A family of novel exact solutions to \((2+1)\)-dimensional KdV equation. (English) Zbl 1474.35576

Summary: We introduce two subequations with different independent variables for constructing exact solutions to nonlinear partial differential equations. In order to illustrate the efficiency and usefulness, we apply this method to \((2+1)\)-dimensional KdV equation, which was first derived by M. Boiti et al. [Inverse Probl. 2, 271–279 (1986; Zbl 0617.35119)] using the idea of the weak Lax pair. As a result, we obtained many new exact solutions.


35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
35C08 Soliton solutions
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems


Zbl 0617.35119


Full Text: DOI


[1] Bullough, R. K.; Caudrey, P. J., Solitons (1980), Berlin, Germany: Springer, Berlin, Germany · Zbl 0428.00010
[2] Dodd, R. K.; Eilbeck, J. C.; Morris, H. C., Solitons and Nonlinear Equations (1984), London, UK: Academic Press, London, UK
[3] Drazin, P. G.; Johnson, R. S., Solitons: An Introduction (1989), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0661.35001
[4] Gu, C. H., Soliton Theory and Its Applications (1995), New York, NY, USA: Springer, New York, NY, USA · Zbl 0834.35003
[5] Infeld, E.; Rowlands, G., Nonlinear Waves, Solitons and Chaos (2000), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0726.76018
[6] Rogers, C.; Schief, W. K., Bäcklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory. Bäcklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory, Cambridge Texts in Applied Mathematics (2002), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1019.53002 · doi:10.1017/CBO9780511606359
[7] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering. Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note, 149 (1991), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0762.35001 · doi:10.1017/CBO9780511623998
[8] Wadati, M., Stochastic Korteweg-de Vries equation, Journal of the Physical Society of Japan, 52, 8, 2642-2648 (1983) · doi:10.1143/JPSJ.52.2642
[9] Wadati, M.; Konno, K.; Ichikawa, Y. H., A generalization of inverse scattering method, Journal of the Physical Society of Japan, 46, 6, 1965-1966 (1979) · Zbl 1334.81106 · doi:10.1143/JPSJ.46.1965
[10] Wadati, M.; Sanuki, H.; Konno, K., Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws, Progress of Theoretical Physics, 53, 419-436 (1975) · Zbl 1079.35506 · doi:10.1143/PTP.53.419
[11] Konno, K.; Wadati, M., Simple derivation of Bäcklund transformation from Riccati form of inverse method, Progress of Theoretical Physics, 53, 6, 1652-1656 (1975) · Zbl 1079.35505 · doi:10.1143/PTP.53.1652
[12] Lou, S. Y.; Huang, G. X.; Ruan, H. Y., Exact solitary waves in a convecting fluid, Journal of Physics A: Mathematical and General, 24, 11, L587-L590 (1991) · Zbl 0735.76057 · doi:10.1088/0305-4470/24/11/003
[13] Malfliet, W., Solitary wave solutions of nonlinear wave equations, American Journal of Physics, 60, 7, 650-654 (1992) · Zbl 1219.35246 · doi:10.1119/1.17120
[14] Parkes, E. J.; Duffy, B. R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Computer Physics Communications, 98, 3, 288-300 (1996) · Zbl 0948.76595 · doi:10.1016/0010-4655(96)00104-X
[15] Duffy, B. R.; Parkes, E. J., Travelling solitary wave solutions to a seventh-order generalized KdV equation, Physics Letters A, 214, 5-6, 271-272 (1996) · Zbl 0972.35528 · doi:10.1016/0375-9601(96)00184-3
[16] Parkes, E. J.; Duffy, B. R., Travelling solitary wave solutions to a compound KdV-Burgers equation, Physics Letters A, 229, 4, 217-220 (1997) · Zbl 1043.35521 · doi:10.1016/S0375-9601(97)00193-X
[17] Li, Z. B.; Liu, Y. P., RATH: a Maple package for finding travelling solitary wave solutions to nonlinear evolution equations, Computer Physics Communications, 148, 2, 256-266 (2002) · Zbl 1196.35008 · doi:10.1016/S0010-4655(02)00559-3
[18] Fan, E., Extended tanh-function method and its applications to nonlinear equations, Physics Letters A, 277, 4-5, 212-218 (2000) · Zbl 1167.35331 · doi:10.1016/S0375-9601(00)00725-8
[19] Fan, E., Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos, Solitons and Fractals, 16, 5, 819-839 (2003) · Zbl 1030.35136 · doi:10.1016/S0960-0779(02)00472-1
[20] Yan, Z. Y., Constructive Theory of Complex Nonlinear Waves and Applications (2007), Beijing, China: Science Press, Beijing, China
[21] Xie, F. D.; Gao, X. S., Applications of computer algebra in solving nonlinear evolution equations, Communications in Theoretical Physics, 41, 3, 353-356 (2004) · Zbl 1167.35300
[22] Zhang, Y.; Wang, C.; Zhou, Z., Inherent randomicity in 4-symbolic dynamics, Chaos, Solitons and Fractals, 28, 1, 236-243 (2006) · Zbl 1083.37502 · doi:10.1016/j.chaos.2005.05.041
[23] Ma, W. X., Complexiton solutions to the Korteweg-de Vries equation, Physics Letters A, 301, 1-2, 35-44 (2002) · Zbl 0997.35066 · doi:10.1016/S0375-9601(02)00971-4
[24] Boiti, M.; Leon, J. Jp.; Manna, M.; Pempinelli, F., On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions, Inverse Problems, 2, 3, 271-279 (1986) · Zbl 0617.35119
[25] Lou, S. Y.; Ruan, H. Y., Revisitation of the localized excitations of the \((2 + 1)\)-dimensional KdV equation, Journal of Physics A. Mathematical and General, 34, 2, 305-316 (2001) · Zbl 0979.37036 · doi:10.1088/0305-4470/34/2/307
[26] Lou, S. Y.; Hu, X. B., Infinitely many Lax pairs and symmetry constraints of the KP equation, Journal of Mathematical Physics, 38, 12, 6401-6427 (1997) · Zbl 0898.58029 · doi:10.1063/1.532219
[27] Radha, R.; Lakshmanan, M., Singularity analysis and localized coherent structures in \((2 + 1)\)-dimensional generalized Korteweg-de Vries equations, Journal of Mathematical Physics, 35, 9, 4746-4756 (1994) · Zbl 0818.35109 · doi:10.1063/1.530812
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.