Nonlocal symmetries and exact solutions of variable coefficient AKNS system

In this paper, nonlocal symmetries of variable coefficient Ablowitz-Kaup-Newell-Segur(AKNS) system are discussed for the first time. With lax pair of time-dependent coefficient AKNS system, the nonlocal symmetries are obtained, and they are successfully localized to a Lie point symmetries by introducing a suitable auxiliary dependent variable. Furthermore, using the obtained Lie point symmetries of closed system, we give out two types of symmetry reduction and explicit analytic solutions. For some interesting solutions, the figures are given out to show their dynamic behavior.

With the development of symmetry theory, a lot of studies have been devoted to seeking the generalized Lie point symmetry. P.J.Olver [23] has construct a new type of symmetry by using recursion operator which was called nonlocal symmetry. Compared with the local symmetry, nonlocal symmetry was not easy to construct the solutions of differential equations [6,15], because the nonlocal symmetries contain some auxiliary variables. G.W. Bluman et al. [2,3] have presented many methods to find nonlocal symmetries of the partial differential equations (PDEs)by using potential systems. F. Galas [10]obtained the nonlocal symmetries by using the pseudo-potentials of PDEs, and construct exact solutions by the obtained nonlocal symmetries. Recently, Lou et al. [14,21,[29][30][31][32][33] found that Painlevé analysis can also be used to construct the nonlocal symmetries which was called residual symmetries.
In this paper, we consider the variable coefficient AKNS system. With the help of lax pair, the nonlocal symmetries of this system are obtained which be transformed into local symmetries by introducing new variables variables, and analytic solutions are constructed by using the Lie group theory. In [36], the Lie group method is used to study the AKNS system with constant coefficients and it can be found that the results in this paper are special cases of our article. In [19], the constant coefficients AKNS system is studied by using the residue symmetry method, because the nonlocal symmetries obtained in this article are different from our article, so the results are also very different. By comparing with the conclusions of the two articles, we can see that our results are new.
This paper is arranged as follows: In Sec.2, the nonlocal symmetries were constructed by using the Lax pair of variable coefficient AKNS system. In Sec.3, the process of transforming from nonlocal symmetries to local symmetries was introduced in detail. In Sec.4 some symmetry reductions and analytic solutions were obtained by using the Lie point symmetry of closed system. Finally, some conclusions and discussions are given in Sec.6.

Nonlocal symmetries of variable coefficient AKN-S system
The time-dependent coefficient AKNS system [16] reads where u = u(x, t) and v = v(x, t) are the real functions, δ = δ(t) is a real function of t. The system(2.1) was obtained via the variable transformation from timedependent Whitham-Broer-Kaup equations, which is used for the shallow water under the Boussinesq approximation. Lax pair,infinitely-many conservation laws and soliton solutions are given in [16]. When δ = 1, α = i/2 Eq.(2.1) reduces to the well-known AKNS system, where i 2 = −1, nonlocal symmetries and exact solutions for the constant coefficient AKNS system have been obtained [22]. To our knowledge, nonlocal symmetries for Eq.(2.1) have not been obtained and discussed, which will be the goal of this paper. The corresponding Lax pair has been obtained in [16], To seek the nonlocal symmetries of variable coefficient AKNS system(2.1), one must solve the following linearized equations, with the infinitesimal parameter . Be different from Lie point symmetries, we assume nonlocal symmetries of the system(2.1)have the following form, (2.5) Then, one can using Lie group method to find their solutions of σ 1 , σ 2 , σ 3 . By substituting Eq.(2.5) into Eq.(2.3) and eliminating u t , v t , φ 1x , φ 1t , φ 2x , φ 2t in terms of the lax pair(2.3), it yields a system of determining equations for the functions X,T ,Ū ,V ,∆ , solving these determining equations can obtain, where c i (i = 1, ..., 4) are four arbitrary constants and F 1 (t) is arbitrary function of t.
Remark 2.1. It is show that the results(2.6) are local symmetries of variable coefficient AKNS system when c 4 = 0, and they are nonlocal symmetries when c 4 = 0.
Nonlocal symmetries need to be transformed into local ones [14,21] before construct analytic solutions, so we construct a closed system whose Lie symmetries contain above nonlocal symmetries.

