. 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]

$$G(|\varphi {|}^2)=\frac{{(3+|\varphi {|}^2)}^2}{9|\varphi {|}^2},w(|\varphi {|}^2)=\frac{2}{3}(1+|\varphi {|}^2).$$ 4.3

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

$$E\ge |\int F_{12}\hspace{0.17em}\mathrm{d}x|.$$ 4.4

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

$$F_{12}\pm \frac{3|\varphi {|}^2(|\varphi {|}^2-1)}{(3+|\varphi {|}^2)\sqrt{1-|\varphi {|}^2{(|\varphi {|}^2-1)}^2/b^2}}=0$$ 4.5

and

$$D_1\varphi \pm \mathrm{i}{D}_2\varphi =0,$$ 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 ${\mathbb{R}}^2$, respectively.

In view of the structure of the potential (4.2), on ${\mathbb{R}}^2$ finite energy implies that there are two kinds of boundary conditions for ϕ at space infinity,

$$|\varphi |\to 1\text{as}\hspace{0.17em}|x|\to \mathrm{\infty }$$ 4.7

and

$$|\varphi |\to 0\text{as}\hspace{0.17em}|x|\to \mathrm{\infty },$$ 4.8

which are called topological and non-topological, respectively, as that for the Chern–Simons model in [45]. Then, on ${\mathbb{R}}^2,$ 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

$$N<\frac{3{\mathrm{\lambda }}_0|\mathit{\Omega }|}{2\pi },$$ 4.9

where λ₀ defined by (4.18) is a positive constant depending only on the Born–Infeld parameter b.

(Sufficient condition) If

$$N<\frac{3|\mathit{\Omega }|}{32\pi }$$ 4.10

and |Ω| is suitably large, the problem (4.5)–(4.6) has a solution over Ω.

Furthermore, there also holds the magnetic flux and energy formula (3.11).

The second one is about the existence of topological solutions for (4.5)–(4.6).

Theorem 4.2 —

Consider the BPS equations (4.5)–(4.6) for the configurations (A₁,A₂,ϕ) over ${\mathbb{R}}^2$ with any prescribed zero set of ϕ given by (3.7). There exists a solution for the coupled system (4.5)–(4.6) realizing the boundary condition (4.7). Moreover, for this solution, there holds the following decay estimate at infinity,

$$|D_1\varphi |+|D_2\varphi |,F_{12}\mathrm{and}|\varphi {|}^2-1=\mathrm{O}({\mathrm{e}}^{-(1-\epsilon )\sqrt{3/2}|x|}),$$ 4.11

where ε∈(0,1) is arbitrarily small.

Furthermore, this solution carries minimum energy, and the magnetic flux and energy formula (3.11) also holds.

For the existence of non-topological solution, because it is more involved, here we deal only with the case where all vortices concentrate at one point.

Theorem 4.3 —

Consider the BPS equations (4.5)–(4.6) for the configurations (A₁,A₂,ϕ) over ${\mathbb{R}}^2$. For any $p\in {\mathbb{R}}^2$, integer N≥0, and $\alpha \ge \ln 2$, there exists a solution for the coupled system (4.5)–(4.6) such that p is the only zero of ϕ, with multiplicities N, which realizes the boundary condition (4.8) and is radially symmetric about p, with $\underset{x\in {\mathbb{R}}^2}{max}|\varphi {|}^2={\mathrm{e}}^{-\alpha }$.

Moreover, there hold the asymptotic estimates at infinity,

$$|\varphi {|}^2=\mathrm{O}(|x-p{|}^{-2\beta }),|D_j\varphi {|}^2=\mathrm{O}(|x-p{|}^{-2(\beta +1)})\mathrm{and}{F}_{12}=\mathrm{O}(|x-p{|}^{-2\beta }),$$ 4.12

where β>N+2 is a constant depending on α.

Furthermore, for the above solution, the magnetic flux and energy read as

$$\mathit{\Phi }={\int }_{{\mathbb{R}}^2}{F}_{12}\hspace{0.17em}\mathrm{d}x=2\pi (N+\beta ),E={\int }_{{\mathbb{R}}^2}\mathcal{E}\hspace{0.17em}\mathrm{d}x=2\pi (N+\beta ).$$ 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].

