1. Introduction
In this paper, we consider the Chern-Simons-Schrödinger (CSS) system
(1)iD0ϕ+D1D1ϕ-N2ϕ+2κ|ϕ|2ϕ=0,κ∂1N+|ϕ|2=0,κF01-2N|ϕ|2=0,κ∂0N-2Im(ϕ-D1ϕ)=0
and the 𝒩=2 supersymmetric Chern-Simons-Schrödinger (s-CSS) system
(2)iD0ϕ+D1D1ϕ-N2ϕ+2λ1|ϕ|2ϕ+λ2|ψ|2ϕ=0,iD0ψ+D1D1ψ-ψ∂1N-N2ψ+λ2|ϕ|2ψ=0,κ∂1N+(|ϕ|2+|ψ|2)=0,κF01+∂1(|ψ|2)-2N(|ϕ|2+|ψ|2)=0,κ∂0N-2Im(ϕ-D1ϕ)-2Im(ψ-D1ψ)=0,
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 Dαϕ=∂αϕ-iAαϕ, (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
(3)ℒ=κNF01+iϕ-D0ϕ-|D1ϕ|2-N2|ϕ|2+1κ|ϕ|4,
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:
(4)i∂tϕ+∂x∂xϕ+A0ϕ-iϕ∂xA1 -2iA1∂xϕ-A12ϕ-N2ϕ+2κ|ϕ|2ϕ=0,
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:
(5)i∂tϕ+∂x∂xϕ+2κ|ϕ|2ϕ=0.

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:
(6)i∂tϕ+∂x∂xϕ+2λ1|ϕ|2ϕ+(λ2+1κ)|ψ|2ϕ=0,i∂tψ+∂x∂xψ+2κ|ψ|2ψ+(λ2+1κ)|ϕ|2ψ=0.

