A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The Bäcklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.
Recently high dimensional nonlinear partial differential or difference equations attract much interest. Both inte- grable and non-integrable equations have their physical and mathematical values but the former posses some special properties such as infinite conservation laws and symmetries, multi-soliton solutions, Bäcklund and Darboux transformation (c.f. [1] - [3] ). Among these high dimensional equations some are deduced from physics phenomenon originally, say KP equation, but others are deduced firstly from (1 + 1) dimensional equation mathematically ( [4] - [8] ). However, the findings of new solutions or special constructions of these equations makes nonlinearity of equations be realized clearly, which helps the development of subject of nonlinear science. In this paper we will consider a (2 + 1) dimensional KdV-mKdV equation as follows
where subscript means a partial derivative such as and. It is obvious
that if the equation becomes a mixed KdV-mKdV equation, which is widely researched by many authors (see [7] - [10] ). The related negative KdV equation and (2 + 1)-dimensional KdV equation were also discussed by several authors (c.f. [11] - [14] ). Now we set
to treat the integral appearing in equation. The Equation (1) is then rewritten as
We will prove it has Painlevé property firstly, then deduce a Bäcklund transformation and bilinear equation. Using bilinear equation we can construct Wronskian solutions and present some exact solutions finally.
2. Painlevé Test
Painlevé analysis method is an important method for testing integrability [15] - [19] . As we know, the basic Painlevé test consists of the following steps (taking (1 + 1) dimensional case as an example) [15] [19] .
Step 1. Expanding the solution of a PDE as Laurent series of a singular manifold
where is constant to be determined and coefficients Then substitute it into PDE to find all dominant balances.
Step 2. If all exponents are integers, find the resonances where arbitrary constants can appear.
Step 3. If all resonances are integers, check the resonance conditions in each Laurent expansion.
Conclusion. If no obstruction is found in Steps 1 - 3 for every dominant balances, then the Painlevé test is satisfied.
The situation of high dimensional case is similar. For step 1, we can simply let
Substituting them into (2, 3) gives us
where Thus
Insert them into (2, 3) and equal coefficients of both side of in (3), in (2) we have
From them we work out
To get resonances we collect the coefficient of in (3), in (2) for general term number r respec- tively, we have
where F, G are functions of and their derivatives. This gives the resonances and means the singular manifold
Now we proceed to verify the resonance conditions. First we consider For this purpose we extract in (3) and set it be zero. We readily have
or equivalently
The part of in (2) gives
and it is true by employing obtained above. This result shows that an arbitrary appears in reso- nance, i.e. resonance condition is satisfied. Further, we verify resonance condition for Collecting the terms of in (3) reads
where
In a similar way, collecting the terms of in (2) makes us have
where
we need to verify
because is a resonance, i.e. By inserting (13) into and through a dull calculation
we can complete the proof of compatible condition. It is a turn to consider which emerge from in (3) and in (2). They are
where
and
where
Its resonance condition is verified similarly but is more complex. Thus we prove that (2 + 1) dimensional KdV-mKdV equation passes Painlevé test.
Now we consider to truncate the series (6). To meet this end we must let Thus we will have
and combine the equation satisfied by we obtain a Bäcklund transformation actually. In fact, if we take then (13) gives
Furthermore, If we continue to set we get following relations from (15, 16)
and
The condition produces another identity
Using (20)-(23) we may truncate the series. Thus we indeed get a Bäcklund transformation by noting (22, 23). But it is more important pointing that the identities (20)-(23) have only two independent expressions, say (22, 23). Applying the definition of Schwartzian derivative
we simplify them as a concise form, i.e. so called Schwartzian derivative equation
It is the condition satisfied by function in Bäcklund transformation (19).
3. Hirota Method for Finding Exact Solutions
In this section we will give the bilinear equation of Equation (1) and present some exact solutions from it. The truncation form (19) suggests us to try the transformation
We first take an integral with respect to x on Equation (1). Then eliminate the remaining integral operator by setting
where D is bilinear operator. Thus we can transfer Equation (1) into
Equations (26, 27) are bilinear equations of (1). To find its solutions we set further, where * means complex conjugation. Expanding f as perturbation series
and substituting it into bilinear equations, equaling coefficients of power of yields
Take
where and are all real constants (the similar condition will be imposed on later text but omitting), we know the relation immediately
The coefficient of can take as zero according to this result. So we get a single solution solution as follows
If we take
then after substituting it into (29, 30) we know relations
are valid. Again compare coefficient of, we have
When employing (34),
are obtained. After that we consider coefficient of
The r.h.s is computed to zero. Thus we may truncate the perturbation series and 2-soliton solution is got as
Further, keeping these results in mind we can conjecture the N-soliton solution taking on
where and
4. Wronskian Solutions
Wronskian technique is one of the powerful methods in finding exact solutions of nonlinear integrable evolution equation [20] [21] . It can be used to solve whole integrable evolution equation hierarchy (c.f. [22] [23] ) and its application had been extended to negative nonlinear evolution equation (c.f. [23] [24] ), high dimensional nonlinear evolution equation [25] , etc. The generalization of this method can obtain several types of exact solutions (c.f. [26] [27] ). Here we use the Nimmo's brief notation to denote Wronskia determinants:
and
where and
Supposing that vectors satisfies the following conditions
where A is a non-singular real constant matrix. We will prove that is the solution of bilinear Equations (26) and (27). We first point out that in this situation, can be expressed by related Wronskia determinant:
To get down to our work we need the help of two Lemmas, we list out them first.
