Remarks on Chern-Simons-Higgs Equations in

Abstract

We prove global existence of solutions to Chern-Simons-Higgs equations under the gauge condition . We also find stationary solutions.

1. Introduction

We are interested in Chern-Simons-Higgs equations in . The model was proposed in [1] to study BPS domain wall solutions of the Chern-Simons-Higgs system. The Lagrangian density of the dimensional Chern-Simons-Higgs system is given bywhere , , are real fields, is a complex scalar field, and . The covariant derivative is defined by and the potential is given bywhere is a coupling constant.

The Chern-Simons-Higgs system [2, 3] in is derived from the following Lagrangian density:where is totally skew symmetric tensor with . The Lagrangian density (1) is obtained by the dimensional reduction of (3). More precisely, we consider the Lagrangian (3) independent of the coordinate and renaming as to get (1).

The Euler-Lagrange equations of (1) areThe conservation of the total energy implies thatwhere the energy density corresponding to the Lagrangian density (1) is given by

For the static configuration, this model with self-dual coupling constant admits a system of first order self-dual equations whose solutions minimize the total static energy (8). The solutions to the self-dual equations and their nonrelativistic limit have been studied in [1].

The system of (4)–(7) is invariant under the following gauge transformations:where is a real-valued smooth function on . Here we will fix the gauge by imposing the condition . Note that temporal gauge condition is well known. The motivation considering the gauge condition comes from a standing wave solution of (4)–(7). As shown in Section 2 the usual ansatz of standing wave solution leads to . To study stability it seems natural to study the initial value problem of (4)–(7) with the condition . We have studied Chern-Simons-Schrödinger [4] and Chern-Simons-Dirac [5] equations in the same gauge condition. Our first result is concerned with standing wave solutions of (4)–(7).

Theorem 1. There are standing wave solutions to (4)–(7) of the formwhere and is real-valued function. In fact, we find solutions (26) satisfying the boundary condition and solutions (32) satisfying the boundary condition or .

Low regularity local solutions have been recently constructed in [6] where it is proved that Chern-Simons-Higgs system under the Lorenz gauge is locally well posed with the regularity : In [6], they observed that the Chern-Simons-Higgs system has null form structures corresponding to the Lorenz gauge and used several null form estimates. Moreover the global energy solution has been constructed when . We study the initial value problem of Chern-Simons-Higgs equations under the gauge condition to obtain the following result.

Theorem 2. For the initial data and , the initial value problem for (44) has a global solution which belongs to

In Section 2 we find standing wave solutions of CSH to prove Theorem 1. Theorem 2 is proved in Section 3. We conclude this section by giving a few notations. We denote by the D’Alembertian operator . We use to denote various constants. When we are interested in local solutions, we may assume that . Thus we shall replace smooth function of , by . We use to denote an estimate of the form .

2. Standing Wave Solutions

In this section we consider standing wave solutions of the formwhere is a real constant and , , , are real-valued functions. From (6) we derive which leads us to . Then we can rewrite (4)–(7) as follows:where denotes a derivative .

Using (16) and (17), we may checkfrom which we setThen (15) becomes

The energy of (20) becomesWe consider two cases and .

(1) The case of : to make the energy (21) finite, we consider a boundary condition which is called “symmetric phase.” Multiplying (20) by and considering symmetry phase boundary condition, we obtain

For , changing of variables and using the following formula we obtain To guarantee a smooth solution, we choose + sign and impose to getwhere and .

Considering and (16), we have which can be written asIntegrating (27), we obtain which can be rewritten as can be obtained by (19). We can treat the case in the same manner.

(2) The case of : to make the energy (21) finite, we consider a boundary condition which is called “asymmetric phase.” We can check that if is a solution of (20), then is also a solution of (20).

Considering asymmetry phase boundary condition, we obtain which implieswhere denotes . Since and , we have , which gives us for some constant . We also have .

3. Proof of Theorem 2

Here we study the initial value problem of Chern-Simons-Higgs system in one space dimension under the gauge condition . We are interested in the symmetry phase boundary condition and assume for a simpler presentation.

3.1. Local Existence

The system (4)–(7) with the gauge condition can be rewritten asWe obtain from (35)

From (19) and (29), we can check and . We will impose a boundary condition , , and . Let us check compatibility of (37) with other equations (34)–(36). Using (34) and (36), we can check that which implies .

Now we obtain, from (35) and (36),Multiplying (40), (41) by , , respectively, and integrating with a boundary condition and , we obtain and as follows:From (37) and (38), we havewhere a boundary condition is used. In the following, we assume for simple presentation. In fact, the boundary condition does not affect the proof of Theorem 2.

We have shown that the initial value problem of (34)–(37) reduces to the study of the following equation:where , , and are defined by (42)-(43).

We are ready to prove the existence of local solutions to (44). Proof follows by standard arguments from a priori estimates of the following propositions.

Proposition 3. Let be a solution of (44) in a strip withDefine Then there exist constants and , depending only on , such that if then .

Proposition 4. Let and be two solutions of (44) verifying the hypothesis of Proposition 3 in a strip and let as in Proposition 3 and be the corresponding quantity for the primed solution. Also define Then there exist constants and , depending only on and , such that if then .

Proposition 4 follows by the similar argument to Proposition 3. We will only present the proof of Proposition 3. We define It is easily shown that by applying energy estimate. We will derive the inequalityThen a bootstrap argument completes the proof of Proposition 3.

From the representation (42), we have the following bounds: Considering (43), we have Applying the above bounds, the integrals in can be treated as follows: Combining the above estimates, we get the relation (49).

3.2. Global Existence of the Energy Solution

Here we prove the global existence of the solution to (44) for the initial data and . The existence of global energy solutions is proved by showing that the corresponding norm of the local solution is uniformly bounded on each finite time interval.

The energy conservation (8) implies that the quantity is bounded by . First of all, we haveTo control norm of , we considerTaking into account , it follows thatFrom (36), we have the following bound of : where is a polynomial of and . Then we obtain from (35) Making use of the above information of and , we can control the norm of as follows:Combining (54)–(59), we conclude that where is a bounded function on finite time interval .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this article and regarding the funding that they have received.

Acknowledgments

H. Huh was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2053747).