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 $u=\ln |\varphi {|}^2$. Then, equations (4.5)–(4.6) are reduced into the nonlinear elliptic equation

$$\mathrm{\Delta }u=\frac{6{\mathrm{e}}^u({\mathrm{e}}^u-1)}{(3+{\mathrm{e}}^u)\sqrt{1-{\mathrm{e}}^u{({\mathrm{e}}^u-1)}^2/b^2}}+4\pi \sum _{s=1}^N{\delta }_{p_s}.$$ 4.14

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

$$u\to 0\text{as}|x|\to \mathrm{\infty }$$ 4.15

and

$$u\to -\mathrm{\infty }\text{as}|x|\to \mathrm{\infty },$$ 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

$$\mathrm{\Delta }v=\frac{6{\mathrm{e}}^{u_0+v}({\mathrm{e}}^{u_0+v}-1)}{(3+{\mathrm{e}}^{u_0+v})\sqrt{1-{\mathrm{e}}^{u_0+v}{({\mathrm{e}}^{u_0+v}-1)}^2/b^2}}+\frac{4\pi N}{|\mathit{\Omega }|}.$$ 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

$${\mathrm{\lambda }}_0\equiv \underset{0\le t\le 1}{max}\frac{t(1-t)}{(3+t)\sqrt{1-t{(t-1)}^2/b^2}}>0.$$ 4.18

Then integrating equation (4.17) over Ω gives

$$\begin{array}{rl}0& ={\int }_{\mathit{\Omega }}\frac{6{\mathrm{e}}^{u_0+v}({\mathrm{e}}^{u_0+v}-1)}{(3+{\mathrm{e}}^{u_0+v})\sqrt{1-{\mathrm{e}}^{u_0+v}{({\mathrm{e}}^{u_0+v}-1)}^2/b^2}}\hspace{0.17em}\mathrm{d}x+4\pi N\\ & >-6{\mathrm{\lambda }}_0|\mathit{\Omega }|+4\pi N,\end{array}$$

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 super- and subsolutions.

First, we observe that $\overline{u}\equiv 0$ 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

$$\mathrm{\Delta }w=\frac{3}{2}{\mathrm{e}}^w({\mathrm{e}}^w-1)+4\pi \sum _{s=1}^N{\delta }_{p_s}.$$ 4.19

By [45,69,71,72], we know that (4.19) admits a solution w with w<0 if

$$\frac{3}{2}>\frac{16\pi N}{|\mathit{\Omega }|}$$ 4.20

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

$$\begin{array}{rl}\mathrm{\Delta }w& =\frac{3}{2}{\mathrm{e}}^w({\mathrm{e}}^w-1)+4\pi \sum _{s=1}^N{\delta }_{p_s}\\ & \ge \frac{6{\mathrm{e}}^w({\mathrm{e}}^w-1)}{({\mathrm{e}}^w+3)\sqrt{1-{\mathrm{e}}^w{({\mathrm{e}}^w-1)}^2/b^2}}+4\pi \sum _{s=1}^N{\delta }_{p_s}\end{array}$$ 4.21

holds in the sense of distribution. Hence, $\underset{\_}{u}\equiv w$ is a subsolution of equation (4.14) over Ω in the sense of distribution.

As a consequence, $\underset{\_}{v}\equiv -u_0+w$ and $\overline{v}\equiv -u_0$, satisfying $\underset{\_}{v}<\overline{v}$, are a pair of sub- and super-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 ${\mathbb{R}}^2$ with boundary condition (4.15).

It follows from maximum principle that any solutions of (4.14) over ${\mathbb{R}}^2$ 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

$$\mathrm{\Delta }v=\frac{6{\mathrm{e}}^{u_0+v}({\mathrm{e}}^{u_0+v}-1)}{(3+{\mathrm{e}}^{u_0+v})\sqrt{1-{\mathrm{e}}^{u_0+v}{({\mathrm{e}}^{u_0+v}-1)}^2/b^2}}+g$$ 4.22

and the boundary condition (4.15) becomes

$$v\to 0\text{as}|x|\to \mathrm{\infty }.$$ 4.23

We directly see that $\overline{u}=0$ 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 ${\mathbb{R}}^2$ the problem

