Reduction of Chern-Simons-Schrödinger Systems in One Space Dimension

Abstract

We study Chern-Simons-Schrödinger systems in one space dimension. We show that Chern-Simons-Schrödinger and 𝒩=2 supersymmetric Chern-Simons-Schrödinger equations can be reduced, under the gauge condition 𝐴₁≡0, to equations of 𝜙, 𝜓 only which are coupled cubic Schrödinger systems.

1. Introduction

In this paper, we consider the Chern-Simons-Schrödinger (CSS) system 𝑖𝐷₀𝜙+𝐷₁𝐷₁𝜙−𝑁²2𝜙+𝜅𝜙²𝜙=0,𝜅𝜕₁𝑁+𝜙²=0,𝜅𝐹₀₁−2𝑁𝜙²=0,𝜅𝜕₀𝑁−2Im𝜙𝐷₁𝜙=0 (1) and the 𝒩=2 supersymmetric Chern-Simons-Schrödinger (s-CSS) system 𝑖𝐷₀𝜙+𝐷₁𝐷₁𝜙−𝑁²𝜙+2𝜆₁𝜙²𝜙+𝜆₂𝜓²𝜙=0,𝑖𝐷₀𝜓+𝐷₁𝐷₁𝜓−𝜓𝜕₁𝑁−𝑁²𝜓+𝜆₂𝜙²𝜓=0,𝜅𝜕₁𝑁+𝜙²+𝜓²=0,𝜅𝐹₀₁+𝜕₁𝜓²−2𝑁𝜙²+𝜓²=0,𝜅𝜕₀𝑁−2Im𝜙𝐷₁𝜙−2Im𝜓𝐷₁𝜓=0, (2) on ℝ¹⁺¹. Here, 𝐴₀, 𝐴₁, and 𝑁 are real fields, 𝜙, 𝜓 are complex scalar fields, and 𝐹₀₁=𝜕₀𝐴₁−𝜕₁𝐴₀. The covariant derivative is defined by 𝐷_𝛼𝜙=𝜕_𝛼𝜙−𝑖𝐴_𝛼𝜙, (𝛼=0,1). 𝜆₁, 𝜆₂, and 𝜅>0 are coupling constants. The space-time derivatives of a function 𝑓 are denoted by (𝜕₀𝑓,𝜕₁𝑓)=(𝜕_𝑡𝑓,𝜕_𝑥𝑓).

The CSS model was proposed in [1] to study BPS domain wall solutions. The Lagrangian density of the (1+1) dimensional Chern-Simons-Schrödinger system is given by ℒ=𝜅𝑁𝐹₀₁+𝑖𝜙𝐷₀𝜙−𝐷₁𝜙²−𝑁²𝜙²+1𝜅𝜙⁴, (3) which is obtained by the dimensional reduction of the Lagrangian density of (2+1) dimensional Chern-Simons-Schrödinger system in [2]. The (s-CSS) system is derived, by the dimensional reduction, from the paper [3].

An important property of the systems CSS and s-CSS is the gauge invariance. Therefore, a solution to the systems CSS and s-CSS is formed by a class of gauge equivalent pairs (𝜙,𝐴₀,𝐴₁,𝑁) and (𝜙,𝜓,𝐴₀,𝐴₁,𝑁), respectively. Here, we will fix the gauge by imposing the condition 𝐴₁≡0. Note that temporal gauge condition 𝐴₀≡0 is well known.

The motivation considering the gauge condition 𝐴₁≡0 comes from standing wave solutions of CSS. As shown in Section 2.1, the usual ansatz of standing wave leads to 𝐴₁≡0. To study stability, it seems natural to study the initial value problem of CSS with the condition 𝐴₁≡0. The other motivation is that the Schrödinger part in CSS system is written, under the Lorenz gauge condition 𝜕₀𝐴₀−𝜕₁𝐴₁=0, as follows: 𝑖𝜕_𝑡𝜙+𝜕_𝑥𝜕_𝑥𝜙+𝐴₀𝜙−𝑖𝜙𝜕_𝑥𝐴₁−2𝑖𝐴₁𝜕_𝑥𝜙−𝐴²₁𝜙−𝑁²2𝜙+𝜅𝜙²𝜙=0, (4) where we have a singular derivative nonlinear term 𝐴₁𝜕_𝑥𝜙. The gauge condition 𝐴₁≡0 removes troublesome nonlinearity automatically. Note that Lorenz gauge condition was made use of in previous studies [4, 5] on Maxwell-Schrödinger equations in one space dimension.

The initial value problem of the Chern-Simons-Schrödinger system in ℝ²⁺¹ was investigated in [6–9]. Blow-up solutions in finite time have been studied in [6] by deriving a virial identity and in [10] by the use of a pseudoconformal transformation. The existence of standing wave solutions has been studied in [11, 12]. Global energy solutions of Chern-Simons-Higgs equations in one space dimension have been studied in [13].

