×

zbMATH — the first resource for mathematics

Seven common errors in finding exact solutions of nonlinear differential equations. (English) Zbl 1221.35342
Summary: We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We show that many popular methods in finding the exact solutions are equivalent each other. We demonstrate that some authors look for the solitary wave solutions of nonlinear ordinary differential equations and do not take into account the well-known general solutions of these equations. We illustrate several cases when authors present some functions for describing solutions but do not use arbitrary constants. As this fact takes place, redundant solutions of differential equations are found. A few examples of incorrect solutions by some authors are presented. Several other errors in finding the exact solutions of nonlinear differential equations are also discussed.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Zabusky, N.J.; Kruskal, M.D., Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys rev lett, 15, 240-243, (1965) · Zbl 1201.35174
[2] Gardner, C.S.; Green, J.M.; Kruskal, M.D.; Miura, R.M., Method for solving the Korteweg-de Vries equations, Phys rev lett, 19, 1095-1097, (1967) · Zbl 1061.35520
[3] Lax, P.D., Integrals of nonlinear equations of evolution and solitary waves, Commun pure appl math, 21, 467-490, (1968) · Zbl 0162.41103
[4] Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H., The inverse scattering transform - Fourier analysis for nonlinear problems, Stud appl math, 53, 249-315, (1974) · Zbl 0408.35068
[5] Ablowitz, M.J.; Clarkson, P.A., Solitons nonlinear evolution equations and inverse scattering, (1991), Cambridge University Press Cambridge, MA · Zbl 0762.35001
[6] Hirota, R., Exact solution of the Korteweg-de Vries for multiple collisions of solutions, Phys rev lett, 27, 1192-1194, (1971) · Zbl 1168.35423
[7] Kudryashov NA. Analitical theory of nonlinear differential equations. Moskow-Igevsk. Institute of Computer Investigations; 2004 [in Russian].
[8] Polyanin, A.D.; Zaitsev, V.F.; Zhyrov, A.I., Methods of nonlinear equations of mathematical physics and mechanics, Moscow fizmatlit, (2005)
[9] Weiss, J.; Tabor, M.; Carnevalle, G., The painleve property for partial differential equations, J math phys, 24, 522-526, (1983) · Zbl 0514.35083
[10] Weiss, J., The painleve property for partial differential equations. II: backlund transformation, Lax pairs, and the Schwarzian derivative, J math phys, 24, 1405-1413, (1983) · Zbl 0531.35069
[11] Conte, R., The painleve property, one century later, CRM series in mathematical physics, (1999), Springer-Verlag New York, p. 77-180
[12] Kudryashov, N.A., Exact soliton solutions of the generalized evolution equation of wave dynamics, J appl math mech, 52, 3, 360-365, (1988)
[13] Conte, R.; Musette, M., Painlevé analysis and backlund transformations in the kuramoto – sivashinsky equation, J phys A math gen, 22, 169-177, (1989) · Zbl 0687.35087
[14] Kudryashov, N.A., Exact solutions of the generalized kuramoto – sivashinsky equation, Phys lett A, 147, 287-291, (1990)
[15] Kudryashov, N.A., Exact soliton solutions of nonlinear wave equations arising in mechanics, J appl math mech, 54, 3, 450-453, (1990)
[16] Kudryashov, N.A., On types of nonlinear nonintegrable equations with exact solutions, Phys lett A, 155, 269-275, (1991)
[17] Kudryashov, N.A., Partial differential equations with solutions having movable first-order ingularities, Phys lett A, 169, 237-242, (1992)
[18] Kudryashov, N.A., Truncated expansions and nonlinear integrable partial differential equations, Phys lett A, 178, 99-104, (1993)
[19] Kudryashov, N.A., From singular manifold equations to integrable evolution equations, J phys A math gen, 27, 2457-2470, (1994) · Zbl 0839.35119
[20] Kudryashov, N.A.; Zargaryan, E.D., Solitary waves in active-dissipative dispersive media, J phys A math gen, 29, 8067-8077, (1996) · Zbl 0901.35090
[21] Peng, Y.Z.; Krishnan, E.V., The singular manifold method and exact periodic wave solutions to a restricted BLP dispersive long wave system, Rep math phys, 56, 367-378, (2005) · Zbl 1090.35159
[22] Kudryashov, N.A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos soliton fract, 24, 1217-1231, (2005) · Zbl 1069.35018
