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 [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] .
The AKNS (Ablowitz-Kaup-Newell-Segur) equation is one of the most important physical models [17] - [19] . In 1997, Lou and Hu have obtained the (2 + 1)-dimensional AKNS equation from the inner parameter dependent symmetry constraints of the KP equation [20] . Moreover, Lou et al. have studied Painlev integrability of the (2 + 1)-dimensional AKNS equation [21] . In this paper, we will apply the Hirota method and the Wronskian technique to obtain new exact solutions of the (2 + 1)-dimensional AKNS equation.
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.
2. N-Soliton Solutions of the (2 + 1)-Dimensional AKNS Equation
We consider the following (2 + 1)-dimensional AKNS equation [21]
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 yields
Taking
we can obtain
Letting then, ,. 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).
where
, and take over all possible combinations of and satisfy the following condition
3. 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
where and The following two determinantal identities were often used [4] [5] . The one is
where D is a matrix and and d represent N column vectors. The other is
where are N column vectors and denotes.
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 and 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.
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 and are arbitrary real constants.
Taking the double Wronskian solution of Equation (2.3) is obtained as follows:
Letting and gives
then one-soliton solution of Equation (2.1) is
Choosing and yields
So, we have
Similarly, when and, we get
In the following, we will prove that Equation (2.3) has the generalized double Wronskian solution. First, we give the following lemma [19] .
Lemma 1. Assume that is an operator matrix and its entries are differential operators. is an function matrix with column vector set and row vector set , then
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 and satisfy the following conditions
is an 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.
(1) If setting
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 we can consider this as a limit case where tends to zero. Then (3.15)-(3.17) become
Using (3.18), Equation (3.12) still satisfies Equation (2.3).
From Equation (3.13), we can get the general solution
where and are real constant vectors. Thus, we have the fol
lowing result.
Theorem 3. is an arbitrary real matrix independent of x and t. Equation (2.3) has double Wronskian solution (3.12), where and are constructed by (3.19). The corresponding solution of Equation (2.1) can be expressed as
4. Rational Solutions
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 Thus (4.1) can be truncated as
The components of and are
In (4.6), taking then (4.6) becomes
Thus, we can calculate some rational solutions of Equation (2.1).
5. Matveev Solutions
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 is an unite matrix. We have
i.e.,
Substituting (5.2) into (4.1), we get
The components of and are
Specially, taking then (5.5) becomes
Thus, Matveev solutions of Equation (2.1) can be obtained, where
In (5.7), taking
where and are generated from (5.6), we can obtain the Matveev solution of Equation (2.1).
Similarly, choosing
and we get
When we have
Assume that
letting gives
Similarly, taking yields
6. Complexitions of the (2 + 1)-Dimensional AKNS Equation
In the following, we would like to consider that A is a real Jordan matrix.
where
and are real constants. Then, from (4.1), complexitons can be obtained.
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, we have
Similarly,
Further, we consider the matrix A as a Jordan block
where the symbol denotes tensor product of matrices. Noting that, we get
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 the partial derivative with respect to can be replaced by the
partial derivative with respect to in (6.10) and (6.11).
For example, taking (dropping the subscript) and we have
7. 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.
Acknowledgements
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.
