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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.

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 [

The AKNS (Ablowitz-Kaup-Newell-Segur) equation is one of the most important physical models [

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.

We consider the following (2 + 1)-dimensional AKNS equation [

Through the dependent variable transformation

Equation (2.1) is transformed into the following bilinear form

where D is the well-known Hirota bilinear operator defined by

Expanding f, g and h as the series

substituting Equation (2.4) into (2.3) and comparing the coefficients of the same power of

Taking

we can obtain

Letting

where

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

where

Let us first specify some properties of the Wronskian determinant. As is well known, the double Wronskian determinant is

where

where D is a

where

Employing the Wronskian technique, we have the following result.

Theorem 1. The (2 + 1)-dimensional AKNS Equation (2.3) has the double Wronskian solution

where

Proof. In the following, we use the abbreviated notation of Freeman and Nimmo for the Wronskian and its derivatives [

First, we calculate various derivatives of g and f with respect to x and t.

Then a direct calculation gives

Utilizing Equation (3.2) and Equation (3.4), we get

Noting

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.

In the following, we give some exact solutions. From Equation (3.4), we deduce that

where

Taking

Letting

then one-soliton solution of Equation (2.1) is

Choosing

So, we have

Similarly, when

In the following, we will prove that Equation (2.3) has the generalized double Wronskian solution. First, we give the following lemma [

Lemma 1. Assume that

where

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

where

In fact, similar the proof of Theorem 1, we only need to verify that identities (3.7) hold.

(1) If

from Lemma 1, we can get

Using Equation (3.13), the left-hand side of (3.14) is equal to

Therefore,

From (3.15), we derive further

It is obvious that (3.7) hold.

(2) If

Using (3.18), Equation (3.12) still satisfies Equation (2.3).

From Equation (3.13), we can get the general solution

where

lowing result.

Theorem 3.

In the section, we will give rational solutions of the (2 + 1)-dimensional AKNS Equation (2.1).

Expanding (3.19) leads to

If

we can obtain solution solutions of Equation (2.3), where

If

it is obvious to know that

The components of

In (4.6), taking

Thus, we can calculate some rational solutions of Equation (2.1).

In the following, we will discuss Matveev solutions of the (2 + 1)-dimensional AKNS equation.

Let A be a Jordan matrix

Without loss of generality, we observe the following Jordan block (dropping the subscript of k)

where

i.e.,

Substituting (5.2) into (4.1), we get

The components of

Specially, taking

Thus, Matveev solutions of Equation (2.1) can be obtained, where

In (5.7), taking

where

Similarly, choosing

and

When

Assume that

letting

Similarly, taking

In the following, we would like to consider that A is a real Jordan matrix.

where

and

In order to prove that, we first observe the simplest case when

Substituting (6.2) into (4.1a) yields

Expanding the above φ and taking advantage of

Similarly,

Further, we consider the matrix A as a Jordan block

where the symbol

Employing the following formula

then (6.6) can be written as

Substituting (6.8) into (4.1) yields

or

(6.10a) (6.10b)

where

According to (6.4), Equation (6.10) can be expressed as the following explicit form:

Thus, the double Wronskian (3.12) is the complextion of Equation (2.3), where

On the other hand, for

partial derivative with respect to

For example, taking

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.

The author would like to express his thanks to the Editor and the referee for their comments. This work is supported by the Natural Science Foundation of Shandong Province of China (Grant No. ZR2014AM001), and the youth teacher development program of Shandong Province of China.

Yepeng Sun, (2015) New Exact Solutions of the (2 + 1)-Dimensional AKNS Equation. Journal of Applied Mathematics and Physics,03,1391-1405. doi: 10.4236/jamp.2015.311167