Integrability and Multisoliton Solutions of the Reverse Space or/and Time Nonlocal Fokas-Lenells (FL) Equation

This paper studies reverse space or/and time nonlocal Fokas-Lenells (FL) equation, which describes the propagation of nonlinear light pulses in monomode optical fibers when certain higher-order nonlinear effects are considered, by Hirota bilinear method. Firstly, variable transformations from reverse space nonlocal FL equation to reverse time and reverse space-time nonlocal FL equations are constructed. Secondly, the one-, twoand three-soliton solutions of the reverse space nonlocal FL equation are derived through Hirota bilinear method, and the soliton solutions of reverse time and reverse space-time nonlocal FL equations are given through variable transformations. Dynamical behaviors of the multisoliton solutions are discussed in detail by analyzing their wave structures. Thirdly, asymptotic analysis of twoand three-soliton solutions of reverse space nonlocal FL equation is used to investigated the elastic interaction and inelastic interaction. At last, the Lax integrability and conservation laws of three types of nonlocal FL equations is studied. The results obtained in this paper possess new properties that different from the ones for FL equation, which are useful in exploring novel physical phenomena of nonlocal systems in nonlinear media.


Introduction
The Fokas-Lenells (FL) equation was derived as an integrable generalization of the nonlinear Schrödinger (NLS) equation using bi-Hamiltonian methods [1], which is a completely integrable nonlinear partial differential equation (here means it admits a Lax pair). The FL equation where u(x, t) is a complex valued function for the independent spatial variable x and temporal variable t, and u * (−x, t) denotes complex conjugate of u(x, t). The subscript x (or t) denotes partial derivative with respect to x (or t). Through the method in [20], the variable transformations from reverse space nonlocal FL equation to reverse time and reverse space-time nonlocal FL equation can be derived as follows Through these variable transformations, reverse time and reverse space-time nonlocal FL equation are presented subsequently where u = u(x, t) is a complex-valued function of x and t, and the * denotes complex conjugation.
In this paper, we use the Hirota bilinear method to get one-, two-and three-soliton solutions of the reverse space nonlocal FL equation (1), then study multisoliton solutions of the reverse time and inverse space-time nonlocal FL equations through variable transforms. Asymptotic analysis is used to investigate the elastic interactions and inelastic interactions of the two solitons and the three solitons solutions, and dynamical behaviors of the multisoliton solutions are investigated by analyzing their wave structures. Finally, the Lax pairs and conservation laws of three types of nonlocal FL equations are obtained. The outline of this paper is presented as follows. In Section 2, the one-, two-and threesoliton solutions of three types of nonlocal FL equations are obtained by using Hirota bilinear method and the variable transformations (2) and (3). And some figures are given to describe the dynamic characteristics of these soliton solutions. In Section 3, the asymptotic analysis on two-and three-soliton solutions of the reverse space nonlocal FL equation is given. In Section 4, we exhibit the Lax pairs of three types of nonlocal FL equations. Meanwhile, based on the Lax pairs, the infinitely many conservation laws of these equations (1), (4) and (5) are derived. Finally, the conclusions of this paper are stated in Section 5. In order to receive one-soliton solution of reverse space nonlocal FL equation, the Hirota bilinear method [30][31][32][33] and symbolic computation are used. By introducing the dependent variable transformations u(x, t) = G(x, t) F (x, t) , u * (−x, t) = G * (−x, t) F * (−x, t) , (6) where G(x, t), G * (−x, t), F (x, t) and F * (−x, t) are complex functions, the nonlocal FL equation (1) converts into the following bilinear equation This equation can be decoupled into the following system of bilinear equations for the functions F and G, where the D x and D t are bilinear operators. These operators defined as where m and n are non-negative integers. Solving the above series of bilinear equations (8)- (9) and combining (6), some soliton solutions can be obtained. We expand the unknown functions (x, t), G * (−x, t), F (x, t) and F * (−x, t)as a polynomial of small parameter ǫ as follows G(x, t) = ǫG 1 + ǫ 3 G 3 + ǫ 5 G 5 + · · · , G * (−x, t) = ǫG * 1 + ǫ 3 G * 3 + ǫ 5 G * 5 + · · · , F (x, t) = 1 + ǫ 2 F 2 + ǫ 4 F 4 + ǫ 6 F 6 + · · · , where the G 1 , F 2 , etc. are functions with spatial variable x and temporal variable t, the functions G * 1 , F * 2 , etc. with variables −x and t. Substituting the above expansions into Eqs. (8)- (9), and comparing the coefficients of ǫ, the unknown functions G(x, t), G * (−x, t), F (x, t) and F * (−x, t) can be obtained by selecting appropriate functions G 1 , G * 1 , F 2 , F * 2 . In this section, the unknown functions G(x, t), G * (−x, t), F (x, t) and F * (−x, t) are expanded in terms of a small parameter ǫ as follows Substituting (12) into bilinear equation (8)- (9), we obtain a set of equations by comparing the coefficients of same powers of ǫ to zero where G 1 , G * 1 , F 2 and F * 2 are given rise to as follows We suppose that , and k 1 , k * 1 are arbitrary complex constants. Form Eqs. (13)- (14), the relations about ω 1 , A 1 and k 1 are given as follows . Since the ω * 1 is the complex conjugate of ω 1 and the A * 1 is the complex conjugate of A 1 , the expressions for ω * 1 and A * 1 are presented as follows Then, the general one-soliton solution of the reverse space nonlocal FL equation (1) is According to the bilinear form of parity transformed complex conjugate equation, the parity transformed complex conjugate field is derived in the form where Here, some figures are provided to describe the one-soliton solutions Eqs. (20)-(25) of three types of nonlocal FL equations (see Fig.1-Fig.3). In Fig.1, (a), (b) and (c) are the profiles of |u|, and (d), (e) and (f) are the profiles of |u * |. The results show that the solutions of three types of FL equations are periodic wave, and the periodic oscillations have exponential growth trend. It is obvious that |u| and |u * | of the reverse space/time nonlocal FL equation have the same shapes as spatial/time evolution, but their enhancing shapes are antipodal. In order to intuitively observe one-soliton solutions' difference between the reverse space/time nonlocal FL equation and the reverse space-time nonlocal FL equation, more figures (Fig.2, Fig.3) are provided. These figures have the same parameters k 1 , k * 1 , η 10 , η * 10 for different equations.

