The Bäcklund transformations and abundant exact explicit solutions for a general nonintegrable nonlinear convection-diffusion equation. (English) Zbl 1254.35045

Huang, Yong; Shang, Yadong

Abstr. Appl. Anal. 2012, Article ID 489043, 11 p. (2012).

Summary: The Bäcklund transformations and abundant exact explicit solutions for a class of nonlinear wave equations are obtained by the extended homogeneous balance method. These solutions include the solitary wave solution of rational function, the solitary wave solutions, singular solutions, and the periodic wave solutions of triangle function type. In addition to rederiving some known solutions, some entirely new exact solutions are also established. Explicit and exact particular solutions of many well-known nonlinear evolution equations which are of important physical significance, such as Kolmogorov-Petrovskii-Piskunov equation, FitzHugh-Nagumo equation, Burgers-Huxley equation, Chaffee-Infante reaction diffusion equation, Newell-Whitehead equation, Fisher equation, Fisher-Burgers equation, and an isothermal autocatalytic system, are obtained as special cases.

MSC:

35C05	Solutions to PDEs in closed form
35K57	Reaction-diffusion equations
58J72	Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
35C08	Soliton solutions

Keywords:

extended homogeneous balance method; solitary wave solution; periodic wave solutions; Chaffee-Infante reaction diffusion equation; Newell-Whitehead equation; Kolmogorov-Petrovskii-Piskunov equation; FitzHugh-Nagumo equation; Burgers-Huxley equation

References:

[1]	M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0762.35001 · doi:10.1017/CBO9780511623998
[2]	B. Grammaticos, A. Ramani, and J. Hietarinta, “A search for integrable bilinear equations: the Painlevé approach,” Journal of Mathematical Physics, vol. 31, no. 11, pp. 2572-2578, 1990. · Zbl 0729.35133 · doi:10.1063/1.529005
[3]	G. B. Whitham, “Comments on periodic waves and solitons,” IMA Journal of Applied Mathematics, vol. 32, no. 1-3, pp. 353-366, 1984. · Zbl 0565.35090 · doi:10.1093/imamat/32.1-3.353
[4]	G. B. Whitham, “On shocks and solitary waves,” Scripps Institution of Oceanography Reference Series, pp. 91-124, 1991.
[5]	J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522-526, 1983. · Zbl 0531.35069 · doi:10.1063/1.525875
[6]	J. D. Gibbon, A. C. Newell, M. Tabor, and Y. B. Zeng, “Lax pairs, Bäcklund transformations and special solutions for ordinary differential equations,” Nonlinearity, vol. 1, no. 3, pp. 481-490, 1988. · Zbl 0682.34011 · doi:10.1088/0951-7715/1/3/005
[7]	J. Weiss, “Bäcklund transformation and the Painlevé property,” in Partially Integrable Evolution Equations in Physics, R. Conte and N. Boccara, Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
[8]	J. Hietarinta, “Hirota’s bilinear method and partial integrability,” in Partially Integrable Evolution Equations in Physics, R. Conte and N. Boccara, Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. · Zbl 0717.58024
[9]	N. G. Berloff and L. N. Howard, “Solitary and periodic solutions of nonlinear nonintegrable equations,” Studies in Applied Mathematics, vol. 99, no. 1, pp. 1-24, 1997. · Zbl 0880.35105 · doi:10.1111/1467-9590.00054
[10]	M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, no. 1-5, pp. 67-75, 1996. · Zbl 1125.35401 · doi:10.1016/0375-9601(96)00283-6
[11]	M. L. Wang, “Exact solutions for a compound KdV-Burgers equation,” Physics Letters A, vol. 213, no. 5-6, pp. 279-287, 1996. · Zbl 0972.35526 · doi:10.1016/0375-9601(96)00103-X
[12]	M. L. Wang, Y. B. Zhou, and H. Q. Zhang, “A nonlinear transformation of the shallow water wave equations and its application,” Advances in Mathematics, vol. 28, no. 1, pp. 72-75, 1999. · Zbl 1054.35518
[13]	E. G. Fan and H. Q. Zhang, “A new approach to Bäcklund transformations of nonlinear evolution equations,” Applied Mathematics and Mechanics, vol. 19, no. 7, pp. 645-650, 1998. · Zbl 0923.35157 · doi:10.1007/BF02452372
[14]	Y. D. Shang, “Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1286-1297, 2007. · Zbl 1112.76010 · doi:10.1016/j.amc.2006.09.038
[15]	Y. D. Shang, J. Qin, Y. Huang, and W. Yuan, “Abundant exact and explicit solitary wave and periodic wave solutions to the Sharma-Tasso-Olver equation,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 532-538, 2008. · Zbl 1151.65076 · doi:10.1016/j.amc.2008.02.034
[16]	C. Roman and P. Oleksii, “New conditional symmetries and exact solutions of nonlinear reaction-diffusion-convection equations,” Journal of Physics A, vol. 40, no. 33, pp. 10049-10070, 2007. · Zbl 1128.35358 · doi:10.1088/1751-8113/40/33/009
[17]	J. F. Zhang, “Exact and explicit solitary wave solutions to some nonlinear equations,” International Journal of Theoretical Physics, vol. 35, no. 8, pp. 1793-1798, 1996. · Zbl 0862.35110 · doi:10.1007/BF02302272
[18]	A. M. Wazwaz, “Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 639-656, 2005. · Zbl 1078.35109 · doi:10.1016/j.amc.2004.09.081
[19]	R. Cherniha, “New Q symmetries and exact solutions of some reaction-diffusion-convection equations arising in mathematical biology,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 783-799, 2007. · Zbl 1160.35031 · doi:10.1016/j.jmaa.2006.03.026
[20]	J. H. Merkin, R. A. Satnoianu, and S. K. Scott, “Travelling waves in a differential flow reactor with simple autocatalytic kinetics,” Journal of Engineering Mathematics, vol. 33, no. 2, pp. 157-174, 1998. · Zbl 0899.92038 · doi:10.1023/A:1004292023428