Cite this paper
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
ReferencesGardner, C.S., Greene, J.M., Kruskal, M.D. and Miura, R.M. (1967) Method for Solving the Korteweg de Vries Equation. Physical Review Letters, 19, 1095-1097. http://dx.doi.org/10.1103/PhysRevLett.19.1095Matveev, V.B. and Salle, M.A. (1991) Darboux Transformations and Solitons. Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-3-662-00922-2Hirota, R. (1971) Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons. Physical Review Letters, 27, 1192-1194. http://dx.doi.org/10.1103/PhysRevLett.27.1192Freeman, N.C. and Nimmo, J.J.C. (1983) Soliton Solutions of the KdV and KP Equations: The Wronskian Technique. Physics Letters A, 95, 1-3. http://dx.doi.org/10.1016/0375-9601(83)90764-8Nimmo, J.J.C. and Freeman, N.C. (1983) A Method of Obtaining the N-Soliton Solution of the Boussinesq Equation in Terms of a Wronskian. Physics Letters A, 95, 4-6. http://dx.doi.org/10.1016/0375-9601(83)90765-XHu, X.B. and Wang, H.Y. (2006) Construction of dKP and BKP Equations with Self-Consistent Sources. Inverse Problems, 22, 1903-1920. http://dx.doi.org/10.1088/0266-5611/22/5/022Hu, X.B. and Wang, H.Y. (2007) New Type of Kadomtsev-Petviashvili Equation with Self-Consistent Soureces and Its Blinear Bäcklund Transformation. Inverse Problems, 23, 1433-1444. http://dx.doi.org/10.1088/0266-5611/23/4/005Satsuma, J. (1979) A Wronskian Representation of N-Soliton Solutions of Nonlinear Evolution Equations. Journal of the Physical Society of Japan, 46, 359-360. http://dx.doi.org/10.1143/JPSJ.46.359Matveev, V.B. (1992) Generalized Wronskian Formula for Solutions of the KdV Equation: First Applications. Physics Letters A, 166, 205-208. http://dx.doi.org/10.1016/0375-9601(92)90362-PMa, W.X. (2002) Complexiton Solutions to the Korteweg-de Vries Equation. Physics Letters A, 301, 35-44. http://dx.doi.org/10.1016/S0375-9601(02)00971-4Ma, W.X. (2004) Wronskians, Generalized Wronskians and Solutions to the Korteweg-de Vries Equation. Chaos, Solitons & Fractals, 19, 163-170. http://dx.doi.org/10.1016/S0960-0779(03)00087-0Ma, W.X. and Maruno, K. (2004) Complexiton Solutions of the Toda Lattice Equation. Physica A, 343, 219-237. http://dx.doi.org/10.1016/j.physa.2004.06.072Ma, W.X. and You, Y. (2005) Solving the Korteweg-de Vries Equation by Its Bilinear Form: Wronskian Solutions. Transactions of the American Mathematical Society, 357, 1753-1778. http://dx.doi.org/10.1090/S0002-9947-04-03726-2Ma, W.X. (2005) Complexiton Solutions to Integrable Equations. Nonlinear Analysis: Theory, Methods & Applications, 63, 2461-2471. http://dx.doi.org/10.1016/j.na.2005.01.068Ma, W.X. (2005) Complexiton Solutions of the Korteweg-de Vries Equation with Self-Consistent Sources. Chaos, Solitons & Fractals, 26, 1453-1458. http://dx.doi.org/10.1016/j.chaos.2005.03.030Ma, W.X., Li, C.X. and He, J.S. (2009) A Second Wronskian Formulation of the Boussinesq Equation. Nonlinear Analysis: Theory, Methods & Applications, 70, 4245-4258. http://dx.doi.org/10.1016/j.na.2008.09.010Ablowitz, M.J., Kaup, D.J., Newell, A.C. and Segur, H. (1974) The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems. Studies in Applied Mathematics, 53, 249-315. http://dx.doi.org/10.1002/sapm1974534249Liu, Q.M. (1990) Double Wronskian Solution of the AKNS and Classical Boussinesq Hierarchies. Journal of the Physical Society of Japan, 59, 3520-3527. http://dx.doi.org/10.1143/JPSJ.59.3520Chen, D.Y., Zhang, D.J. and Bo, J.B. (2008) New Double Wronskian Solutions of the AKNS Equation. Science in China Series A: Mathematics, 51, 55-69. http://dx.doi.org/10.1007/s11425-007-0165-6Lou, S.Y. and Hu, X.B. (1997) Infinitely Many Lax Pair and Symmetry Constraints of the KP Equation. Journal of Mathematical Physics, 38, 6407-6427. http://dx.doi.org/10.1063/1.532219Lou, S.Y., Chen, C.L. and Tang, X.Y. (2002) (2 + 1)-Dimensional (M + N)-Component AKNS System: Painlevé Integrability, Infinitely Many Symmetries, Similarity Reductions and Exact Solutions. Journal of Mathematical Physics, 43, 4078-4109. http://dx.doi.org/10.1063/1.1490407
(6.10a)
(6.10b)