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Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations. (English) Zbl 1205.35262
Summary: If a partial differential equation is reduced to an ordinary differential equation in the form u ' (ξ)=G(u,θ 1 ,,θ m ) under the traveling wave transformation, where θ 1 ,,θ m are parameters, its solutions can be written in the integral form ξ-ξ 0 =du G(u,θ 1 ,,θ m ). Therefore, the key steps are to determine the parameter and to solve the corresponding integral. When G is related to a polynomial, a mathematical tool named complete discrimination system for a polynomial is applied to this problem so that the parameter scopes can be determined easily. The complete discrimination system for a polynomial is a natural generalization of the discrimination Δ=b 2 -4ac of the second degree polynomial ax 2 +bx+c. For example, the complete discrimination system for the third degree polynomial F(w)=w 3 +d 2 w 2 +d 1 w+d 0 is given by Δ=-27(2d 1 3 27+d 0 -d 1 d 2 3) 2 -4(d 1 -d 2 2 3) 3 and D 1 =d 1 -d 2 2 3. In the paper, we give some new applications of the complete discrimination system for polynomials, that is, we give the classifications of traveling wave solutions to some nonlinear differential equations through solving the corresponding integrals. Finally, as a result, we give a partial answer to a problem on Fan’s expansion method.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35A20Analytic methods, singularities (PDE)
35A24Methods of ordinary differential equations for PDE
Software:
RATH; RAEEM; MACSYMA
References:
[1]Hereman, W.: Comput. phys. Commun., Comput. phys. Commun. 65, 143 (1991)
[2]Parkes, E. J.; Duffy, B. R.: Comput. phys. Commun., Comput. phys. Commun. 98, 288 (1996)
[3]Fan, E. G.: Comput. phys. Commun., Comput. phys. Commun. 153, 17 (2003)
[4]Li, Z.; Liu, Y.: Comput. phys. Commun., Comput. phys. Commun. 148, 256 (2002)
[5]Liu, Y.; Li, Z.: Comput. phys. Commun., Comput. phys. Commun. 155, 65 (2003)
[6]Baldwin, D.; Göktas, Ü.; Hereman, W.: Comput. phys. Commun., Comput. phys. Commun. 162, 203 (2004)
[7]Li, Zhi-Bin; Liu, Yin-Ping: Comput. phys. Commun., Comput. phys. Commun. 163, 191 (2004)
[8]Liu, S. K.; Fu, Z. T.; Liu, S. D.; Zhao, Q.: Phys. lett. A, Phys. lett. A 290, 72 (2001)
[9]Fan, E. G.: Chaos solitons fractals, Chaos solitons fractals 16, 819 (2003)
[10]Liu, Cheng-Shi: Acta phys. Sinica, Acta phys. Sinica 54, 2505 (2005)
[11]Liu, Cheng-Shi: Acta phys. Sinica, Acta phys. Sinica 54, 4506 (2005)
[12]Liu, Cheng-Shi: Comm. theoret. Phys., Comm. theoret. Phys. 44, 799 (2005)
[13]Liu, Cheng-Shi: Comm. theoret. Phys., Comm. theoret. Phys. 46, 395 (2006)
[14]Yang, L.; Hou, X. R.; Zeng, Z. B.: Sci. China ser. E, Sci. China ser. E 39, 628 (1996)
[15]Liu, Cheng-Shi; Du, X. H.: Acta phys. Sinica, Acta phys. Sinica 54, 1709 (2005)
[16]Liu, Cheng-Shi: Chin. phys. Lett., Chin. phys. Lett. 21, 1369 (2004)