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)=eiω1tu(x) and ψ(x,t)=eiω2tv(x), one can check that u and v satisfy the following system:
(7)∂x∂xu-ω1u+2λ1|u|2u+(λ2+1κ)|v|2u=0,∂x∂xv-ω2v+2κ|v|2v+(λ2+1κ)|u|2v=0.
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
(8)ϕ(t,x)=eiωtu(x), A0(t,x)=A0(x),A1(t,x)=A1(x), N(t,x)=N(x),
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:
(9)u′′-ωu+A0u-N2u+2κ|u|2u=0,(10)κN′+u2=0,(11)κA0′+2Nu2=0,
where ′ denotes a derivative d/dx. From (10) and (11), we may have the following expressions, with a boundary condition N(-∞)=0=A0(-∞):
(12)N(x)=-1κ∫-∞x|u(y)|2dy,A0(x)=-2κ∫-∞xN(y)|u(y)|2dy.
A simple calculation shows that (d/dx)(A0-N2)=0 which implies (A0-N2)(x)=(A0-N2)(-∞)=0. Then, (9) becomes
(13)u′′-ωu+2κ|u|2u=0.
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:
(14)ϕ⟶ϕeiχ, Aα⟶Aα+∂αχ, N⟶N,
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:
(15)i∂tϕ+A0ϕ+∂x∂xϕ-N2ϕ+2κ|ϕ|2ϕ=0,(16)κ∂xN+|ϕ|2=0,(17)κ∂xA0+2N|ϕ|2=0,(18)κ∂tN-2Im(ϕ-∂xϕ)=0.
From (16) and (17), we have, with a boundary condition N(-∞)=0=A0(-∞), the following representations:
(19)N(x,t)=-1κ∫-∞x|ϕ(y,t)|2dy,A0(x,t)=-2κ∫-∞xN(y,t)|ϕ(y,t)|2dy.
Let us check the compatibility of (18) with other (15)–(17). Multiplying (15) by ϕ- and taking imaginary part, we have
(20)∂t|ϕ|2+2Im(ϕ-∂x∂xϕ)=0.
Taking time derivative of N in (19) and considering (20), we have
(21)κ∂tN(x,t)=-∫-∞x∂t|ϕ|2(y,t)dy=∫-∞x2∂yIm(ϕ-∂yϕ)(y,t)dy=2Im(ϕ-∂xϕ),
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:
(22)i∂tϕ+A0ϕ+∂x∂xϕ-N2ϕ+2κ|ϕ|2ϕ=0,
where N and A0 are defined by (19). Moreover, we can check that, using (19):
(23)κ∂x(A0-N2)=-2N|ϕ|2-2N(-|ϕ|2)=0,
which implies (A0-N2)(x,t)=(A0-N2)(-∞,t)=0. Therefore, (22) reduces finally to
(24)i∂tϕ+∂x∂xϕ+2κ|ϕ|2ϕ=0,
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
(25)ℒ=κNF01+iϕ-D0ϕ-|D1ϕ|2+iψ-D0ψ-|D1ψ|2-|ψ|2∂1N-N2(|ϕ|2+|ψ|2)+λ1|ϕ|4+λ2|ϕ|2|ψ|2,
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:
(26)ϕ⟶ϕeiχ, ψ⟶ψeiχ, Aα⟶Aα+∂αχ, N⟶N,
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:
(27)i∂tϕ+A0ϕ+∂x∂xϕ-N2ϕ+2λ1|ϕ|2ϕ+λ2|ψ|2ϕ=0,(28)i∂tψ+A0ψ+∂x∂xψ-ψ∂xN-N2ψ+λ2|ϕ|2ψ=0,(29)κ∂xN+(|ϕ|2+|ψ|2)=0,(30)κ∂xA0-∂x(|ψ|2)+2N(|ϕ|2+|ψ|2)=0,(31)κ∂tN-2Im(ϕ-∂xϕ)-2Im(ψ-∂xψ)=0.
From (29) and (30), we have, with a boundary condition N(-∞)=0=A0(-∞), the following representations:
(32)κN(x,t)=-∫-∞x(|ϕ|2+|ψ|2)(y,t)dy,κA0(x,t)=|ψ(x,t)|2-2∫-∞xN(|ϕ|2+|ψ|2)(y,t)dy.
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
(33)∂t|ϕ|2+2Im(ϕ-∂x∂xϕ)=0,∂t|ψ|2+2Im(ψ-∂x∂xψ)=0.
Taking time derivative of N in (32) and considering (33), we have
(34)κ∂tN(x,t)=-∫-∞x(∂t|ϕ|2+∂t|ψ|2)(y,t)dy=∫-∞x(2∂yIm(ϕ-∂yϕ)+2∂yIm(ψ-∂yψ)) ×(y,t)dy=2Im(ϕ-∂xϕ)+2Im(ψ-∂xψ).

We have showed that the study of (27)–(31) reduces to the following system:
(35)i∂tϕ+A0ϕ+∂x∂xϕ-N2ϕ+2λ1|ϕ|2ϕ+λ2|ψ|2ϕ=0,i∂tψ+A0ψ+∂x∂xψ-ψ∂xN-N2ψ+λ2|ϕ|2ψ=0,
where N and A0 are defined by (32). Now we can check that ∂x(κA0-κN2-|ψ|2)=0, which implies
(36)(κA0-κN2-|ψ|2)(x,t)=0.
Taking (32) and (36) into account, we can check that
(37)(A0-∂xN-N2)(x,t)=1κ(|ϕ|2+2|ψ|2)(x,t).
Then, considering (36) and (37), the system (35) reduces to
(38)i∂tϕ+∂x∂xϕ+2λ1|ϕ|2ϕ+(λ2+1κ)|ψ|2ϕ=0,(39)i∂tψ+∂x∂xψ+2κ|ψ|2ψ+(λ2+1κ)|ϕ|2ψ=0,
which proves Theorem 2.