Two-soliton solutions of three types of nonlocal FL equations
The two-soliton solution of the reverse space nonlocal FL equation (1) can also be obtained with Hirota bilinear method. We consider the truncating of the following expansions G(x, t) = . Substituting these expansions into the bilinear equations (8)- (9), and equating the coefficients of same powers of ǫ to zero, a set of equations can be derived where G 1 , G * 1 , F 2 and F * 2 are given rise to as follows In the above expressions, , and k 1 , k * 1 , k 2 and k * 2 are arbitrary complex constants. Form Eqs. (26)-(28), we know and Thus, a set of equations for unknown functions G 1 (x, t), G * 1 (−x, t), F 2 (x, t) and F * 2 (−x, t) are obtained. Substituting the expressions for G 1 and F 2 into the Eq. (27), the function G 3 and its parity transformed complex conjugate G * 3 are given in the form where Then substituting the expressions of G 1 , G * 1 , G 3 , G * 3 , F 2 and F * 2 into Eq. (29), the functions F 4 and F * 4 are derived as follows where The general nonlocal two-soliton solution of the reverse space nonlocal FL equation (1) is given as follows According to the bilinear form of parity transformed complex conjugate equation, the parity transformed complex conjugate field is derived in the form Through the transformations x = −ix, t = it and x = −x, t = it, the two-soliton solutions (36)-(37) of reverse space nonlocal FL equation transform into two-soliton solutions of the reverse time nonlocal FL equation (4) and the reverse space-time nonlocal FL equation (5). The solutions are presented as follows where η 10 , η * 10 , η 20 , η * 20 for different equations. Through these pictures, we could observe two-soliton solutions' differences intuitively. The shapes of two-soliton solutions of the reverse space/time FL equation are parallel with the x or t axis, however two-soliton solution of the reverse space-time FL equation is not parallel with neither x axis nor t axis, which can be viewed as a parallel superposition of time and space local solitons.

Three-soliton solutions of three types of nonlocal FL equations
Through Hirota bilinear method, the three-soliton solution of the reverse space nonlocal FL equation (1) can be obtained. The truncating expansions of G(x, t), G * (−x, t), F (x, t) and F * (−x, t) are given as follows Substituting these expansions into the bilinear equations (8)- (9) and equating the coefficients of same powers of ǫ to zero, a set of equations can be derived where G 1 , G * 1 , F 2 and F * 2 are given rise to as follows +A 8 e η 3 +η * 2 + A 9 e η 3 +η * 3 , In these equations, , and k 1 , k 2 and k 3 are arbitrary complex constants. Form Eqs. (42)-(45), we know and .
Thus, a set of equations for the unknown functions G 1 (x, t), G * 1 (−x, t), F 2 (x, t) and F * 2 (−x, t) are obtained. In order to get the function G 3 and its parity transformed complex conjugate G * 3 , substituting the expressions for G 1 and F 2 into Eq. (43), G 3 and G * 3 are given in the form Then substituting the expressions of G 1 , G * 1 , G 3 , G * 3 , F 2 and F * 2 into Eq. (46), the functions F 4 and F * 4 are derived as follows where So as to derive the expression of G 5 , we substitute the expressions for G 1 , G * 1 , G 3 , G * 3 , F 2 , F * 2 , F 4 and F * 4 into Eq. (44), the functions G 5 and G * 5 are given as follows where into Eq. (47), the functions F 6 and F * 6 are given as follows where The general nonlocal three-soliton solution of the reverse space nonlocal FL equation (1) is given as follows According to the bilinear form of parity transformed complex conjugate equation, the parity transformed complex conjugate field is derived in the form In order to derive three-soliton solutions of the reverse time and reverse space-time nonlocal FL equation, we substitute transformations x → −ix, t → it into three-soliton solutions Eqs.