Lemma 1 ( [26] [27] ) Assuming that M is a matrix and are n-dimensional vectors, then the following determinantal identity is valid:
Lemma 2 ( [23] [24] ) Assuming P is a matrix, are the columns of another matrix, then we have the following foluma
We first treat bilinear Equations (26). Computing derivatives of Wronskians and substituting them into (26) yields
When apply Lemma 2 into Wronskians we get an identity as follows
Then adding it to (44) gives us
which equals zero by using Lemma 1. Now we can focus our attention on the bilinear Equation (27). We also calculate the derivative of Wronskians prior to carrying out our procedure. For example, we have
Then becomes as
Again using Lemma 2, we produce two identities as follows:
The substitution of (48, 49) into (47) yields
To vanish r.h.s of this equation we apply Lemma 1 again, which give us a valuable identity
Multiply to this identity we work out another relation as follows:
It is because of
In a same way, we deduce
Thus we complete the proof that
Now we present some exact solutions as examples. Firstly, we may write out the expression of spectral vector:
where are two real constant vectors and
where I is unit matrix. If we choose A as diagonal matrix then soliton solutions of equation (1) can be got again. In fact, supposing
and
then spectral vector adopts the following formula
The solutions given by (25) are solitons solutions in this situation. In fact, when, it is exactly the solution (33). When consider, we compute out
This gives the same solution as (41) or simplified form:
which is a two-soliton solution. We can also take into account other solutions. For instance, let
Then we find in this situation:
Taking, the spectral vector is got then:
The correspondent solution of Equation (1) is
or simplified form
This is known as a complexiton solution (c.f. [26] ).
5. Conclusion
Utilizing Painlevé test we prove the integrability of a (2 + 1) dimensional KdV-mKdV equation in the sense of Painlevé. And in the mean time a Bäcklund transformation is produced. Through bilinear equation we get several exact solutions by Hirota method and generalized Wronskian method. Some explicit formulas of exact solutions are obtained. Particularly, 2-soliton solution and complexiton solutions are presented as examples.
Acknowledgements
The authors are grateful to editors and referees for their very careful works. In the mean time, the authors thank to the referees for giving helpful advices.
Support
This work is partly supported by Chinese National Social Science Foundation (Grant Number: CNSSF: 13CJY037) Research on the indemnificatory Apartment Construction Based on Residential Integration.
NOTESReferencesAblowitz, M.J. and Clarkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, 149, Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/cbo9780511623998Ablowitz, M.J., Biondini, G. and Prinari, B. (2007) Inverse Scattering Transform for the Integrable Discrete Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions. Inverse Problems, 23, 1711-1758.
http://dx.doi.org/10.1088/0266-5611/23/4/021Nimmoa, J.J.C. and Yilmaz, H. (2014) On Darboux Transformations for the Derivative Nonlinear Schrödinger Equation. Journal of Nonlinear Mathematical Physics, 21, 278-293. http://dx.doi.org/10.1080/14029251.2014.905301Kakei, S., Ikeda, T. and Takasaki, K. (2002) Hierarchy of (2+1)-Dimensional Nonlinear Schrödinger Equation, Self-Dual Yang-Mills Equation and Toroidal Lie Algebras. Annales Henri Poincaré, 3, 817-845.
http://dx.doi.org/10.1007/s00023-002-8638-1Chen, Y., Yan, Z. and Zhang, H. (2003) New Explicit Solitary Wave Solutions for (2 + 1)-Dimensional Boussinesq Equation and (3 + 1)-Dimensional KP Equation. Physics Letters A, 307, 107-113.
http://dx.doi.org/10.1016/S0375-9601(02)01668-7Toda, K. and Kobayashi, T. (2005) Integrable Nonlinear Partial Diferential Equations with Variable Coefficients from the Painlevé Test. Proceedings of 10th International Conference in Modern Group Analysis, 214-221.Zhang, Y., Song, Y. and Cheng, L. (2012) Exact Solutions and Painlevé Analysis of a New (2 + 1)-Dimensional Generalized KdV Equation. Nonlinear Dynamics, 68, 445-458. http://dx.doi.org/10.1007/s11071-011-0228-7Wazwaz, A.M. (2005) Variants of the Two-Dimensional Boussinesq Equation with Compactons, Solitons, and Periodic Solutions. Computers & Mathematics with Applications, 49, 295-301.
