Lie symmetry analysis, conservation laws and analytical solutions for chiral nonlinear Schrödinger equation in (2 + 1)-dimensions

In this work, we consider the chiral nonlinear Schrödinger equation in (2 + 1)-dimensions, which describes the envelope of amplitude in many physical media. We employ the Lie symmetry analysis method to study the vector field and the optimal system of the equation. The similarity reductions are analyzed by considering the optimal system. Furthermore, we find the power series solution of the equation with convergence analysis. Based on a new conservation law, we construct the conservation laws of the equation by using the resulting symmetries. 


Introduction
The nonlinear Schrödinger equation, which plays a very important role in nonlinear evolution equations (NLEEs), has been fully applied in many phenomena, such as fluid dynamics, nonlinear fiber, molecular biology, quantum mechanics, deep water modeling, etc. [9,10,30]. Up to now, finding for the exact solution of NLEEs still plays a very important role in the study dynamics of nonlinear phenomena. In the last few decades, the exact solutions of NLEEs have been extensively studied. The main methods used are Darboux transformation, the inverse scattering method, Hirota bilinear method, Lie symmetry group method [1,4,11,19,23,26]. Among them, the Lie symmetry group method can simultaneously obtain the symmetry, exact solutions and conservation laws of NLEEs through some effective calculations [3,7,12,16,29,[38][39][40][41][42][43][44][45]49].
In the past few decades, the conservation laws has played an increasingly important role in the research of NLEEs. At the same time, various methods for solving conservation law of the NLEE are also produced, such as Noether's theorem, characteristic method, variational approach, conservation theorem [5,8,15,24,25,48,50], etc. The famous Noether's theorem establishes the connection between symmetries of NLEEs and conservation laws. But the disadvantage of the Noether's theorem is that it is not suitable for solving NLEE without Lagrangian. In order to solve this NLEE without Lagrangian, Ibragimov entered a new method for solving the conservation law in 2007. This new method relies on the notion of Lie symmetry generators, the adjoint equation and formal Lagrangians of NLEEs. Therefore, this new conservation law will also play an important role in solving the conservation laws of NLEEs.
In this work, we mainly study the chiral nonlinear Schrödinger (NLS) equation in (2 + 1)-dimensions iq t + a(q xx + q yy ) where q = q(x, y, t), a means the coefficient of dispersion term, and b 1 , b 2 are the coefficients of nonlinear coupling terms. In [6], the bright and dark soliton solution of the chiral NLS equation in (2 + 1)-dimensions (1) is obtained using the constant coefficient method. In [14], the singular periodic solution of the chiral NLS equation in (2 + 1)dimensions (1) is obtained by using the trial solution method. As we all know, the Lie symmetry and conservation laws of equation (1) have not been studied. Therefore, in this work, we will mainly study the Lie symmetry and conservation laws of equation (1). The rest of the paper is structured as follows. In Section 2, we first transform equation (1) into a form of equations, and then vector field and optimal system are constructed by using the Lie symmetry analysis method. In Section 3, the symmetry reductions of equation (1) is obtained by using the optimal system. In Section 4, we obtain the power series solution of the system by using the power series method. In Section 5, the conservation law of the equation is obtained by the new conservation law. In Section 6, we give some summaries and discussions.

Lie symmetries analysis
In this section, Lie symmetry analysis will be performed on the chiral NLS equation in (2 + 1)-dimensions (1). Firstly, we consider the complex-valued function q(x, y, t) in the following form: q(x, y, t) = u(x, y, t) + iv(x, y, t), where u(x, y, t) and v(x, y, t) are real-valued functions, and q * represents the conjugate of q. Substituting equation (2) into equation (1) and equating the real and imaginary parts, we can obtain To construct the point symmetry of equation (1), we first introduce a Lie group with a one-parameter Lie transformation group where ε 1 means a group parameter, and ξ 1 , ξ 2 , ξ 3 , η 1 and η 2 are the infinitesimal generators. The vector field corresponding to the above group of transformation as where ξ 1 (x, y, t, u, v), ξ 2 (x, y, t, u, v), ξ 3 (x, y, t, u, v), η 1 (x, y, t, u, v) and η 2 (x, y, t, u, v) are functions of coefficient to be determined. For system (3), pr 2 will be the second prolongation, then its invariance condition is where On the basis of Lie's theory, the second prolongation of equation (4) can be written as Combining (5) and (6), we can get an equivalent condition of (5) as Substituting (6) into (7) and then simplifying, we can get the determining equations of system (3) as Then we can obtain a very important theorem through further calculations as follows.
Theorem 1. The Lie algebra of infinitesimal symmetry of equation (1) are spanned by the following six linear independent operators:  (3). Table 1).
Based on the commutator relations in Table 1, we want to get the adjoint representations of the vector fields by using the following Lie series: Then we have http://www.journals.vu.lt/nonlinear-analysis Therefore, we can get the optimal system of (3) according to the adjoint representation of the vector field (8), then have get

