×

A Maple package for finding interaction solutions of nonlinear evolution equations. (English) Zbl 1368.35004

Summary: Based on Wu’s elimination method, an algorithm about the consistent Riccati expansion (CRE) method is presented to find different types of interaction wave solutions for nonlinear partial differential equations. Furthermore, a Maple package is developed to entirely implement the algorithm, and several examples are given to illustrate the effectiveness of the package.

MSC:

35-04 Software, source code, etc. for problems pertaining to partial differential equations
35Q53 KdV equations (Korteweg-de Vries equations)
68W30 Symbolic computation and algebraic computation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Li, Z. B.; Liu, Y. P., RATH: a maple package for finding travelling solitary wave solutions to nonlinear evolution equations, Comput. Phys. Comm., 148, 256-266 (2002) · Zbl 1196.35008
[2] Parkes, E. J.; Duffy, B. R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Comm., 98, 288-300 (1996) · Zbl 0948.76595
[3] Fan, E. G., Extended tanh-function method and its application to nonlinear equations, Phys. Lett. A, 277, 212-218 (2000) · Zbl 1167.35331
[4] Yao, R. X.; Li, Z. B., New exact solutions for three nonlinear evolution equations, Phys. Lett. A, 297, 196-204 (2002) · Zbl 0995.35003
[5] Yan, Z. Y., New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. Lett. A, 292, 100-106 (2001) · Zbl 1092.35524
[6] Jia, M.; Lou, S. Y., New deformation relations and exact solutions of the high-dimensional \(\phi^6\) field model, Phys. Lett. A, 353, 407-415 (2006) · Zbl 1181.81087
[7] Jia, M.; Lou, S. Y., New types of exact solutions for \((N + 1)\)-dimensional \(\phi^4\) model, Commun. Theor. Phys., 46, 91-96 (2006)
[8] Adem, A. R.; Lü, X., Travelling wave solutions of a two-dimensional generalized Sawada-Kotera equation, Nonlinear Dynam., 84, 915-922 (2016) · Zbl 1354.35009
[9] Huang, F.; Tang, X. Y.; Lou, S. Y., Exact solutions for a higher-order nonlinear Schrödinger equation in atmospheric dynamics, Commun. Theor. Phys., 45, 573-576 (2006)
[10] Chen, Y.; Yan, Z. Y., The weierstrass elliptic function expansion method and its applications in nonlinear wave equations, Chaos Solitons Fractals, 29, 4, 393-398 (2006)
[11] Parkes, E. J.; Duffy, B. R.; Abbott, P. C., The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations, Phys. Lett. A, 295, 280-286 (2002) · Zbl 1052.35143
[12] Fan, E. G.; Zhang, J., Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A, 305, 383-392 (2002) · Zbl 1005.35063
[13] Fan, E. G., Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos Solitons Fractals, 16, 819-839 (2003) · Zbl 1030.35136
[14] Li, Z. B.; Liu, Y. P., RAEEM: a maple package for finding a series of exact traveling wave solutions for nonlinear evolution equation, Comput. Phys. Comm., 163, 191-201 (2004) · Zbl 1196.35009
[15] 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
[16] Adem, A. R.; Muatjetjeja, B., Conservation laws and exact solutions for a 2D Zakharov-Kuznetsov equation, Appl. Math. Lett., 48, 109-117 (2015) · Zbl 1326.35014
[17] Ma, W. X.; Huang, T. W.; Zhang, Y., A multiple exp-function method for nonlinear differential equations and its application, Phys. Scr., 82, Article 065003 pp. (2010), (8pp) · Zbl 1219.35209
[18] Ma, W. X.; Zhu, Z. N., Solving the \((3 + 1)\)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Appl. Math. Comput., 218, 11871-11879 (2012) · Zbl 1280.35122
[19] Adem, A. R., A \((2 + 1)\)-dimensional Korteweg-de Vries type equation in water waves: Lie symmetry analysis; multiple exp-function method; conservation laws, Internat. J. Modern Phys. B, 1640001 (2016) · Zbl 1351.35166
[20] Adem, A. R., The generalized \((1 + 1)\)-dimensional and \((2 + 1)\)-dimensional Ito equations: multiple exp-function algorithm and multiple wave solutions, Comput. Math. Appl., 71, 1248-1258 (2016) · Zbl 1443.35120
[21] Fan, E. G., Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems, Phys. Lett. A, 300, 243-249 (2002) · Zbl 0997.34007
[22] Zhou, Y. B.; Li, C., Application of modified G’/G-expansion method to traveling wave solutions for Whitham-Broer-Kaup-like equations, Commun. Theor. Phys., 51, 4, 664-670 (2009) · Zbl 1181.35223
[23] Chen, C. L.; Tang, X. Y.; Lou, S. Y., Solutions of a \((2 + 1)\)-dimensional dispersive long wave equation, Phys. Rev. E, 66, Article 036605 pp. (2002)
[24] Lou, S. Y., Consistent Riccati expansion for integrable systems, Stud. Appl. Math., 134, 372-402 (2015) · Zbl 1314.35145
[25] Chen, C. L.; Lou, S. Y., CTE solvability and exact solution to the Broer-Kaup system, Chinese Phys. Lett., 30, 11, Article 110202 pp. (2013)
[26] Jin, Y.; Jia, M.; Lou, S. Y., Bäcklund transformations and interaction solutions of the Burgers equation, Chinese Phys. Lett., 30, 2, Article 020203 pp. (2013)
[27] Lou, S. Y.; Cheng, X. P.; Tang, X. Y., Dressed dark solitons of the defocusing nonlinear Schrödinger equation, Chinese Phys. Lett., 31, Article 070201 pp. (2014)
[28] Wang, Y. H., CTE method to the interaction solutions of Boussinesq-Burgers equations, Appl. Math. Lett., 38, 100-105 (2014) · Zbl 1314.35153
[29] Yang, D.; Lou, S. Y.; Yu, W. F., Interactions between solitons and cnoidal periodic waves of the Boussinesq equation, Commun. Theor. Phys., 60, 387-390 (2013) · Zbl 1277.35300
[30] Jiao, X. L.; Lou, S. Y., CRE method for solving mKdV equation and new interactions between solitons and cnoidal periodic waves, Commun. Theor. Phys., 63, 7-9 (2015) · Zbl 1305.35008
[31] Yu, W. F.; Lou, S. Y.; Yu, J.; Hu, H. W., Interactions between solitons and cnoidal periodic waves of the Gardner equation, Chinese Phys. Lett., 31, 7, Article 070203 pp. (2014)
[32] Yu, W. F.; Lou, S. Y.; Yu, J.; Yang, D., Interactions between solitons and cnoidal periodic waves of the \((2 + 1)\)-dimensional Konopelchenko-Dubrovsky equation, Commun. Theor. Phys., 62, 3, 297-300 (2014) · Zbl 1298.35037
[33] Baldwin, D.; Göktaş, U.; Hereman, W.; Hong, L.; Martino, R. S.; Miller, J. C., Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs, J. Symbolic Comput., 37, 6, 669-705 (2004) · Zbl 1137.35324
[34] Wang, D. M., Elimination Practice: Software Tools and Applications (2004), Imperial College Press: Imperial College Press London · Zbl 1099.13047
[35] Gao, X. S.; Wang, D. M., Mathematics Mechanization and Applications (2000), Academic Press · Zbl 0968.68201
[36] Wang, J. Y.; Tang, X. Y.; Lou, S. Y.; Gao, X. N.; Jia, M., Nanopteron solution of the Korteweg-de Vires equation, Europhys. Lett., 108, 20005 (2014)
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.