A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B？cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.
References
[1]
Ablowitz, 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/cbo9780511623998
[2]
Ablowitz, 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/10.1088/0266-5611/23/4/021
[3]
Nimmoa, 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/10.1080/14029251.2014.905301
[4]
Kakei, 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/s00023-002-8638-1
[5]
Chen, 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)01668-7
[6]
Toda, 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.
[7]
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/s11071-011-0228-7
[8]
Wazwaz, 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.2004.06.029
[9]
Radha, 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-G
[10]
Lou, 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/10.1088/0253-6102/26/4/487
[11]
Qiao, 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.86.016601
[12]
Lou, 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/10.1088/0305-4470/34/2/307
[13]
Hu, 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/s10409-012-0070-2
[14]
Guo, 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.2012.07.038
[15]
Weiss, 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/10.1063/1.525721
[16]
Weiss, 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/10.1063/1.525875
[17]
Joshi, 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)90184-8
[18]
Mohammad, 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.1996.3.1-2.15
[19]
Baldwin, 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.2006.13.1.8
[20]
Nimmo, 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-X
[21]
Freeman, 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)90764-8
[22]
Liu, 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.59.3520
[23]
Liu, 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.2011.07.014
[24]
Liu, 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/29/8/080202
[25]
Yao, 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.2007.10.096
[26]
Ma, 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)00971-4
[27]
Chen, 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