Symmetry reductions
In the previous section, we mainly obtained the vector field and the optimal system of equation (1). Therefore, in this section, we will mainly do the symmetry reduction of these optimal systems.
where ξ = x and τ = y. Inserting (9) into (3), we can get system (3) of ordinary differential equations (ODEs) in which F and G satisfy For the generator V 2 , we can obtain where ξ = y and τ = t. Inserting (11) into (3), we can get system (3) of ODEs in which F and G satisfy where F = dF/dξ, F = d 2 F/dξ 2 and G = dG/dτ . Solving system (12), we can get Therefore, we get the solution of equation (1) as where c 1 and c 2 are arbitrary functions.

The generator
For the generator V 3 , we can obtain where ξ = x and τ = t. Inserting (13) into (3), we can get system (3) of ODEs in which F and G satisfy where F = dF/dξ, F = d 2 F/dξ 2 and G = dG/dτ . Solving system (14), we can get Therefore, we get the solution of equation (1) as where c 1 and c 2 are arbitrary functions.

The power series solutions
Based on the symbolic calculation methods [13, 17, 18, 22, 27, 28, 31-37, 46, 47], we study the analytical solution of ODE by through the power series method. When we get the analytical solution of ODE, we can easily obtain the power series solutions of the original partial differential equation.
According to (10), we can get Below we will use the following hypothetical form to calculate the solution of (23) where the coefficients P n and Q n (n = 0, 1, . . . ) are all constants.
Inserting (24) into (23) yields When n = 0, we compare coefficients of ξ to get Generally, when n 1, we can obtain with (n−k+1)Q j P k−j Q n−k+1 , http://www.journals.vu.lt/nonlinear-analysis Then we can get the following results: when n = 1, when n = 2, From the above derivation we can see that all the coefficients (P n , Q n ) in the power series solution of (24) can be represented by a, b 1 , b 2 , Q 0 , Q 1 , P 0 , P 1 , where a, b 1 , b 2 , Q 0 , Q 1 , P 0 , P 1 are arbitrary constants. Besides, on the basis of [2,21], we can also prove the convergence of the coefficients determined by (25)- (26). Thus, we obtain that a power series solution (24) is the power series solution of (23). Then a power series solution of (24) can be rewritten into Then the power series solution of equation (1) is where a, b 1 , b 2 , Q 0 , Q 1 , P 0 , P 1 are arbitrary constants, and other coefficients determined by (25)- (26). Based on the previous detailed derivation, we can obtain the following theorem.
Next, by choosing the appropriate parameters, we draw the graph of the power series solution and thus illustrate its properties (see Figs. 1, 2).