Localization of the nonlocal symmetry
For simplicity, let c 1 = c 2 = c 3 = 0, c 4 = 1, F 1 (t) = 0 in formula (2.6) i.e., To localize the nonlocal symmetry (3.1), we have to solve the following linearized equations, which is form invariant under the following transformation, with the infinitesimal parameter , and σ 1 , σ 2 , σ 3 given by (3.1). It is not difficult to verify that the solutions of (3.2) have the following forms, where f satisfies the following equations, it is easy to obtain the following result, One can see that the nonlocal symmetry (3.1) in the original space {x, t, u, v, δ} has been successfully localized to a Lie point symmetry in the enlarged space {x, t, u, v, δ, φ1, φ2, f }. It is not difficult to verify that the auxiliary dependent variable f just satisfies the Schwartzian form of the variable coefficient AKNS system is the Schwartzian derivative.
After we successfully transform the nonlocal symmetries(3.1) into local symmetries. New analytic solutions can be constructed naturally by Lie group theory. With the Lie point symmetry(3.1),(3.4),(3.6), by solving the following initial value problem, where is the group parameter, we arrive at the symmetry group theorem as follows: with is an arbitrary group parameter.
Here we give a simple example, starting from a soliton solution of (2.1) it's not difficult to derive the special solutions for the variables φ 1 , φ 2 , f from(2.2)and (3.5), (3.11) Using theorem 1, it's not hard to verify  To search for more similarity reductions and analytic solutions of Eq.(2.1), we use classical Lie group method. Assume the symmetries of whole prolonged system have the vector form, where X, T, U, ∆, P, Q, F are the functions with respect to x, t, u, δ, φ 1 , φ 2 , f , which means that the closed system is invariant under the transformations (x, t, u, v, δ, φ 1 , φ 2 , f ) → (x+ X, t+ T, u+ U, v+ V, δ+ ∆, φ 1 + P, φ 2 + Q, f + F ), (3.13) with a small parameter . Symmetries in the vector form (3.12) can be assumed as where X, T, U, ∆, P, Q, F are the functions with respect to {x, t, u, δ, φ 1 , φ 2 , f }. And σ i , (i = 1, ..., 6) satisfy the linearized equations of the prolonged system, i.e., (2.3),(3.2),and ) and eliminating u t , v t , φ 1x , φ 1t , φ 2x , φ 2t , f x , f t in terms of the closed system, determining equations for the functions X, T, U, V, ∆, P, Q, F can be obtained, by solving these equations, one can get where c i , (i = 1, 2, ..., 5) are arbitrary constants, F 2 (t) is arbitrary function of t.

