# zbMATH — the first resource for mathematics

A transformed rational function method and exact solutions to the $$3+1$$ dimensional Jimbo-Miwa equation. (English) Zbl 1198.35231
Summary: A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the $$F$$-expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3+1 dimensional Jimbo-Miwa equation is treated, together with a Bäcklund transformation.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35A30 Geometric theory, characteristics, transformations in context of PDEs
RATH
Full Text:
##### References:
 [1] Lan, H.B.; Wang, K.L., Exact solutions for two nonlinear equations: I, J phys A: math gen, 23, 3923-3928, (1990) · Zbl 0718.35020 [2] Malfliet, W., Solitary wave solutions of nonlinear wave equations, Am J phys, 60, 650-654, (1992) · Zbl 1219.35246 [3] Malfliet, W.; Hereman, W., The tanh method I: exact solutions of nonlinear evolution and wave equations, Phys scripta, 54, 563-568, (1996) · Zbl 0942.35034 [4] Parkes, E.J.; Duffy, B.R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput phys commun, 98, 288-300, (1996) · Zbl 0948.76595 [5] Li, Z.B.; Liu, Y.P., RATH: a Maple package for finding travelling solitary wave solutions to nonlinear evolution equations, Comput phys commun, 148, 256-266, (2002) · Zbl 1196.35008 [6] Ma, W.X., Travelling wave solutions to a seventh order generalized KdV equation, Phys lett A, 180, 221-224, (1993) [7] Ma, W.X.; Zhou, D.T., Solitary wave solutions to a generalized KdV equation, Acta phys sinica, 42, 1731-1734, (1993) · Zbl 0810.35109 [8] Duffy, B.R.; Parkes, E.J., Travelling solitary wave solutions to a seventh-order generalized KdV equation, Phys lett A, 214, 271-272, (1996) · Zbl 0972.35528 [9] Parkes, E.J.; Zhu, Z.; Duffy, B.R.; Huang, H.C., Sech-polynomial travelling solitary-wave solutions of odd-order generalized KdV equations, Phys lett A, 248, 219-224, (1998) [10] Wang, M.L., Solitary wave solutions for variant Boussinesq equations, Phys lett A, 199, 169-172, (1995) · Zbl 1020.35528 [11] Wang, M.L., Exact solutions for a compound KdV-Burgers equation, Phys lett A, 213, 279-287, (1996) · Zbl 0972.35526 [12] Ma, W.X.; Fuchssteiner, B., Explicit and exact solutions to a kolmogorov – petrovskii – piskunov equation, Int J non-linear mech, 31, 329-338, (1996) · Zbl 0863.35106 [13] Fan, E.G., Extended tanh-function method and its applications to nonlinear equations, Phys lett A, 277, 212-218, (2000) · Zbl 1167.35331 [14] Han, T.W.; Zhuo, X.L., Rational form solitary wave solutions for some types of high order nonlinear evolution equations, Ann differential equations, 16, 315-319, (2000) · Zbl 0974.35108 [15] Yan, C.T., A simple transformation for nonlinear waves, Phys lett A, 224, 77-84, (1996) · Zbl 1037.35504 [16] Wazwaz, A.M., The tanh – coth method for solitons and kink solutions for nonlinear parabolic equations, Appl math comput, 188, 1467-1475, (2007) · Zbl 1119.65100 [17] Liu, S.K.; Fu, Z.T.; Liu, S.D.; Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys lett A, 289, 69-74, (2001) · Zbl 0972.35062 [18] He, J.H.; Wu, X.H., Exp-function method for nonlinear wave equations, Chaos, solitons & fractals, 30, 700-708, (2006) · Zbl 1141.35448 [19] Zhou, Y.; Wang, M.L.; Wang, Y.M., Periodic wave solutions to a coupled KdV equations with variable coefficients, Phys lett A, 308, 31-36, (2003) · Zbl 1008.35061 [20] Peng, Y.Z., A mapping method for obtaining exact travelling wave solutions to nonlinear evolution equations, Chin J phys, 41, 103-110, (2003) [21] Liu, J.B.; Yang, K.Q., The extended $$F$$-expansion method and exact solutions of nonlinear pdes, Chaos, solitons & fractals, 22, 111-121, (2004) · Zbl 1062.35105 [22] Chen, Y.; Yan, Z.Y., New exact solutions of $$(2 + 1)$$-dimensional gardner equation via the new sine-Gordon equation expansion method, Chaos, solitons & fractals, 26, 399-406, (2005) · Zbl 1070.35058 [23] Jimbo, M.; Miwa, T., Solitons and infinite-dimensional Lie algebras, Publ res inst math sci, 19, 943-1001, (1983) · Zbl 0557.35091 [24] Dorizzi, B.; Grammaticos, B.; Ramani, A.; Winternitz, P., Are all the equations of the kadomtsev – petviashvili hierarchy integrable?, J math phys, 27, 2848-2852, (1986) · Zbl 0619.35086 [25] Rubin, J.; Winternitz, P., Point symmetries of conditionally integrable nonlinear evolution equations, J math phys, 31, 2085-2090, (1990) · Zbl 0722.58026 [26] Tian, B.; Gao, Y.T., Beyond travelling waves: a new algorithm for solving nonlinear evolution equations, Comput phys commun, 95, 139-142, (1996) · Zbl 0923.65068 [27] Senthilvelan, M., On the extended applications