In this study, we consider smooth solutions which satisfy equations in the classical sense and decay properly at spatial infinity. Our first result says that CSS system can be reduced, under the gauge condition 𝐴₁≡0, to the equation of 𝜙 only which is a cubic Schrödinger equation.

Theorem 1. Let one consider a smooth solution (𝜙,𝐴₀,𝑁) of (15)–(18) satisfying 𝜙∈𝐶([0,𝑇];𝐻²(ℝ)). Then, the scalar field 𝜙 is also a solution to the following Schrödinger equation: 𝑖𝜕_𝑡𝜙+𝜕_𝑥𝜕_𝑥2𝜙+𝜅𝜙²𝜙=0. (5)

The s-CSS system can be reduced, under the gauge condition 𝐴₁≡0, to the system of 𝜙 and 𝜓 only.

Theorem 2. Let one consider a smooth solution (𝜙,𝜓,𝐴₀,𝑁) of the system (27)–(31) satisfying 𝜙,𝜓∈𝐶([0,𝑇];𝐻²(ℝ)). Then, the scalar fields 𝜙 and 𝜓 are also a solution to the following coupled Schrödinger equations: 𝑖𝜕_𝑡𝜙+𝜕_𝑥𝜕_𝑥𝜙+2𝜆₁𝜙²𝜙+𝜆₂+1𝜅𝜓²𝜙=0,𝑖𝜕_𝑡𝜓+𝜕_𝑥𝜕_𝑥2𝜓+𝜅𝜓²𝜓+𝜆₂+1𝜅𝜙²𝜓=0. (6)

Remark 3. (i) The model (5) is a cubic Schrödinger equation with attractive potential, and the system (6) is the two coupled Schrödinger equations. In particular, when 𝜆₂=−1/𝜅, the equations are two versions of a single nonlinear Schrödinger equation which is integrable.
(ii) Looking for standing wave solutions of (6), 𝜙(𝑥,𝑡)=𝑒^{𝑖𝜔1𝑡}𝑢(𝑥) and 𝜓(𝑥,𝑡)=𝑒^{𝑖𝜔2𝑡}𝑣(𝑥), one can check that 𝑢 and 𝑣 satisfy the following system: 𝜕_𝑥𝜕_𝑥𝑢−𝜔₁𝑢+2𝜆₁|𝑢|²𝑢+𝜆₂+1𝜅|𝑣|²𝜕𝑢=0,_𝑥𝜕_𝑥𝑣−𝜔₂2𝑣+𝜅|𝑣|²𝑣+𝜆₂+1𝜅|𝑢|²𝑣=0. (7) The existence of standing waves and their properties have been studied extensively, for instance, in [14–16].
Theorem 1 is proved in Section 2, and Theorem 2 is proved in Section 3. We give concluding remark in Section 4. We use the standard Sobolev space 𝐻²(ℝ) which denotes the set of weakly differentiable functions 𝑢 on ℝ such that 𝑢, 𝜕_𝑥𝑢, and 𝜕_𝑥𝜕_𝑥𝑢 are square integrable.

2. Reduction of Chern-Simons-Schrödinger System

Here, we consider the reduction of Chern-Simons-Schrödinger system in one space dimension. In Section 2.1, we investigate standing wave solutions of CSS system, and Theorem 1 is proved in Section 2.2.

2.1. Standing Wave Solutions of CSS System

In this section, we look for standing wave solutions of the form 𝜙(𝑡,𝑥)=𝑒^𝑖𝜔𝑡𝑢(𝑥),𝐴₀(𝑡,𝑥)=𝐴₀(𝑥),𝐴₁(𝑡,𝑥)=𝐴₁(𝑥),𝑁(𝑡,𝑥)=𝑁(𝑥), (8) where 𝜔 is a real constant and 𝑢 is a real-valued function. The fourth equation in (1) leads us to 𝐴₁≡0. Then, we have from (1) the following: 𝑢^′′−𝜔𝑢+𝐴₀𝑢−𝑁²2𝑢+𝜅|𝑢|²𝑢=0,𝜅𝑁^′+𝑢²=0,𝜅𝐴^′₀+2𝑁𝑢²=0, (9)(10)(11) where ′ denotes a derivative 𝑑/𝑑𝑥. From (10) and (11), we may have the following expressions, with a boundary condition 𝑁(−∞)=0=𝐴₀(−∞): 𝑁(𝑥)1=−𝜅^𝑥_−∞𝑢𝑦²𝐴𝑑𝑦,₀(𝑥)2=−𝜅^𝑥_−∞𝑁𝑦𝑢𝑦²𝑑𝑦. (12) A simple calculation shows that (𝑑/𝑑𝑥)(𝐴₀−𝑁²)=0 which implies (𝐴₀−𝑁²)(𝑥)=(𝐴₀−𝑁²)(−∞)=0. Then, (9) becomes 𝑢^′′2−𝜔𝑢+𝜅|𝑢|²𝑢=0. (13) We may obtain a solution 𝑢(𝑥)=√𝜅𝜔sech(√𝜔𝑥) and 𝑁(𝑥)=−√𝜔(1+tanh(√𝜔𝑥)).