http://dx.doi.org/10.1016/j.camwa.2004.06.029Radha, R. and Lakshmanan, M. (1995) Dromion like Structures in the (2+1)-Dimensional Breaking Soliton Equation. Physics Letters A, 197, 7-12. http://dx.doi.org/10.1016/0375-9601(94)00926-GLou, S.Y. (1996) On the Dromion Solutions of a (2 + 1)-Dimensional KdV-Type Equation. Communications in Theoretical Physics, 26, 487-490. http://dx.doi.org/10.1088/0253-6102/26/4/487Qiao, Z.J. and Fan, E.G. (2012) Negative-Order Korteweg-De Vries Equations. Physical Review E, 86, Article ID: 016601. http://dx.doi.org/10.1103/PhysRevE.86.016601Lou, S.Y. and Ruan, H.Y. (2001) Revisitation of the Localized Excitations of the (2+1)-Dimensional KdV Equation. Journal of Physics A: Mathematical and General, 34, 305-316. http://dx.doi.org/10.1088/0305-4470/34/2/307Hu, W.P., Deng, Z.C., Qin, Y.Y. and Zang, W.L. (2012) Multi-Symplectic Method for the Generalized (2+1)-Dimensional KdV-mKdV Equation. Acta Mechanica Sinica, 28, 793-800. http://dx.doi.org/10.1007/s10409-012-0070-2Guo, H., Fu, Z.T. and Liu, S.K. (2013) Exact Coherent Structures in the (2+1)-Dimensional KdV Equations. Applied Mathematical Modelling, 37, 3102-3111. http://dx.doi.org/10.1016/j.apm.2012.07.038Weiss, J., Tabor, M. and Carnevale, G. (1983) The Painleveé Property for Partial Differential Equations. Journal of Mathematical Physics, 24, 522-526. http://dx.doi.org/10.1063/1.525721Weiss, J. (1983) The Painleveé Property for Partial Differential Equations. II: Baäcklund Transformation, Lax Pairs, and the Schwarzian Derivative. Journal of Mathematical Physics, 24, 1405-1413. http://dx.doi.org/10.1063/1.525875Joshi, N. (1987) Painlevé Property of General Variable-Coefficient Versions of the Korteweg de Vries and Nonlinear Schrödinger Equations. Physics Letters A, 125, 456-460. http://dx.doi.org/10.1016/0375-9601(87)90184-8Mohammad, A.A. and Can, M. (1996) Painleve Analysis and Symmetries of the Hirota–Satsuma Equation. Journal of Nonlinear Mathematical Physics, 3, 152-155. http://dx.doi.org/10.2991/jnmp.1996.3.1-2.15Baldwin, D. and Hereman, W. (2006) Symbolic Software for the Painlevé Test of Nonlinear Ordinary and Partial Differential Equations. Journal of Nonlinear Mathematical Physics, 13, 90-110.
http://dx.doi.org/10.2991/jnmp.2006.13.1.8Nimmo, J.J.C. (1983) Soliton solution of Three Differential-Difference Equations in Wronskian Form. Physics Letters A, 99, 281-286. http://dx.doi.org/10.1016/0375-9601(83)90885-XFreeman, 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)90764-8Liu, Q.M. (1990) Double Wronskian Solution of the AKNS and the Classical Boussinesq Hierarchies. Journal of the Physical Society of Japan, 59, 3520-3527. http://dx.doi.org/10.1143/JPSJ.59.3520Liu, Y.Q., Chen, D.Y. and Hu, C. (2011) The Generalized Wronskian Solutions of a Inverse KdV Hierarchy. Applied Mathematics and Computation, 218, 2025-2035. http://dx.doi.org/10.1016/j.amc.2011.07.014Liu, Y.Q., Chen, D.Y. and Hu, C. (2012) The Generalized Wronskian Solution to a Negative KdV-mKdV Equation. Chinese Physics Letters, 29, Article ID: 080202. http://dx.doi.org/10.1088/0256-307X/29/8/080202Yao, Y.Q., Chen, D.Y. and zhang, D.J. (2008) Multisoliton Solutions to a Nonisospectral (2+1)-Dimensional Breaking Soliton Equation. Physics Letters A, 372, 2017-2025. http://dx.doi.org/10.1016/j.physleta.2007.10.096Ma, 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)00971-4Chen, D.Y., Zhang, D.J. and Bi, 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/s11425-007-0165-6