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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.

MSC:

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

Citations:

Zbl 0617.35119

Software:

RATH; ATFM
PDFBibTeX XMLCite
Full Text: DOI

References:

[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
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