Conservation laws
In this section, if we want to derived the conservation law of equation (1), it is necessary to first find the conservation law of system (3). Therefore, we will use Lie point symmetry (8) to construct the conservation law of system (3). A vector C = (C t , C x , C y ) is called a conserved vector for equation (1) if it satisfy the following conservation equations: In [20], Ibragimov proposes a new conservation theorem, that is, constructing a conservation law without a Lagrangian quantity in a differential equation. Then on the basis of [20], the Lagrangian of system (3) can be written as follows: where φ(x, y, t) and ψ(x, y, t) are two new dependent variables. The adjoint equations of system (3) can be written as following form: Combining with (28) and adjoint equations (29), we can obtain F * = 6b 1 uv x φ + 6b 2 uv y φ + 6b 1 vv x ψ + 6b 2 vv y ψ + 2b 1 uvφ x + 2b 1 v 2 ψ x + 2b 2 uvφ y + 2b 2 v 2 ψ y − ψ t + aφ xx + aφ yy , In the above system (30), if we substitute v instead of φ and u instead of −ψ, we can get system (3). In [20], we know that the conservation vector C = (C 1 , C 2 , C 3 , . . . ) has the following form: where W α = η α − ξ j u α j (α = 1, 2, . . . , m) are shown in [20]. Using the above formula, we can further write about the conservation vector of (28) as in which W u and W v are the Lie characteristic functions. In order to obtain the conservation vector of system (3), we can use the symmetry generators V 1 , V 2 , V 3 , V 4 , V 5 and V 6 as an example to illustrate. Case 1. For the generator V 1 = ∂/∂t, we can get the following Lie characteristic functions: Inserting (32) into (31), we can get the following conserved vector: http://www.journals.vu.lt/nonlinear-analysis After calculation, we can find the following equation: Thus, we know that (33) is a conservation law of system (3). Inserting u = (q + q * )/2 and v = (q − q * )/(2i) into (33), we can obtain conservation laws of equation (1) as Case 2. For the generator V 2 = ∂/∂x, we can get the following Lie characteristic functions: Inserting (34) into (31), we can get the following conserved vector: After calculation, we can find the following equation: Thus, we know that (35) is a conservation law of system (3). Inserting u = (q + q * )/2 and v = (q − q * )/(2i) into (35), we can obtain conservation laws of equation (1) as Case 3. For the generator V 3 = ∂/∂y, we can get the following Lie characteristic functions: Inserting (36) into (31), we can get the following conserved vector: After calculation, we can find the following equation: Thus, we know that (37) is a conservation law of system (3). Inserting u = (q + q * )/2 and v = (q − q * )/(2i) into (37), we can obtain conservation laws of equation (1) as Case 4. For the generator V 4 = v∂/∂u − u∂/∂v, we can get the following Lie characteristic functions: Inserting (38) into (31), we can get the conserved vector After calculation, we can find the following equation: Thus, we know that (39) is a conservation law of system (3). Inserting u = (q + q * )/2 and v = (q − q * )/(2i) into (39), we can obtain conservation laws of equation (1) as Case 5. For the generator V 5 = (1/2)x∂/∂x+(1/2)y∂/∂y +t∂/∂t−(1/4)u∂/∂u− (1/4)v∂/∂v, we can get the following Lie characteristic functions: Inserting (40) into (31), we can get the following conserved vector: C t 5 = atvu xx + atvu yy − atuv xx − atuv yy http://www.journals.vu.lt/nonlinear-analysis After calculation, we can find the following equation: Thus, we know that (41) is a conservation law of system (3). Inserting u = (q + q * )/2 and v = (q − q * )/(2i) into (41), we can obtain conservation laws of equation (1) as )/(2ab 2 )∂/∂v, we can get the following Lie characteristic functions: Inserting (42) into (31), we can get the following conserved vector: After calculation, we can find the following equation: Thus, we know that (43) is a conservation law of system (3). Inserting u = (q + q * )/2 and v = (q − q * )/(2i) into (43), we can obtain conservation laws of equation (1) as at 2i (qq * yy − q * q yy ) − 1 2 qq * + ab 1 t 2ib 2 (q y q * x − q x q * y − q * q xy + qq * xy ), (q y q * x − q x q * y + q * q xy − qq * xy ).

Conclusions and discussions
As we mentioned above, the bright and dark soliton solutions of the chiral NLS equation (1) have been obtained using the constant coefficient method in [6]. The singular periodic solution of equation (1) has been obtained by using the trial solution method in [14]. Compared with previous literatures [6,14], we have obtained some new results, such as vector field, optimal system, similarity reduction solutions, power series solutions with convergence analysis, and conservation laws of equation (1). Firstly, we have transformed the complex model (1) to the real system (3) by using the transformation q(x, y, t) = u(x, y, t) + iv(x, y, t). Then, through the Lie symmetry analysis method, we have constructed the optimal systems and symmetry reductions of system (3). In addition, we have also obtained the power series solution of equation (1) by the power series method. In Figs. 1, 2, when n = 4, 5, we have obtained a perspective view of the real part of the power series solution and the wave propagation pattern of the wave along the x-axis by selecting the appropriate parameter values. Subsequently, we have obtained the conservation law related to the lie symmetry of equation (1) by using the new conservation law method introduced by Ibragimov in [20]. The new results presented in this work can be used to describe soliton dynamics in nuclear physics and other optical experiments. Therefore, it is hoped that all the research results in this work can be used to enrich the dynamic behavior of nonlinear Schrödinger-type equations in engineering and mathematical physics.