Symmetry reduction and analytic solutions of variable coefficient AKNS system
In this section, we will give two nontrivial similarity reductions and group invariant solutions of variable coefficient AKNS system(2.1)under consideration c 3 = 0. case 1: c 5 = 0.
Without loss of generality, let c 2 = c 4 = 0, c 1 = c 3 = 1, c 5 = k 1 , F 2 (t) = k 2 , with k 1 , k 2 are two arbitrary constants. By solving the following characteristic equations, Substituting Eqs.(4.2)into the prolonged system yields, where C is arbitrary constant. One can see that through the Eqs.(4.2)and (4.3), if we know the form of F 1 (ξ), then u, v can be obtained directly. We known that auxiliary dependent variable f satisfies the Schwartzian form, by substituting f = √ k 1 tanh(Θ) into (3.7), one can get, It is not difficult to verify that the above equation is equivalent to the following elliptic equation, C 1 , C 2 are arbitrary constants. It is know that the general solution of Eq.(4.5) can be written in terms of Jacobi elliptic functions. Hence, expression of solution (4.2) reflects the wave interaction between the soliton and the Elliptic function periodic wave. A simple solution of Eq.(4.5) is given as, substituting Eq.(4.6) into Eq.(4.5) yields with k 3 , λ, α ∈ R, 0 ≤ n ≤ 1. Substituting Eqs.(4.7),(4.6) and F 1ξ = F into Eq.(4.3), one can obtain the solutions of u, v. Because the expression is too prolix, it is omitted here. In order to study the properties of these solutions of AKNS system, we give some pictures of u, v as following, In Fig.1, we plot the interaction solutions between solitary waves and elliptic function waves expressed by (4.2) with parameters C = 5, C 1 = 2, k 1 = 0.18, k 2 = 10, λ = 0.1, α = 1, n = 0.1.
We can see that the component u exhibits a soliton propagates on a Jacobi elliptic sine function wave background. In Fig.1, the first picture(a) shows that the height of the soliton is approximately 0.03 at t = −10. With the development of time, soliton produces elastic collisions with other waves, and the height increases continuously. Picture(e) shows that soliton is roughly in line with its adjacent wave at t = 14. After the collision, the soliton reverts to the original height and continues to collide with the adjacent waves see the pictures(f → i). The corresponding 3d image is given below, exhibits a soliton propagating on period waves background. As one can see from the expression(4.2), u, v possess similar form, so there is no more detailed discussion here.
In order to study the properties of the solutions, we draw the corresponding 3-D images using Maple software,(see Fig.2) and the parameters used in the figures are selected same as Fig.1.
In fact, it is of interest to study these types of solutions, for example, in describing localized states in optically refractive index gratings. In the ocean, there are some typical nonlinear waves such as the solitary waves and the cnoidal periodic waves. case 2: c 5 = 0. We let c 1 = k 1 , c 2 = 2k 2 , c 3 = k 3 , c 4 = c 5 = 0, F 1 (t) = 1, with k 1 , k 2 , k 3 are arbitrary constants. By solving the following characteristic equation, where ς = x − k 1 t,C is a arbitrary constant. Substituting Eqs.(4.9)into the prolonged system yields, , (4.10) whereC 1 is a arbitrary constant and F = F (ς) = F 1ς satisfies the following equatioñ the equation(4.11) is equivalent to the following elliptic equation, Cα . (4.12) To solve the equation(4.12), we assume a solution with the following form, substituting Eq.(4.13) into Eq.(4.11) yields the following eight sets of solutions, 14) Remark 4.1. Substituting Eqs.(4.14), (4.13) and (4.10) into Eq.(4.9) yields the analytic solutions of variable coefficient AKNS system(2.1). It can be known from the expression (4.9) that u, v are rational function form solutions. If take k 2 = 0, then solutions are transformed into elliptic function solutions.

Summary and Discussion
In this paper, we have studied nonlocal symmetries and analytic solutions of the variable coefficient AKNS system for the first time. First of all, starting from the known Lax pairs of the variable coefficient AKNS system, nonlocal symmetries are derived directly through a particular assumption. To take advantage of nonlocal symmetries, an auxiliary variable is introduced. Then, the primary nonlocal symmetries are equivalent to a Lie point symmetries of a prolonged system. Applying the Lie group theorem to these local symmetries, the corresponding group invariant solutions are derived. Secondly, several classes of analytic solutions are provided in this paper, including some special forms of analytic solutions. For example, analytic interaction solutions among solitons and other complicated waves, exponential solution, etc., These forms of solutions display solitons fission and fusions which can be easily applicable to the analysis of physically interesting processes for example the generation process of Rogue waves of variable coefficient AKNS system.
It is very meaningful to study the nonlocal symmetries and analytic solutions of variable coefficient integrable models. However, there is still a lot of work to be done. For example, in a large number of nonlocal symmetries of an integrable model Which one can be localized. Is it possible to apply the nonlocal symmetry theory of constant coefficient differential equation to the variable coefficient differential equation? Above topics will be discussed in the future series research works.