[23] Kudryashov, N.A., Exact solitary waves of the Fisher equations, Phys lett A, 342, 99-106, (2005) · Zbl 1222.35054
[24] Kudryashov, N.A.; Demina, M.V., Polygons of differential equation for finding exact solutions, Chaos soliton fract, 33, 1480-1496, (2007) · Zbl 1133.35084
[25] Kudryashov, N.A., Solitary and periodic solutions of the generalized kuramoto – sivashinsky equation, Regular chaotic dyn, 13, 234-238, (2008) · Zbl 1229.35229
[26] Kudryashov, N.A.; Loguinova, N.B., Extended simplest equation method for nonlinear differential equations, Appl math comput, 205, 396-402, (2008) · Zbl 1168.34003
[27] Lan, H.; Wang, K., Exact solutions for some nonlinear equations, Phys lett A, 137, 369-373, (1989)
[28] Lou, S.Y.; Huang, G.X.; Ruan, H.Y., Exact solitary waves in a convecting fluid, J phys A math gen, 24, 11, L587-L590, (1991) · Zbl 0735.76057
[29] 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
[30] 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
[31] Wang, M.L.; Li, X.; Zhang, J., The \(G^\prime / G\)-expansion method and evolution equations in mathematical physics, Phys lett A, 372, 4, 417, (2008)
[32] Bekir, A., Application of the \(G^\prime / G\)-expansion method for nonlinear evolution equations, Phys lett A, 3400-3406, (2008) · Zbl 1228.35195
[33] Zhang, J.; Wei, X.; Lu, Y., A generalized \(G^\prime / G\)-expansion method and its applications, Phys lett A, 372, 3653-3658, (2008) · Zbl 1220.37070
[34] 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
[35] Abdou, M.A., The extended tanh method and its applications for solving nonlinear physical models, Appl math comput, 199, 988-996, (2007) · Zbl 1123.65103
[36] Wazzan, L., A modified tanh – coth method for solving the general burgers – fisher and the kuramoto – sivashinsky equations, Commun nonlinear sci numer simulat, 14, 2642-2652, (2009) · Zbl 1221.35320
[37] He, J.H.; Wu, X.H., Exp-function method for nonlinear wave equations, Chaos soliton fract, 30, 700-708, (2006) · Zbl 1141.35448
[38] He, J.H.; Abdou, M.A., New periodic solutions for nonlinear evolution equations using exp-function method, Chaos soliton fract, 34, 1421-1429, (2007) · Zbl 1152.35441
[39] Wu, X.H.; He, J.H., Solitary solutions, periodic solutions and compaction-like solutions using the exp-function method, Comput math appl, 54, 966-986, (2007) · Zbl 1143.35360
[40] Ebaid, A., Exact solitary wave solutions for some nonlinear evolution equations via exp-function method, Phys lett A, 365, 213-219, (2007) · Zbl 1203.35213
[41] El-Wakil, S.A.; Madkour, M.A.; Abdou, M.A., Application of the exp-function method for nonlinear evolution equations with variable coefficients, Phys lett A, 369, 62-69, (2007) · Zbl 1209.81097
[42] Kudryashov, N.A.; Loguinova, N.B., Be careful with the exp-function method, Commun nonlinear sci numer simulat, 14, 1881-1890, (2009) · Zbl 1221.35344
[43] Li, W.; Zhang, H., Generalized multiple Riccati equations rational expansion method with symbolic computation to construct exact complex solutions of nonlinear partial differential equations, Appl math comput, 197, 288-296, (2008) · Zbl 1135.65385
[44] Erbas, B.; Yusufogˇlu, E., Exp-function method for constructing exact solutions of sharma – tasso – olver equation, Chaos soliton fract, (2008) · Zbl 1198.81087
[45] Zhang, S., Application of the exp-function method to high-dimensional nonlinear evolution equation, Chaos soliton fract, 38, 270-276, (2008) · Zbl 1142.35593
[46] Korteweg, D.J.; de Vries, G., On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos mag, 39, 422-443, (1895) · JFM 26.0881.02
[47] Kudryashov, N.A., On “new traveling wave solutions of the KdV and the KdV - Burgers equations, Commun nonlinear sci numer simulat, 14, 1891-1900, (2009) · Zbl 1221.35343
[48] Khani, F., Analytic study on the higher order ito equations: new solitary wave solutions using the exp-function method, Chaos soliton fract, (2008) · Zbl 1198.35224
[49] Öziş, T.; Aslan, I., Exact and explicit solutions to the (3+1)-dimensional jimbo – miva equation via the exp-function method, Phys lett A, 372, 7011-7015, (2008) · Zbl 1227.37019
[50] Ganji, Z.Z.; Ganji, D.D.; Bararnia, H., Approximate general and explicit solutions of nonlinear BBMB equations by exp-function method, Appl math model, (2008) · Zbl 1205.35250
