New Exact Solutions of the ( 2 + 1 )-Dimensional AKNS Equation

N-soliton solutions and the bilinear form of the (2 + 1)-dimensional AKNS equation are obtained by using the Hirota method. Moreover, the double Wronskian solution and generalized double Wronskian solution are constructed through the Wronskian technique. Furthermore, rational solutions, Matveev solutions and complexitons of the (2 + 1)-dimensional AKNS equation are given through a matrix method for constructing double Wronskian entries. The three solutions are new.


Introduction
It is one of the most important topics to search for exact solutions of nonlinear evolution equations in soliton theory.Moreover, various methods have been developed, such as the inverse scattering transformation [1], the Darboux transformation [2], the Hirota method [3], the Wronskian technique [4] [5], source generation procedure [6] [7] and so on.In 1971, Hirota first proposed the formal perturbation technique to obtain N-soliton solution of the KdV equation.Satsuma gave the Wronskian representation of the N-soliton solution to the KdV equation [8].Then the Wronskian technique was developed by Freeman and Nimmo [4] [5].In 1992, Matveev introduced the generalized Wronskian to obtain another kind of exact solutions called Positons for the KdV equation [9].Recently, Ma first introduced a new kind of exact solution called complexitons [10].By using these methods, exact solutions of many nonlinear soliton equations are obtained [11]- [16].
This paper is organized as follows.In Section 2, the bilinear form of the (2 + 1)-dimensional AKNS equation and its N-soliton solutions are obtained through the Hirota method.In Section 3, the double Wronskian solution and generalized double Wronskian solution are constructed by using the Wronskian technique.In Sections 4 and 5, rational solutions and Matveev solutions are given.In Section 6, complexitons of the (2 + 1)-dimensional AKNS equation are provided.Finally, we give some conclusions.
where D is the well-known Hirota bilinear operator defined by Expanding f, g and h as the series ( ) , , 1 , , , , j j g t x y g g g ε ε ε , , , j j h t x y h h h ε ε ε substituting Equation (2.4) into (2.3) and comparing the coefficients of the same power of ε yields , , we can obtain . Thus, the one-soliton solution is given as follows.

(
) where ( )( ) In the same way, we can obtain the following N-soliton solutions of Equation (2.3).

3
A µ take over all possible combinations of ( ) and satisfy the following condition

The Double Wronskian Solution and Generalized Double Wronskian Solution
Let us first specify some properties of the Wronskian determinant.As is well known, the double Wronskian determinant is , , , .
The following two determinantal identities were often used [4] [5].The one is , , , , where D is a ( ) matrix and , , a b c and d represent N column vectors.The other is , where j ϕ and j ψ satisfy the following conditions ( ) Proof.In the following, we use the abbreviated notation of Freeman and Nimmo for the Wronskian and its derivatives [4] [5], then Equation (3.3) becomes First, we calculate various derivatives of g and f with respect to x and t.
Y. P. Sun Utilizing Equation (3.2) and Equation (3.4), we get Using Equation (3.7) and Equation (3.8), then Equation (3.6) becomes According to (3.1), it is easy to see that Equation (3.9) is equal to zero.So, the proof of Equation (2.3a) is completed.Similarly Equations (2.3 b) and (2.3 c) can also be proved.
Lemma 1. Assume that is an l l × operator matrix and its entries ij p are differential operators., , , , , , , . , where j ϕ and j ψ satisfy the following conditions arbitrary real matrix independent of x and t.In fact, similar the proof of Theorem 1, we only need to verify that identities (3.7) hold.
It is obvious that (3.7) hold.
(2) If 0, trA = we can consider this as a limit case where trA tends to zero.Then (3.15)-(3.17)become N ;
it is obvious to know that 2 0.

N M
A + + = Thus (4.1) can be truncated as The components of ϕ and ψ are Thus, we can calculate some rational solutions of Equation (2.1).
x y t x y p q u x y t x y t x y t t x y x y p q u t x y t x y t x y

Matveev Solutions
In the following, we will discuss Matveev solutions of the (2 + 1)-dimensional AKNS equation.
Let A be a Jordan matrix .
Y. P. Sun Without loss of generality, we observe the following Jordan block (dropping the subscript of k) ( ) where i l I is an i i l l × unite matrix.We have ( ) ( ) 1 0

k t kx ky k t kx ky
The components of ( ) Thus, Matveev solutions of Equation (2.1) can be obtained, where T T (5.15c)

Complexitions of the (2 + 1)-Dimensional AKNS Equation
In the following, we would like to consider that A is a real Jordan matrix.
 are real constants.Then, from (4.1), complexitons can be obtained.In order to prove that, we first observe the simplest case when e cos 4 sin 4 .
Further, we consider the matrix A as a Jordan block i J where the symbol ⊗ denotes tensor product of matrices.Noting that Employing the following formula then (6.6) can be written as Substituting (6.8) into (4.1)yields    α can be replaced by the partial derivative with respect to i β in (6.10) and (6.11).

Conclusion
In this paper, we have obtained N-solution solutions and the generalized double Wronskian solution of the (2 + 1)-dimensional AKNS equation through the Hirota method and the Wronskian technique, respectively.Moreover, we have given rational solutions, Matveev solutions and complexitons of the (2 + 1)-dimensional AKNS equation.According to our knowledge, the three solutions are novel.
D a b D c d D a c D b d D a d D b c double Wronskian solution of Equation (2.3) is obtained as follows: Using the Lemma 1 and the Wronskian technique, we construct the following result.Theorem 2. The (2 + 1)-dimensional AKNS Equation (2.3) has the generalized double Wronskian solution According to (6.4), Equation (6.10) can be expressed as the following explicit form:( the partial derivative with respect to i ) )