FL equation
Through asymptotic analysis in [34], it shows that when solitons undergo multiple collisions, there exists possibility of soliton's shape restoration. Asymptotic analysis is used to investigate the elastic and inelastic interactions between the bound solitons and the regular one soliton [35].
Comparing the asymptotic expressions of two-soliton solution between before interaction and after interaction, we find that k 1 , k * 1 , k 2 and k * 2 accord with the conditions the relations of amplitudes can be obtained where Am 1− and Am 2− denote the amplitudes for the two solitons before the interaction, while Am 1+ and Am 2+ denote the amplitudes for the two solitons after the interaction. When k 1 , k * 1 , k 2 and k * 2 do not accord with conditions (75), it can be yields Through expressions (76) and (77), it is obvious that the elastic interaction for two-soliton solution of the reverse space nonlocal FL equation appears under conditions (75), inelastic interaction for two-soliton solution of the reverse space nonlocal FL equation arises beyond conditions (75).

Asymptotic analysis on three-soliton solution of the reverse space FL equation
Considering the above three-soliton solution Eq. (65), without loss of generality, we assume that η 10 = η 20 = η 30 = 0, k 1 /k 2 > 0, k 2 /k 3 > 0 and k 1 /k 3 > 0. For fixed η 1 , note that the asymptotic expressions for the three solitons before interaction can be given by .
i) Taking limit t → −∞: η 1 + η * 1 ∼ +∞, η 2 + η * 2 ∼ +∞, η 3 + η * 3 ∼ 0, the asymptotic expressions for the three solitons before interaction can be given by (1) can be expressed as follows with where Ψ S = (ψ S,1 , ψ S,2 ) T is a column vector function, and Ψ T and Ψ ST below are also column vector functions. The compatibility condition of the Lax pair, which is zero curvature equation (1). These variable transformations (2) and (3) allow us to derive the Lax pair of the reverse time and reverse space-time nonlocal FL equations from that of the reverse space one. The Lax pair for the reverse time nonlocal FL equation (4) is derived as follows with .
The Lax pair for the reverse space-time nonlocal FL equation (5) is shown as follows with .
The transformation relationship between these equations provides an effective method for us to derive the Lax pairs of different equations. In fact, given the solutions of the reverse space nonlocal FL equation, the solutions of reverse time and reverse space-time counterparts can be derived from the principle. However, if not, then the solutions of reverse time and reverses pace-time nonlocal FL equation may be derive desired solutions by other methods.

Conservation laws
Based on the Lax pair, the infinitely many conservation laws are constructed in both positive and negative orders. We consider the associated spectral problem of the reverse space nonlocal FL equation and associate time evolution equation They satisfy the following expression The expression of ψ 2 ψ 1 is given as follows Substituting (96) into Eq. (94), and comparing the coefficients of λ, we obtain It can be easily shown that ψ 1 satisfies (lnψ 1 ) xt = (lnψ 1 ) tx .
Hence, the conservation laws are derived as follows which can be written as (u x P i ) t = −(iuP i−1 ) x , (i = 1, 2, ...), Among thses conservation laws, the first two are listed below [u x (−u * xx + u x u * 2 On the other hand, substituting the expansion into Eq. (95) and comparing the coefficients of λ, one obtains uP j P i+1−j (i = 1, 2, ...).
Then other conservation laws are given as follows Among thses conservation laws, the first two are listed below [u x (iu * t + uu * 2 )] t = −i[u(−u * tt + i(uu * 2 ) t )] x .
The transformations Eqs.(2)-(3) allows us to derive the conversation laws of the reverse time and reverse space-time nonlocal FL equation from those of the reverse space ones. The first two conversation laws for the reverse time nonlocal FL equation (4) are derived as follows [u x (u xx − iu x u * 2 and [u x (u * t + uu * 2 )] t = [u(u * tt + (uu * 2 ) t )] x .
The first two conversation laws for the reverse space-time nonlocal FL equation (5) are derived as follows [u x (u * xx + u x u * 2 and [u x (u * t + uu * 2 )] t = [u(u * tt + (uu * 2 ) t )] x .
So, through the transformation relationship between these equations, it is effective to provide the conversation laws of different equations. However the prerequisite for doing these things is knowing the Lax pairs of these equations.

Conclusions
In this paper, three types of nonlocal Fokas-Lenells equations are considered by means of the Hirota bilinear method. The one-, two-and three-soliton solutions of the reverse time and reverse space-time nonlocal FL equation are converted from those of the reverse space ones. Furthermore, the graphical representations are presented showing the shape of solution more visually, and the physical interpretation of the obtained figures is discussed for different choices of the parameters that occur in the solutions. Then, asymptotic analysis of two-and three-soliton solutions of reverse space nonlocal FL equation are given to understand the long time asymptotic behavior. The Lax integrability of three types of nonlocal FL equations is investigated using variable transformations, and infinitely many conservation laws are constructed based on the Lax pairs of different equations. These results might be useful to comprehend some physical phenomena and inspire some novel physical applications on other nonlinear system.