$$\mathrm{\Delta }w=\frac{3}{2}{\mathrm{e}}^w({\mathrm{e}}^w-1)+4\pi \sum _{s=1}^N{\delta }_{p_s}$$ 4.24

has a solution w, which satisfies w→0 at infinity and w<0 on ${\mathbb{R}}^2$. Then, for this w on ${\mathbb{R}}^2$, there holds

$$\begin{array}{rl}\mathrm{\Delta }w& =\frac{3}{2}{\mathrm{e}}^w({\mathrm{e}}^w-1)+4\pi \sum _{s=1}^N{\delta }_{p_s}\\ & \ge \frac{6{\mathrm{e}}^w({\mathrm{e}}^w-1)}{({\mathrm{e}}^w+3)\sqrt{1-{\mathrm{e}}^w{({\mathrm{e}}^w-1)}^2/b^2}}+4\pi \sum _{s=1}^N{\delta }_{p_s}\end{array}$$ 4.25

in the sense of distribution. In other words, $\underset{\_}{u}\equiv w$ is a subsolution for (4.14) with (4.15) in the sense of distribution.

Therefore, we get a pair of super- and subsolutions, $\overline{v}\equiv -u_0$ and $\underset{\_}{v}\equiv -u_0+w$ 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.

(c). Non-topological solutions

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 p_i being the origin, that is

$$\mathrm{\Delta }u=\frac{6{\mathrm{e}}^u({\mathrm{e}}^u-1)}{(3+{\mathrm{e}}^u)\sqrt{1-{\mathrm{e}}^u{({\mathrm{e}}^u-1)}^2/b^2}}+4\pi N\delta (x),$$ 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

$$u_{rr}+\frac{1}{r}{u}_r=\frac{6{\mathrm{e}}^u({\mathrm{e}}^u-1)}{(3+{\mathrm{e}}^u)\sqrt{1-{\mathrm{e}}^u{({\mathrm{e}}^u-1)}^2/b^2}},r>0,$$ 4.27

and

$$\underset{r\to 0}{lim}\frac{u(r)}{\ln r}=2N,\underset{r\to \mathrm{\infty }}{lim}u(r)=-\mathrm{\infty }.$$ 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

$$u_{tt}=\frac{6{\mathrm{e}}^{2t}{\mathrm{e}}^u({\mathrm{e}}^u-1)}{(3+{\mathrm{e}}^u)\sqrt{1-{\mathrm{e}}^u{({\mathrm{e}}^u-1)}^2/b^2}},t\in \mathbb{R}$$ 4.29

and

$$\underset{t\to -\mathrm{\infty }}{lim}{u}_t=2N,\underset{t\to \mathrm{\infty }}{lim}u=-\mathrm{\infty }.$$ 4.30

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

$$u_{tt}=\frac{6{\mathrm{e}}^{2t}{\mathrm{e}}^u({\mathrm{e}}^u-1)}{(3+{\mathrm{e}}^u)\sqrt{1-{\mathrm{e}}^u{({\mathrm{e}}^u-1)}^2/b^2}},t\in \mathbb{R}$$ 4.31

and

$$u(t_0)=-\alpha ,u_t(t_0)=0,$$ 4.32

where the parameters α>0, $t_0\in \mathbb{R}$.

By the standard ODE theory, we see that the problem (4.31)–(4.32) admits a unique global solution for any α>0, $t_0\in \mathbb{R}$.

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 $-\mathrm{\infty }$ in (4.30).

By integration of (4.31), we obtain

$$u_t(t)={\int }_{t_0}^t\frac{6{\mathrm{e}}^{2s}{\mathrm{e}}^{u(s)}({\mathrm{e}}^{u(s)}-1)}{(3+{\mathrm{e}}^{u(s)})\sqrt{1-{\mathrm{e}}^{u(s)}{({\mathrm{e}}^{u(s)}-1)}^2/b^2}}\hspace{0.17em}\mathrm{d}s.$$ 4.33

Noting u(t₀)=−α<0, we see from (4.33) that u(t)<−α for all t<t₀. Then the limit

$$\gamma (t_0,\alpha )\equiv \underset{t\to -\mathrm{\infty }}{lim}{u}_t(t)$$ 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.

