×

Exact solutions of some nonlinear systems of partial differential equations by using the first integral method. (English) Zbl 1254.35044

Summary: In recent years, many approaches were utilized for finding exact solutions of nonlinear systems of partial differential equations. In this paper, the first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including, KdV, Kaup-Boussinesq and Wu-Zhang systems, analytically. By means of this method, some exact solutions for these systems of equations are formally obtained. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.

MSC:

35C05 Solutions to PDEs in closed form
35A25 Other special methods applied to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Wazwaz, A. M., New solitary wave and periodic wave solutions to the \((2 + 1)\)-dimensional Nizhnik-Novikov-Veselov system, Appl. Math. Comput., 187, 1584-1591 (2007) · Zbl 1114.65354
[2] Abdou, M. A., The extended tanh method and its applications for solving nonlinear physical models, Appl. Math. Comput., 190, 988-996 (2007) · Zbl 1123.65103
[3] Bekir, A., Applications of the extended tanh method for coupled nonlinear evolution equations, Commun. Nonlinear Sci. Numer. Simul., 13, 1748-1757 (2008) · Zbl 1221.35322
[4] Shukri, S.; Al-Khaled, K., The extended tanh method for solving systems of nonlinear wave equations, Appl. Math. Comput., 217, 5, 1997-2006 (2010) · Zbl 1202.65131
[5] Soliman, A. A., The modified extended tanh-function method for solving burgers-type equations, Phys. A, 361, 394-404 (2006)
[6] Abdou, M. A.; Soliman, A. A., Modified extended tanh-function method and its application on nonlinear physical equations, Phys. Lett. A, 353, 487-492 (2006)
[7] Wazwaz, A. M., Single and multiple-soliton and solutions for the \((2 + 1)\)-dimensional KdV equation, Appl. Math. Comput., 204, 20-26 (2008) · Zbl 1160.35531
[8] Wazwaz, A. M.; Mehanna, M. S., A variety of exact travelling wave solutions for the \((2 + 1)\)-dimensional Boiti-Leon-Pempinelli equation, Appl. Math. Comput., 217, 4, 1484-1490 (2010) · Zbl 1203.35247
[9] Zhang, S.; Zhang, H. Q., An exp-function method for new \(N\)-soliton solutions with arbitrary functions of a \((2 + 1)\)-dimensional vcBK system, Comput. Math. Appl., 61, 8, 1923-1930 (2011) · Zbl 1219.35274
[10] Deng, X.; Cao, J.; Li, X., Travelling wave solutions for the nonlinear dispersion Drinfelʼd-Sokolov \((D(m, n))\) system, Commun. Nonlinear Sci. Numer. Simul., 15, 281-290 (2010) · Zbl 1221.35329
[11] Lu, B.; Zhang, H. Q.; Xie, F. D., Travelling wave solutions of nonlinear partial equations by using the first integral method, Appl. Math. Comput., 216, 1329-1336 (2010) · Zbl 1191.35090
[12] Feng, Z., On explicit exact solutions to the compound Burgers-KdV equation, Phys. Lett. A, 293, 57-66 (2002) · Zbl 0984.35138
[13] Feng, Z., Exact solution to an approximate sine-Gordon equation in \((n + 1)\)-dimensional space, Phys. Lett. A, 302, 64-76 (2002) · Zbl 0998.35046
[14] Feng, Z.; Wang, X., The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation, Phys. Lett. A, 308, 173-178 (2003) · Zbl 1008.35062
[15] Feng, Z.; Chenb, G., Solitary wave solutions of the compound Burgers-Korteweg-de Vries equation, Phys. A, 352, 419-435 (2005)
[16] Feng, Z.; Li, Y., Complex traveling wave solutions to the Fisher equation, Phys. A, 366, 115-123 (2006)
[17] Feng, Z., Traveling wave behavior for a generalized Fisher equation, Chaos Solitons Fractals, 38, 481-488 (2008) · Zbl 1146.35380
[18] Tascan, F.; Bekir, A., Travelling wave solutions of the Cahn-Allen equation by using first integral method, Appl. Math. Comput., 207, 279-282 (2009) · Zbl 1162.35304
[19] Tascan, F.; Bekir, A.; Koparan, M., Travelling wave solutions of nonlinear evolution equations by using the first integral method, Commun. Nonlinear Sci. Numer. Simul., 14, 1810-1815 (2009)
[20] Raslan, K. R., The first integral method for solving some important nonlinear partial differential equations, Nonlinear Dynam., 53, 4, 281-286 (2008) · Zbl 1176.35149
[21] Abbasbandy, S.; Shirzadi, A., The first integral method for modified Benjamin-Bona-Mahony equation, Commun. Nonlinear Sci. Numer. Simul., 15, 1759-1764 (2010) · Zbl 1222.35166
[22] Taghizadeh, N.; Mirzazadeh, M.; Farahrooz, F., Exact solutions of the nonlinear Schrödinger equation by the first integral method, J. Math. Anal. Appl., 374, 549-553 (2011) · Zbl 1202.35308
[23] Bourbaki, N., Commutative Algebra (1972), Addison-Wesley: Addison-Wesley Paris
[24] Zhou, J.; Tian, L.; Fan, X., Solitary-wave solutions to a dual equation of the Kaup-Boussinesq system, Nonlinear Anal., 11, 3229-3235 (2010) · Zbl 1196.35197
[25] Zheng, X.; Chen, Y.; Zhang, H., Generalized extended tanh-function method and its application to \((1 + 1)\)-dimensional dispersive long wave equation, Phys. Lett. A, 311, 145-157 (2003) · Zbl 1019.35059
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.