2.2. Reduction of CSS System

The (CSS) system (1) is invariant under the following gauge transformation: 𝜙⟶𝜙𝑒^𝑖𝜒,𝐴_𝛼⟶𝐴_𝛼+𝜕_𝛼𝜒,𝑁⟶𝑁, (14) where 𝜒∶ℝ¹⁺¹→ℝ is a smooth function. Here, we impose the gauge condition 𝐴₁≡0 which reformulates the CSS system (1) as follows: 𝑖𝜕_𝑡𝜙+𝐴₀𝜙+𝜕_𝑥𝜕_𝑥𝜙−𝑁²2𝜙+𝜅𝜙²𝜙=0,𝜅𝜕_𝑥𝑁+𝜙²=0,𝜅𝜕_𝑥𝐴₀+2𝑁𝜙²=0,𝜅𝜕_𝑡𝑁−2Im𝜙𝜕_𝑥𝜙=0. (15)(16)(17)(18) From (16) and (17), we have, with a boundary condition 𝑁(−∞)=0=𝐴₀(−∞), the following representations: 𝑁()1𝑥,𝑡=−𝜅^𝑥_−∞𝜙𝑦,𝑡²𝐴𝑑𝑦,₀()2𝑥,𝑡=−𝜅^𝑥_−∞𝑁𝑦,𝑡𝜙𝑦,𝑡²𝑑𝑦. (19) Let us check the compatibility of (18) with other (15)–(17). Multiplying (15) by 𝜙 and taking imaginary part, we have 𝜕_𝑡𝜙²+2Im𝜙𝜕_𝑥𝜕_𝑥𝜙=0. (20) Taking time derivative of 𝑁 in (19) and considering (20), we have 𝜅𝜕_𝑡𝑁(𝑥,𝑡)=−^𝑥_−∞𝜕_𝑡𝜙²=𝑦,𝑡𝑑𝑦_𝑥−∞2𝜕_𝑦Im𝜙𝜕_𝑦𝜙𝑦,𝑡𝑑𝑦=2Im𝜙𝜕_𝑥𝜙, (21) where Im(𝜙𝜕_𝑥𝜕_𝑥𝜙)=𝜕_𝑥Im(𝜙𝜕_𝑥𝜙) and 𝜙(⋅,𝑡)∈𝐻²(ℝ) are used.

We have showed that the study of (15)–(18) reduces to the following system: 𝑖𝜕_𝑡𝜙+𝐴₀𝜙+𝜕_𝑥𝜕_𝑥𝜙−𝑁²2𝜙+𝜅𝜙²𝜙=0, (22) where 𝑁 and 𝐴₀ are defined by (19). Moreover, we can check that, using (19): 𝜅𝜕_𝑥𝐴₀−𝑁²=−2𝑁𝜙²−2𝑁−𝜙²=0, (23) which implies (𝐴₀−𝑁²)(𝑥,𝑡)=(𝐴₀−𝑁²)(−∞,𝑡)=0. Therefore, (22) reduces finally to 𝑖𝜕_𝑡𝜙+𝜕_𝑥𝜕_𝑥2𝜙+𝜅𝜙²𝜙=0, (24) which proves Theorem 1.