It follows from equation (4.31) that

$$u_{tt}>-2a_0{\mathrm{e}}^{2t}{\mathrm{e}}^u,$$ 4.35

where

$$a_0\equiv \frac{1}{\sqrt{1-(4/27)/b^2}}.$$ 4.36

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

$$v_{tt}>-2a_0{\mathrm{e}}^v.$$ 4.37

Note that v_t=u_t+2 and u_t>0 for t<t₀ ensured by (4.33). Multiplying (4.37) by v_t and by integration, we see that for t<t₀

$$4-{(v_t)}^2>4a_0({\mathrm{e}}^{v(t)}-{\mathrm{e}}^{2t_0-\alpha }),$$ 4.38

which gives

$$0<{(v_t)}^2<4(1+a_0){\mathrm{e}}^{2t_0-\alpha }.$$ 4.39

Then, we obtain

$$0<u_t<2(\sqrt{1+a_0{\mathrm{e}}^{2t_0-\alpha }}-1)\equiv K_0\text{for}t<t_0,$$ 4.40

which implies

$$-\alpha -K_0(t_0-t)<u(t)<-\alpha \text{for}t<t_0.$$ 4.41

It is straightforward to see that the function e^s(e^s−1) is decreasing on $(-\mathrm{\infty },-\ln 2)$. Let $\alpha \ge \ln 2$. We see from (4.41) that

$$u_{tt}<\frac{6{\mathrm{e}}^{2t}{\mathrm{e}}^u({\mathrm{e}}^u-1)}{(3+{\mathrm{e}}^{-\alpha })}\text{for}t<t_0,$$ 4.42

integrating which over $(-\mathrm{\infty },t_0)$ concludes

$$\begin{array}{rl}\gamma (t_0,\alpha )& >-\frac{6}{(3+{\mathrm{e}}^{-\alpha })}{\int }_{-\mathrm{\infty }}^{t_0}{\mathrm{e}}^{2t}{\mathrm{e}}^{-\alpha -K_0(t_0-t)}({\mathrm{e}}^{-\alpha -K_0(t_0-t)}-1)\hspace{0.17em}\mathrm{d}t\\ & =\frac{6{\mathrm{e}}^{2t_0-\alpha }}{(3+{\mathrm{e}}^{-\alpha })}(\frac{1}{K_0+2}-\frac{{\mathrm{e}}^{-\alpha }}{2(K_0+1)})\\ & >\frac{3{\mathrm{e}}^{2t_0-\alpha }}{(3+{\mathrm{e}}^{-\alpha })(K_0+2)}\\ & =\frac{3{\mathrm{e}}^{2t_0-\alpha }}{2(3+{\mathrm{e}}^{-\alpha })\sqrt{1+a_0{\mathrm{e}}^{2t_0-\alpha }}}.\end{array}$$ 4.43

Therefore, it follows from (4.43) that, for given α, there exists some t₀ such that

$$\gamma (t_0,\alpha )>2N.$$ 4.44

From (4.40), we may obtain

$$\gamma (t_0,\alpha )\le 2(\sqrt{1+a_0{\mathrm{e}}^{2t_0-\alpha }}-1).$$ 4.45

Hence, for given α, there exists a suitable t₀, so that

$$\gamma (t_0,\alpha )<2N.$$ 4.46

Consequently, noting (4.44) and (4.46), for any $\alpha \ge \ln 2$, we may find a suitable t₀=t₀(α) such that

$$\gamma (t_0,\alpha )=2N.$$ 4.47

Therefore, we realize the boundary condition at $-\mathrm{\infty }$ in (4.30).

In what follows, we aim to achieve the boundary condition at $\mathrm{\infty }$ in (4.30).

Noting u<0, we see from (4.33) that u_t<0 for all t>t₀. Then, we have u<−α for all $t\in \mathbb{R}\setminus \{t_0\}$, which implies the integral

$${\int }_{t_0}^{\mathrm{\infty }}{\mathrm{e}}^{2s+u(s)}\hspace{0.17em}\mathrm{d}s\text{is convergent}.$$ 4.48

Therefore, the limit

$$-2\beta \equiv \underset{t\to \mathrm{\infty }}{lim}{u}_t={\int }_{t_0}^{\mathrm{\infty }}\frac{6{\mathrm{e}}^{2s}{\mathrm{e}}^{u(s)}({\mathrm{e}}^{u(s)}-1)}{(3+{\mathrm{e}}^{u(s)})\sqrt{1-{\mathrm{e}}^{u(s)}{({\mathrm{e}}^{u(s)}-1)}^2/b^2}}\hspace{0.17em}\mathrm{d}s<0\text{is finite}.$$ 4.49

Then, (4.48) implies β>1. Hence, the boundary condition at $\mathrm{\infty }$ in (4.30) is established.