of homogeneous balance method, Appl math comput, 123, 381-388, (2001) · Zbl 1032.35159 [28] Wazwaz, A.M., Multiple-soliton solutions for the calogero – bogoyavlenskii – schiff, jimbo – miwa and YTSF equations, Appl math comput, 203, 592-597, (2008) · Zbl 1154.65366 [29] Öziş, T.; Aslan, I., Exact and explicit solutions to the $$(3 + 1)$$-dimensional jimbo – miwa equation via the exp-function method, Phys lett A, 372, 7011-7015, (2008) · Zbl 1227.37019 [30] Wazwaz, A.M., New solutions of distinct physical structures to high-dimensional nonlinear evolution equations, Appl math comput, 196, 363-370, (2008) · Zbl 1133.65087 [31] Liu, X.Q.; Jiang, S., New solutions of the $$3 + 1$$ dimensional jimbo – miwa equation, Appl math comput, 158, 177-184, (2004) · Zbl 1061.35082 [32] Tang, X.Y.; Liang, Z.F., Variable separated solutions for the $$(3 + 1)$$-dimensional jimbo – miwa equation, Phys lett A, 351, 398-402, (2006) · Zbl 1187.37099 [33] Feng, Z.S., On explicit exact solutions to the compound Burgers-KdV equation, Phys lett A, 293, 57-66, (2002) · Zbl 0984.35138 [34] Li, J.B.; Wu, J.H.; Zhu, H.P., Traveling waves for an integrable higher order KdV type wave equations, Int J bifur chaos appl sci eng, 16, 2235-2260, (2006) · Zbl 1192.37100 [35] Ma, W.X., An exact solution to two-dimensional korteweg – de vries – burgers equation, J phys A: math gen, 26, L17-L20, (1993) · Zbl 0771.35059 [36] Yomba, E., On exact solutions of the coupled klein – gordon – schrodinger and the complex coupled KdV equations using mapping method, Chaos, solitons & fractals, 21, 209-229, (2004) · Zbl 1046.35105 [37] Chen, Y.Z.; Ding, X.W., Exact travelling wave solutions of nonlinear evolution equations in $$(1 + 1)$$ and $$(2 + 1)$$ dimensions, Nonlinear anal, 61, 1005-1013, (2005) · Zbl 1086.34501 [38] Tian, B.; Gao, Y.T.; Hong, W., The solitonic features of a nonintegrable $$(3 + 1)$$-dimensional jimbo – miwa equation, Comput math appl, 44, 525-528, (2002) · Zbl 1053.37059 [39] El-Wakil, S.A.; Abdou, M.A., The extended mapping method and its applications for nonlinear evolution equations, Phys lett A, 358, 275-282, (2006) · Zbl 1142.35604 [40] Wang, D.; Zhang, H.Q., Further improved $$F$$-expansion method and new exact solutions of konopelchenko – dubovsky equation, Chaos, solitons & fractals, 25, 601-610, (2005) · Zbl 1083.35122 [41] Zhang, S.; Xia, T.C., A generalized $$F$$-expansion method and new exact solutions of konopelchenko – dubrovsky equations, Appl math comput, 183, 1190-1200, (2006) · Zbl 1111.35318 [42] Abdou, M.A., The extended $$F$$-expansion method and its application for a class of nonlinear evolution equations, Chaos, solitons & fractals, 31, 95-104, (2007) · Zbl 1138.35385 [43] Gao L, Ma WX, Xu W. Travelling wave solutions to Zufiria’s higher-order Boussinesq type equations. Nonlinear Anal, in press. · Zbl 1238.35131 [44] Ma, W.X.; You, Y., Solving the korteweg – de Vries equation by its bilinear form: Wronskian solutions, Trans am math soc, 357, 1753-1778, (2005) · Zbl 1062.37077 [45] Aktosun, T.; van der Mee, C., Explicit solutions to the korteweg – de Vries equation on the half line, Inverse probl, 22, 2165-2174, (2006) · Zbl 1105.35099 [46] Li, C.X.; Ma, W.X.; Liu, X.J.; Zeng, Y.B., Wronskian solutions of the Boussinesq equation – solitons, negatons, positons and complexitons, Inverse probl, 23, 279-296, (2007) · Zbl 1111.35044 [47] Ma WX, Li CX, He JS. A second Wronskian formulation of the Boussinesq equation. Nonlinear Anal, in press. [48] Lee, J.H.; Lin, C.K.; Pashaev, O.K., Shock waves, chiral solitons and semiclassical limit of one-dimensional anyons, Chaos, solitons & fractals, 19, 109-128, (2004) · Zbl 1069.81022 [49] Tam, H.W.; Ma, W.X.; Hu, X.B.; Wang, D.L., The hirota – satsuma coupled KdV equation and a coupled ito system revisited, J phys soc jpn, 69, 45-52, (2000) · Zbl 0965.35144 [50] Biondini, G.; Kodama, Y., On a family of solutions of he kadomtsev – petviashvili equation which also satisfy the Toda lattice hierarchy, J phys A: math gen, 36, 10519-10536, (2003) · Zbl 1116.37316 [51] Hietarinta, J.; Hirota, R., Multidromion solutions to the davey – stewartson equation, Phys lett A, 145, 237-244, (1990) [52] Lou, S.Y.; Lu, J.Z., Special solutions from the variable separated approach: the davey – stewartson equation, J phys A: math gen, 29, 4209-4215, (1996) · Zbl 0899.35101 [53] Ma, W.X., Diversity of exact solutions to a restricted boiti – leon – pempinelli dispersive long-wave system, Phys lett A, 319, 325-333, (2003) · Zbl 1030.35021
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.