3. Reduction of s-CSS System

The Lagrangian density of the (1+1) dimensional 𝒩=2 supersymmetric Chern-Simons-Schrödinger system is given by ℒ=𝜅𝑁𝐹₀₁+𝑖𝜙𝐷₀𝜙−𝐷₁𝜙²+𝑖𝜓𝐷₀𝜓−𝐷₁𝜓²−𝜓²𝜕₁𝑁−𝑁²𝜙²+𝜓²+𝜆₁𝜙⁴+𝜆₂𝜙|²𝜓|², (25) which is obtained by the dimensional reduction of the Lagrangian density of (2+1) dimensional 𝒩=2 supersymmetric Chern-Simons-Schrödinger system in [3]. The s-CSS system (2) is invariant under the following gauge transformation: 𝜙⟶𝜙𝑒^𝑖𝜒,𝜓⟶𝜓𝑒^𝑖𝜒,𝐴_𝛼⟶𝐴_𝛼+𝜕_𝛼𝜒,𝑁⟶𝑁, (26) where 𝜒∶ℝ¹⁺¹→ℝ is a smooth function. We consider the gauge condition 𝐴₁≡0 which reformulates the s-CSS system (2) as follows: 𝑖𝜕_𝑡𝜙+𝐴₀𝜙+𝜕_𝑥𝜕_𝑥𝜙−𝑁²𝜙+2𝜆₁𝜙²𝜙+𝜆₂𝜓²𝜙=0,𝑖𝜕_𝑡𝜓+𝐴₀𝜓+𝜕_𝑥𝜕_𝑥𝜓−𝜓𝜕_𝑥𝑁−𝑁²𝜓+𝜆₂𝜙²𝜓=0,𝜅𝜕_𝑥𝑁+𝜙²+𝜓²=0,𝜅𝜕_𝑥𝐴₀−𝜕_𝑥𝜓²+2𝑁𝜙²+𝜓²=0,𝜅𝜕_𝑡𝑁−2Im𝜙𝜕_𝑥𝜙−2Im𝜓𝜕_𝑥𝜓=0. (27)(28)(29)(30)(31) From (29) and (30), we have, with a boundary condition 𝑁(−∞)=0=𝐴₀(−∞), the following representations: 𝜅𝑁(𝑥,𝑡)=−^𝑥_−∞𝜙²+𝜓²𝑦,𝑡𝑑𝑦,𝜅𝐴₀(𝑥,𝑡)=𝜓(𝑥,𝑡)²−2^𝑥_−∞𝑁𝜙²+𝜓²𝑦,𝑡𝑑𝑦. (32) Let us check the compatibility of (31) with other (27)–(30). Multiplying (27) and (28) by 𝜙 and 𝜓, respectively, and taking imaginary part, we have 𝜕_𝑡𝜙²+2Im𝜙𝜕_𝑥𝜕_𝑥𝜙𝜕=0,_𝑡𝜓²+2Im𝜓𝜕_𝑥𝜕_𝑥𝜓=0. (33) Taking time derivative of 𝑁 in (32) and considering (33), we have 𝜅𝜕_𝑡𝑁(𝑥,𝑡)=−^𝑥_−∞𝜕_𝑡𝜙²+𝜕_𝑡𝜓²=𝑦,𝑡𝑑𝑦_𝑥−∞2𝜕_𝑦Im𝜙𝜕_𝑦𝜙+2𝜕_𝑦Im𝜓𝜕_𝑦𝜓×𝑦,𝑡𝑑𝑦=2Im𝜙𝜕_𝑥𝜙+2Im𝜓𝜕_𝑥𝜓. (34)

We have showed that the study of (27)–(31) reduces to the following system: 𝑖𝜕_𝑡𝜙+𝐴₀𝜙+𝜕_𝑥𝜕_𝑥𝜙−𝑁²𝜙+2𝜆₁𝜙²𝜙+𝜆₂𝜓²𝜙=0,𝑖𝜕_𝑡𝜓+𝐴₀𝜓+𝜕_𝑥𝜕_𝑥𝜓−𝜓𝜕_𝑥𝑁−𝑁²𝜓+𝜆₂𝜙²𝜓=0, (35) where 𝑁 and 𝐴₀ are defined by (32). Now we can check that 𝜕_𝑥(𝜅𝐴₀−𝜅𝑁²−|𝜓|²)=0, which implies 𝜅𝐴₀−𝜅𝑁²−𝜓²(𝑥,𝑡)=0. (36) Taking (32) and (36) into account, we can check that 𝐴₀−𝜕_𝑥𝑁−𝑁²()=1𝑥,𝑡𝜅𝜙²+2𝜓²(𝑥,𝑡). (37) Then, considering (36) and (37), the system (35) reduces to 𝑖𝜕_𝑡𝜙+𝜕_𝑥𝜕_𝑥𝜙+2𝜆₁𝜙²𝜙+𝜆₂+1𝜅𝜓²𝜙=0,𝑖𝜕_𝑡𝜓+𝜕_𝑥𝜕_𝑥2𝜓+𝜅𝜓²𝜓+𝜆₂+1𝜅𝜙²𝜓=0, (38)(39) which proves Theorem 2.

4. Concluding Remark

As we pointed out in Section 1, Schrödinger equations with electromagnetic field like Maxwell-Schrödinger and Chern-Simons-Schrödinger have singular derivative nonlinear terms which give difficulties in analysis of the PDEs. Those challenging problems have prompted development of analytic methods and the results [4–12, 17] regarding issues such as existence, blowup, and asymptotic behaviors of the solution. In this aspect, the results of this study seem interesting and quite unique. CSS and s-CSS systems with gauge condition 𝐴₁≡0 reduce to coupled cubic Schrödinger equations which are much easier from analytic and numerical point of view. We could not obtain similar reduction result for the related equations like Chern-Simons-Higgs and Chern-Simons-Dirac in one space dimension which have their own interesting structures [13, 18].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0015866).