Next, we derive a more precise estimate for β.

Multiplying (4.31) by u_t and integrating over $(-\mathrm{\infty },+\mathrm{\infty })$, and using the fact β>1, we have

$$\begin{array}{rl}2({\beta }^2-N^2)& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{6{\mathrm{e}}^{2t}{\mathrm{e}}^{u(t)}({\mathrm{e}}^{u(t)}-1)u_t(t)}{(3+{\mathrm{e}}^{u(t)})\sqrt{1-{\mathrm{e}}^{u(t)}{({\mathrm{e}}^{u(t)}-1)}^2/b^2}}\hspace{0.17em}\mathrm{d}t\\ & ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{2t}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{-\mathrm{\infty }}^{u(t)}\frac{6{\mathrm{e}}^s({\mathrm{e}}^s-1)}{(3+{\mathrm{e}}^s)\sqrt{1-{\mathrm{e}}^s{({\mathrm{e}}^s-1)}^2/b^2}}\hspace{0.17em}\mathrm{d}s\hspace{0.17em}\mathrm{d}t\\ & ={\mathrm{e}}^{2t}{\int }_{-\mathrm{\infty }}^{u(t)}{\frac{6{\mathrm{e}}^s({\mathrm{e}}^s-1)}{(3+{\mathrm{e}}^s)\sqrt{1-{\mathrm{e}}^s{({\mathrm{e}}^s-1)}^2/b^2}}\hspace{0.17em}\mathrm{d}s|}_{-\mathrm{\infty }}^{\mathrm{\infty }}\\ & -2{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{2t}{\int }_{-\mathrm{\infty }}^{u(t)}\frac{6{\mathrm{e}}^s({\mathrm{e}}^s-1)}{(3+{\mathrm{e}}^s)\sqrt{1-{\mathrm{e}}^s{({\mathrm{e}}^s-1)}^2/b^2}}\hspace{0.17em}\mathrm{d}s\hspace{0.17em}\mathrm{d}t\\ & =-2{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{6{\mathrm{e}}^{2t}{\mathrm{e}}^{u(t)}({\mathrm{e}}^{u(t)}-1)}{(3+{\mathrm{e}}^{u(t)})\sqrt{1-{\mathrm{e}}^{u(t)}{({\mathrm{e}}^{u(t)}-1)}^2/b^2}}\hspace{0.17em}\mathrm{d}t+2{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{2t}\tilde{H}(u(t))\hspace{0.17em}\mathrm{d}t,\end{array}$$ 4.50

where

$$\tilde{H}(u)\equiv \frac{6{\mathrm{e}}^u({\mathrm{e}}^u-1)}{(3+{\mathrm{e}}^u)\sqrt{1-{\mathrm{e}}^u{({\mathrm{e}}^u-1)}^2/b^2}}-{\int }_{-\mathrm{\infty }}^u\frac{6{\mathrm{e}}^s({\mathrm{e}}^s-1)}{(3+{\mathrm{e}}^s)\sqrt{1-{\mathrm{e}}^s{({\mathrm{e}}^s-1)}^2/b^2}}\hspace{0.17em}\mathrm{d}s.$$ 4.51

Noting b>1 and u<0, it is direct to check that

$$\frac{\mathrm{d}\tilde{H}(u)}{\mathrm{d}u}=\frac{3{\mathrm{e}}^{2u}(8b^2+3{[{\mathrm{e}}^u-1]}^2[{\mathrm{e}}^{2u}-1])}{b^2{({\mathrm{e}}^u+3)}^2{(1-{\mathrm{e}}^u{({\mathrm{e}}^u-1)}^2/b^2)}^{3/2}}>0$$ 4.52

and $\tilde{H}(-\mathrm{\infty })=0$, which yields $\tilde{H}(u)>0$. Then, we conclude from (4.50) that

$$2({\beta }^2-N^2)>4(\beta +N),$$ 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.

• (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].

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.

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.