[51] Soliman, A.A., Exact solutions of the KdV - Burgers equation by exp-function method, Chaos soliton fract, (2008) · Zbl 1198.35241
[52] Cole, J.D., On a quasilinear parabolic equation occurring in aerodynamics, Quart appl math, 9, 225-236, (1951) · Zbl 0043.09902
[53] Hopf, E., The partial differential equation \(u_t + \mathit{uu}_x = \mu u_{\mathit{xx}}\), Commun pure appl math, 13, 201-230, (1950) · Zbl 0039.10403
[54] Abdou, M.A., Generalized solitonary and periodic solutions for nonlinear partial differential equations by the exp-function method, Nonlinear dyn, 52, 1-9, (2008) · Zbl 1173.35647
[55] Wazwaz, A.M., Multiple-soliton solutions for the generalized (1+1)-dimensional and the generalized (2+1)-dimensional ito equations, Appl math comput, 202, 840-849, (2008) · Zbl 1147.65085
[56] Clarkson, P.A.; Kruskal, M.D., New similarity reductions of the Boussinesq equation, J math phys, 30, 2201-2213, (1989) · Zbl 0698.35137
[57] Ugurlu, Y.; Kaya, D., Solutions of the cahn – hillard equation, Comput math appl, (2008)
[58] Bekir, A., On travelling wave solutions to combined kdv – mkdv and modified Burgers - KdV equation, Commun nonlinear sci numer simulat, 14, 1038-1042, (2009) · Zbl 1221.35323
[59] Bekir, A., Applications of the extended tanh method for coupled nonlinear evolution equations, Commun nonlinear sci numer simulat, 13, 1748-1757, (2008) · Zbl 1221.35322
[60] Xie, F.; Zhang, Y.; Lü, Z., Symbolic computation in nonlinear evolution equation: application to (3+1)-dimensional kadomttsev – petviasvili equation, Chaos soliton fract, 24, 257-263, (2005)
[61] Zhang, S., Symbolic computation and new families of exact non-travelling wave solutions of (2+1)-dimensional konopelchenko—dubrovsky equations, Chaos soliton fract, 31, 951-959, (2007) · Zbl 1139.35392
[62] Wazwaz, A.M., New solitons and kinks solutions to the sharma – tasso – olver equation, Appl math comput, 188, 1205-1213, (2007) · Zbl 1118.65113
[63] Wazwaz, A.M., New solitary wave solutions to the kuramoto – sivashinsky and the Kawahara equations, Appl math comput, 182, 164250, (2006)
[64] Kuramoto, Y.; Tsuzuki, T., Persistent propagation of concentration waves in dissipative media far from thermal equlibrium, Prog theor phys, 55, 2, 356-369, (1976)
[65] Chen, H.; Zhang, H., New multiple soliton solutions to the general burgers – fisher equation and the kuramoto – sivashinsky equation, Chaos soliton fract, 19, 71-76, (2004) · Zbl 1068.35126
[66] Yusufogˇlu, E., New solitary solutions for the MBBM equations using exp-function method, Phys lett A, 372, 442-446, (2008) · Zbl 1217.35156
[67] Chun, C., Soliton and periodic solutions for the fifth-order KDV equation with the exp-function method, Phys lett A, 372, 2760-2766, (2008) · Zbl 1220.35148
[68] Abdou, M.A.; Soliman, A.A.; El-Basony, S.T., New application of exp-function method for improved Boussinesq equation, Phys lett A, 369, 469-475, (2007) · Zbl 1209.81091
[69] Bekir, A.; Cevikel, A.C., Solitary wave solutions of two nonlinear physical models by tanh – coth method, Commun nonlinear sci numer simulat, 14, 1804-1809, (2009)
[70] Öziş, T.; Körogˇlu, C., A novel approach for solving the Fisher equation using exp-function method, Phys lett A, 372, 3836-3840, (2008) · Zbl 1220.83011
[71] Chun, C., Application of exp-function method to the generalized burgers – huxley equation, J phys conf ser, 96, 012217, (2008)
[72] Efimova, O.Yu.; Kudryashov, N.A., Exact solutions of the burgers – huxley equation, J appl math mech, 68, 3, 413-420, (2004) · Zbl 1092.35084
[73] El-Wakil, S.A.; Abdou, M.A.; Hendi, A., New periodic wave solutions via exp-function method, Phys lett A, 372, 830-840, (2008) · Zbl 1217.37070
[74] Zhang, S., Exp-function method for klein – gordon equation with quadratic nonlinearity, J phys conf ser, 96, 012002, (2008)
[75] Noor, M.A.; Mohyud-Din, S.T.; Waheed, A., Exp-function method for solving kuramoto – sivashinsky and Boussinesq equations, J appl math comput, (2008) · Zbl 1145.65050
[76] Dai, C.Q.; Wang, Y.Y., New exact solutions of the (3+1)-dimensional Burgers system, Phys lett A, (2008)
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.