Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T04:11:47.526Z Has data issue: false hasContentIssue false
  • English
  • Français

New Non-Travelling Wave Solutions of Calogero Equation

Part of: Representations of solutions Equations of mathematical physics and other areas of application Algorithms - Computer Science

Published online by Cambridge University Press:  19 September 2016

Xiaoming Peng ,
Yadong Shang  and
Xiaoxiao Zheng
Xiaoming Peng*
Affiliation:
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
Yadong Shang*
Affiliation:
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China
Xiaoxiao Zheng*
Affiliation:
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
*
*Corresponding author. Email:pengxm1987@163.com (X. M. Peng), gzydshang@126.com (Y. D. Shang), xiaoxiaozheng87@163.com (X. X. Zheng)
*Corresponding author. Email:pengxm1987@163.com (X. M. Peng), gzydshang@126.com (Y. D. Shang), xiaoxiaozheng87@163.com (X. X. Zheng)
*Corresponding author. Email:pengxm1987@163.com (X. M. Peng), gzydshang@126.com (Y. D. Shang), xiaoxiaozheng87@163.com (X. X. Zheng)
Get access

Abstract

In this paper, the idea of a combination of variable separation approach and the extended homoclinic test approach is proposed to seek non-travelling wave solutions of Calogero equation. The equation is reduced to some (1+1)-dimensional nonlinear equations by applying the variable separation approach and solves reduced equations with the extended homoclinic test technique. Based on this idea and with the aid of symbolic computation, some new explicit solutions can be obtained.

Keywords

Variable separation approachextended homoclinic test approachnon-travelling wave solution

MSC classification

Secondary: 35C09: Trigonometric solutions 35Q53: KdV-like equations (Korteweg-de Vries) 68W30: Symbolic computation and algebraic computation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 6 , December 2016 , pp. 1036 - 1049
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Zhdanov, R., Separation of variables in the nonlinear wave equation, J. Phys. A, 27 (1994), pp. L291–L297.CrossRefGoogle Scholar
[2] Lou, S. and Chen, L., Formal variable separation approach for nonintegrable models, J. Math. Phys., 40 (1999), pp. 6491–6500.Google Scholar
[3] Estevez, P. G. and Qu, C., Separation of variables in a nonlinear wave equation with a variable wave speed, Theoret. Math. Phys., 133 (2002), pp. 1490–1497.Google Scholar
[4] Zhang, S., Lou, S. and Qu, C., Variable separation and exact solutions to generalized nonlinear diffusion equations, Chin. Phys. Lett., 19 (2002), pp. 1741–1744.Google Scholar
[5] Zhang, S., Lou, S. and Qu, C., New variable separation approach: application to nonlinear diffusion equations, J. Phys. A, 36 (2003), pp. 12223–12242.Google Scholar
[6] Zhang, S. and Lou, S., Derivative-dependent functional separable solutions for the KdV-type equations, Phys. A, 335 (2004), pp. 430–444.CrossRefGoogle Scholar
[7] Lou, S. and Lu, J., Special solutions from the variable separation approach: the Davey-Stewartson equation, J. Phys. A, 29 (1996), pp. 4209–4215.Google Scholar
[8] Ying, J. and Lou, S., Multilinear variable separation approach in (3+1)-dimensions: the Burgers equation, Chin. phys. Lett., 20 (2003), pp. 1448.Google Scholar
[9] Tang, X. and Lou, S., A variable separation approach to solve the integrable and nonintegrable models: coherent structures of the (2+1)-dimensional KdV equation, Commun. Theor. Phys., 38 (2002), pp. 1–8.Google Scholar
[10] Calogero, F. and Degasperis, A., Nonlinear evolution equations solvable by the inverse spectral transform I, Nuovo Cimento B, (11)(32) (1976), pp. 201–242.CrossRefGoogle Scholar
[11] Calogero, F. and Degasperis, A., Nonlinear evolution equations solvable by the inverse spectral transform II, Nuovo Cimento B, (11)(39) (1977), pp. 1–54.Google Scholar
[12] Bogoyavlenskii, O., Overturning solitons in new two-dimensional integrable equations, Math. USSR-Izv., 34 (1990), pp. 245–259.Google Scholar
[13] Bogoyavlenskii, O., Breaking solitons III, Math. USSR-Izv., 36 (1991), pp. 129–137.CrossRefGoogle Scholar
[14] Schiff, J., Painlevé Transendent, Their Asymptotics and Physical Applications, Plenum, New York, 1992, pp. 393.Google Scholar
[15] Bekir, A., Painlevé test for some (2+1)-dimensional nonlinear equations, Chaos Solitons Fractals, 32 (2007), pp. 449–455.Google Scholar
[16] Fan, E. and Chow, K., Darboux covariant Lax pairs and infinite conservation laws of the (2+1)-dimensional breaking soliton equation, J. Math. Phys., 52 (2011), pp. 1–10.Google Scholar
[17] Alagesan, T., Chung, Y. and Nakkeeran, K., Painlevé test for the certain (2+1)-dimensional nonlinear evolution equations, Chaos Solitons Fractals, 26 (2005), pp. 1203–1209.Google Scholar
[18] Yong, X., Zhang, Z. and Chen, Y., Bäcklund transformation, nonlinear superposition formula and solutions of the Calogero equation, Phys. Lett. A, 372 (2008), pp. 6273–6279.Google Scholar
[19] Gao, Y. and Tian, B., New family of overturning soliton solutions for a typical breaking soliton equation, Comput. Math. Appl., 30 (1995), pp. 97–100.Google Scholar
[20] Yan, Z. and Zhang, H., Constructing families of soliton-like solutions to a (2+1)-dimensional breaking soliton equation using symbolic computation, Comput. Math. Appl., 44 (2002), pp. 1439–1444.Google Scholar
[21] Xian, D., Symmetry reduction and new non-traveling wave solutions of (2+1)-dimensional breaking soliton equation, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), pp. 2061–2065.Google Scholar
[22] Tian, B., Zhao, K. and Gao, Y., Symbolic computation in engineering: application to a breaking soliton equation, Internat. J. Eng. Sci., 35 (1997), pp. 1081–1083.CrossRefGoogle Scholar
[23] Fan, E. and Zhang, H., A note on the homogeneous balance method, Phys. Lett. A, 246 (1998), pp. 403–406.Google Scholar
[24] Lü, Z., Duan, L. and Xie, F., Cross soliton-like waves for the (2+1)-dimensional breaking soliton equation, Chin. Phys. Lett., 27 (2010), pp. 1–3.Google Scholar
[25] Feng, Q. and Zheng, B., Exact traveling wave solution for the (2+1) dimensional breaking Soliton equation, Proceedings of the 2010 American Conference on Applied Mathematics, WSEAS, 2010, pp. 440–442.Google Scholar
[26] Zhao, Z., Dai, Z. and Han, S., The EHTA for nonlinear evolution equations, Appl. Math. Comput., 217 (2010), pp. 4306–4310.Google Scholar
[27] Hirota, R. and Satsuma, J., N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Japan, 40 (1976), pp. 611–612.Google Scholar
[28] Clarkson, P. and Mansfield, E. L., On a shallow waterwave equation, Nonlinearity, 7 (1997), pp. 975–1000.Google Scholar