[17]Liu, Cheng-Shi: Comm. theoret. Phys., Comm. theoret. Phys. 43, 787 (2005)
[18]Liu, Cheng-Shi: Comm. theoret. Phys., Comm. theoret. Phys. 46, 991 (2006)
[19]Liu, Cheng-Shi: Chin. phys., Chin. phys. 16, 1832 (2007)
[20]Du, Xing-Hua; Liu, Cheng-Shi: Comm. theoret. Phys., Comm. theoret. Phys. 46, 787 (2006)
[21]Liu, Cheng-Shi: Comm. theoret. Phys., Comm. theoret. Phys. 45, 219 (2006)
[22]Liu, Cheng-Shi: Chin. phys., Chin. phys. 14, 1710 (2005)
[23]Liu, Cheng-Shi: Direct integral method, complete discrimination system for polynomial and applications to classifications of all single traveling wave solutions to nonlinear differential equations: A survey
[24]Wang, D. S.; Li, H. B.: J. math. Anal. appl., J. math. Anal. appl. 343, 273-298 (2008)
[25]Ablowitz, M. J.; Clarkson, P. A.: Solitons, nonlinear evolutions and inverse scattering, (1991) · Zbl 0762.35001
[26]Dodd, R. K.: Solitons and nonlinear wave equations, (1982)
[27]Dodd, R. K.; Bullough, R. K.: Proc. roy. Soc. A, Proc. roy. Soc. A 351, 459 (1976)
[28]Conte, R.; Musette, M.: J. phys. A, J. phys. A 150, 14 (1990)
[29]Panigrahi, M.; Dash, P. C.: Phys. lett. A, Phys. lett. A 321, 330 (2004)
[30]Fan, E. G.: Phys. lett. A, Phys. lett. A 305, 383 (2002)
[31]Peng, Y. Z.: Phys. lett. A, Phys. lett. A 314, 402 (2003)
[32]Sirendaoreji; Sun, J.: Phys. lett. A, Phys. lett. A 298, 133 (2002)
[33]Chou, K. W.: Wave motion, Wave motion 35, 71 (2002)
[34]Fu, Z. T.; Yao, Z. H.; Liu, S. K.; Liu, S. D.: Comm. theoret. Phys., Comm. theoret. Phys. 44, 23 (2005)
[35]Zhang, J.: Int. J. Theor. phys., Int. J. Theor. phys. 37, 1541 (1998)
[36]Chen, H. T.: Prog. theor. Phys., Prog. theor. Phys. 109, 509 (2003)
[37]Ghosh, S.; Kundu, A.; Nandy, S.: J. math. Phys., J. math. Phys. 40, 1993 (1999)
[38]Konopelchenko, B.; Dubrovsky, V.: Phys. lett. A, Phys. lett. A 102, 15 (1984)
[39]Seyler, C. E.: Phys. fluids, Phys. fluids 27, 4 (1984)
[40]Lamb, G. L.: Rev. mod. Phys., Rev. mod. Phys. 43, 99 (1971)
[41]Zhou, Feng-Wu; Liu, Zhong-Zhu; Zhou, Huai-Chun: Table of integrals, (1992)
[42]Gradsbteyn, L. S.; Ryzbik, L. M.: Table of integrals, series and products, (2004)
[43]Bullough, R. K.; Caudrey, P. J.; Gibbs, H. M.: B.r.kulloughp.j.caudreysolitons, Solitons, 107-143 (1980)
[44]Yang, J. S.; Lou, S. Y.: Z. naturforsch. A, Z. naturforsch. A 54, 195 (1999)
[45]Yang, J. S.; Lou, S. Y.: Chin. phys. Lett., Chin. phys. Lett. 21, 608 (2004)
[46]Lakshmanan, M.; Kalianppan, P.: J. math. Phys., J. math. Phys. 24, 795 (1983)
[47]Li, B. A.; Wang, M. L.: Chin. phys., Chin. phys. 14, 1698 (2005)
[48]Fu, Z. T.; Liu, S. K.; Liu, S. D.: Acta phys. Sinica, Acta phys. Sinica 53, 43 (2002)
[49]Hirota, H.: J. math. Phys., J. math. Phys. 27, 1499 (1986)
[50]Calogero, F.; Degasperis, A.: J. math. Phys., J. math. Phys. 22, 23 (1981)
[51]Focas, A. S.; Fuchssteiner, B.: Physica D, Physica D 4, 47 (1981)