Lie Symmetries and Conservation Laws of Fokas–Lenells Equation and Two Coupled Fokas–Lenells Equations by the Symmetry/Adjoint Symmetry Pair Method

: The Fokas–Lenells equation and its multi-component coupled forms have attracted the attention of many mathematical physicists. The Fokas–Lenells equation and two coupled Fokas– Lenells equations are investigated from the perspective of Lie symmetries and conservation laws. The three systems have been turned into real multi-component coupled systems by appropriate transformations. By procedures of symmetry analysis, Lie symmetries of the three real systems are obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair method, which depends on Lie symmetries and adjoint symmetries. The relationships between the multiplier and the adjoint symmetry are investigated.

The coupled FL equations have the expression: where U and V are both complex functions dependent on x and t, α and β are real numbers, and α = 0, β = 0 [30]. Analytical solutions of Equation (2) have been obtained by Darboux transformation [31]. Based on the determinant representation of the Darboux transformation, higher-order soliton solutions of Equation (2) have been derived in [32]. When α = β = 1, rogue wave and n-order rogue waves have been studied in [33,34]. Another coupled FL equations have the form: where U and V are both complex functions dependent on x and t, and the asterisk represents a complex conjugation. It was show that Equation (3) has a multi-Hamiltonian integrability [35]. One-soliton solutions, higher-order-soliton solutions and other solutions have been studied in [36][37][38].
To the best of our knowledge, the Lie symmetries and explicit conservation laws of Equations (1)-(3) possess much mathematical and physical significance, and they have not been reported. The framework of the rest is shown in what follows. In Section 2, we first transform the FL equation and two coupled FL equations into real equations, then search for their Lie symmetries by the classical Lie symmetry method. In Section 3, conservation laws related to the three systems will be found by making use of the SA method. In Section 4, some closing words and discussions are presented.

Lie Symmetry Analysis of Equations (1)-(3)
Suppose that: with u and v being real functions dependent on x and t, then the FL Equation (1) is changed to: By taking: where u, v, p and q are all real functions with respect to x and t. By the same transformation with Equation (2), the coupled FL Equation (3) can be transformed into: Next, we will seek the Lie symmetries of Equations (4)-(6) by the classical Lie symmetry method. Suppose the vector field of Equation (4) is: where ξ, τ, φ and ψ are undetermined functions with respect to x, t, u and v. Equation (7) is also called a Lie symmetry of Equation (4). By the procedure of the classical Lie symmetry method, Equation (7) should satisfy the following two conditions: where , and pr (2) V denotes the second prolongation of V and: The above coefficient functions are given by: Here, D x and D t are total differential operators. The exact solutions of Equation (8) can be found with the aid of Maple or Mathematica, and then we can obtain Lie symmetries for Equation (4) as follows: From the definition of the Lie bracket, i.e., [V i , V j ] = V i V j − V j V i , the commutation relations of V 1 , V 2 , V 3 and V 4 are shown in the following: Therefore, V i (i = 1, . . . , 4) are closed under the Lie bracket, and they form a fourdimensional Lie algebra.
Similarly, we can gain the Lie symmetries of Equation (5): The commutation relations of Equation (12) are listed in the following: Therefore, V i (i = 1, . . . , 5) are closed under the Lie bracket, and they form a fivedimensional Lie algebra.
By a similar method as with the systems Equations (4) and (5), we can get the Lie symmetries of Equation (6): The commutation relations of Equation (13) are listed in the following: Therefore, V i (i = 1, 2, . . . , 7) are closed under the Lie bracket, and they form a sevendimensional Lie algebra.

Conservation Laws of Equation (4)
Based on the obtained Lie symmetries in Section 2, we can construct conservation laws for Equations (4)-(6) by the steps of the SA method [5,6].
From the definition of the Fréchet derivative and adjoint Fréchet derivative, one obtains the linearizing operator of Equation (4): and the adjoint linearizing operator of Equation (4): (15) Using the Lie symmetries Equation (11), the symmetry componentsη ρ of Equation (4) can be expressed by: From Equation (15), one obtains the following adjoint linearizing system: with ω 1 and ω 2 being functions dependent on x, t, u and v. After calculation, we find that the only solution to Equation (17) is: with C being a constant. This is the adjoint symmetry of Equations (4), and we take C = 1 in the following for simplicity. (16) and (ω 1 , ω 2 ) in Equation (18) into the conservation laws identity of Theorem 1 in [6], one can obtain the following conservation laws of Equation (4) with respect to V 1 : Using the other pairsη i (i = 2, 3, 4) in Equation (16) and (ω 1 , ω 2 ) in Equation (18), one can obtain the other three conservation laws of Equation (4): (19) as an example, we show that adjoint symmetries are also multipliers corresponding to conservation laws. After some calculation, Equation (19) will reduce to:

Remark 2.
As to the conservation law (X 3 , T 3 ), since D x X 3 + D t T 3 ≡ 0, it is not relevant to the solutions of Equation (4), and it is trivial. The other conservation laws (X 1 , T 1 ), (X 2 , T 2 ) and (X 4 , T 4 ) are nontrivial and can be proved. For example, D x X 2 + D t T 2 = −u∆ 1 + v∆ 2 = 0. The accuracy of the above conservation laws has been verified.

Conservation Laws of Equation (5)
From the definition of the Fréchet derivative and adjoint Fréchet derivative, one obtains the linearizing operator of Equation (5): , and the adjoint linearizing operator of Equation (5): By the symmetries Equation (12), there are five symmetry components of Equation (5), and they can be expressed by: From Equation (23), one can obtain the adjoint linearizing system of Equation (5): with ω i (i = 1, 2, 3, 4) being functions dependent on x, t, u, v, p and q. After complicated calculations, we find that there are two solutions to Equation (25): and: (ω 1 , ω 2 , ω 3 , where C 1 and C 2 are constants. They are adjoint symmetries of Equation (5). We take C 1 = 1 and C 2 = 1 in the following for simplicity.

Remark 3.
There are seven symmetry components of Equation (6), and they are expressed by Equation (30). Using Equation (30) and other adjoint symmetries in Equations (32)-(35), we can obtain twenty-one other conservation laws for Equation (6). The effectiveness of the SA method in seeking the conservation laws of multi-component coupled systems has been illustrated.

Conclusions and Discussions
Recently, the FL equation and coupled FL equations have attracted the interest of many researchers. They can be regarded as integrable analogs of the nonlinear Schrödinger (NLS) equation and its coupled forms in the ultra-short regime. In this paper, Lie symmetry analyses for the FL equation and two coupled FL equations are performed, and we have shown that they all form a closed Lie algebra. Explicit conservation laws for the three FL equations have been obtained by the SA method. The correctness of the derived conservation laws has been tested by a mathematic software. The obtained conservation laws for the FL equation and coupled FL systems may be used to explain some practical physical problems.
The SA method and the multiplier method can both derive explicit conservation laws for PDEs, whether the PDEs have a Lagrangian or not. Taking the FL equation as an example, the relationships between the SA method and the multiplier method are further investigated. We conclude that the adjoint symmetry in the SA method is also the multiplier in the multiplier method. Using the multiplier method, every multiplier can only derive one conservation law. Using the SA method, any pair of symmetry and adjoint symmetry can derive one conservation law, so it can derive more conservation laws than the multiplier method. As illustrated in Sections 3.2 and 3.3, it is very effective to search for the conservation laws of multi-component coupled systems by the SA method. However, when we seek conservation laws by the SA method, the adjoint symmetries cannot be zero. If the adjoint symmetry is zero, the obtained conservation law is trivial, and then we will have to use other methods.