Skip to content
Accessible Unlicensed Requires Authentication Published online by De Gruyter October 23, 2021

Extended homogeneous balance conditions in the sub-equation method

Chenwei Song and Yinping Liu


The sub-equation method is a kind of straightforward algebraic method to construct exact solutions of nonlinear evolution equations. In this paper, the sub-equation method is improved by proposing some extended homogeneous balance conditions. By applying them to several examples, it can be seen that new solutions could indeed be obtained.

MSC 2010: 35G50

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11871328

Funding source: Science and Technology Commission of Shanghai Municipality

Award Identifier / Grant number: 18511103105

Award Identifier / Grant number: 18dz2271000

Funding source: Natural Science Foundation of Shanghai

Award Identifier / Grant number: 19ZR1414000

Funding statement: This work is supported by the National Natural Science Foundation of China (No. 11871328), Key project of Shanghai Municipal Science and Technology Commission (No. 18511103105) and Shanghai Natural Science Foundation (No. 19ZR1414000). It is supported in part by Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).


[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser. 149, Cambridge University, Cambridge, 1991. 10.1017/CBO9780511623998Search in Google Scholar

[2] M. M. A. El-Sheikh, H. M. Ahmed, A. H. Arnous and W. B. Rabie, Optical solitons and other solutions in birefringent fibers with Biswas–Arshed equation by Jacobi’s elliptic function approach, Optik 202 (2020), Article ID 163546. 10.1016/j.ijleo.2019.163546Search in Google Scholar

[3] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000), no. 4–5, 212–218. 10.1016/S0375-9601(00)00725-8Search in Google Scholar

[4] K. A. Gepreel, T. A. Nofal and A. A. Al-Asmari, Abundant travelling wave solutions for nonlinear Kawahara partial differential equation using extended trial equation method, Int. J. Comput. Math. 96 (2019), no. 7, 1357–1376. 10.1080/00207160.2018.1487555Search in Google Scholar

[5] K. Hosseini, M. Inc, M. Shafiee, M. Ilie, A. Shafaroody, A. Yusuf and M. Bayram, Invariant subspaces, exact solutions and stability analysis of nonlinear water wave equations, J. Ocean Eng. Sci. 5 (2020), 35–40. 10.1016/j.joes.2019.07.004Search in Google Scholar

[6] X. B. Hu and H. W. Tam, New integrable differential-difference systems: Lax pairs, bilinear forms and soliton solutions, Inverse Problems 17 (2001), no. 2, 319–327. 10.1088/0266-5611/17/2/311Search in Google Scholar

[7] I. A. Kunin, Elastic Media with Microstructure. I: One-Dimensional Models, Springer Ser. Solid-State Sci. 26, Springer, Berlin, 1982. 10.1007/978-3-642-81748-9Search in Google Scholar

[8] Z. B. Li and Y. P. Liu, RATH: A Maple package for finding travelling solitary wave solutions to nonlinear evolution equations, Comput. Phys. Comm. 148 (2002), no. 2, 256–266. 10.1016/S0010-4655(02)00559-3Search in Google Scholar

[9] X. Liu and C. Liu, The relationship among the solutions of two auxiliary ordinary differential equations, Chaos Solitons Fractals 39 (2009), no. 4, 1915–1919. 10.1016/j.chaos.2007.06.093Search in Google Scholar

[10] Z. Y. Long, L. Y. Ping and L. Z. Bin, A connection between the (G/G)-expansion method and the truncated Painlevé expansion method and its application to the mKdV equation, Chinese Phys. B 19 (2010), no. 3, Article ID 030306. Search in Google Scholar

[11] W. X. Ma, Y. Zhang, Y. Tang and J. Tu, Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput. 218 (2012), no. 13, 7174–7183. 10.1016/j.amc.2011.12.085Search in Google Scholar

[12] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Amer. J. Phys. 60 (1992), no. 7, 650–654. 10.1119/1.17120Search in Google Scholar

[13] A. P. Márquez and M. S. Bruzón, Travelling wave solutions of a one-dimensional viscoelasticity model, Int. J. Comput. Math. 97 (2020), no. 1–2, 30–39. 10.1080/00207160.2019.1634262Search in Google Scholar

[14] V. B. Matveev and V. B. Matveev, Darboux Transformations and Solitons, Springer, Berlin, 1991. 10.1007/978-3-662-00922-2Search in Google Scholar

[15] A. V. Mikhailov, The reduction problem and the inverse scattering method, Phys. D 3 (1981), no. 1–2, 73–117. 10.1016/0167-2789(81)90120-2Search in Google Scholar

[16] R. C. Mittal and S. Pandit, Sensitivity analysis of shock wave Burgers’ equation via a novel algorithm based on scale-3 Haar wavelets, Int. J. Comput. Math. 95 (2018), no. 3, 601–625. 10.1080/00207160.2017.1293820Search in Google Scholar

[17] Y. Z. Peng, Exact solutions for some nonlinear partial differential equations, Phys. Lett. A 314 (2003), no. 5–6, 401–408. 10.1016/S0375-9601(03)00909-5Search in Google Scholar

[18] S. Sahoo and S. Saha Ray, Solitary wave solutions for time fractional third order modified KdV equation using two reliable techniques (G/G)-expansion method and improved (G/G)-expansion method, Phys. A 448 (2016), 265–282. 10.1016/j.physa.2015.12.072Search in Google Scholar

[19] M. Wadati, K. Konno and Y. H. Ichikawa, A generalization of inverse scattering method, J. Phys. Soc. Japan 46 (1979), no. 6, 1965–1966. 10.1143/JPSJ.46.1965Search in Google Scholar

[20] A. M. Wazwaz, The Hirota’s bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada–Kotera–Kadomtsev–Petviashvili equation, Appl. Math. Comput. 200 (2008), no. 1, 160–166. 10.1016/j.amc.2007.11.001Search in Google Scholar

[21] G. Q. Xu, New types of exact solutions for the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation, Appl. Math. Comput. 217 (2011), no. 12, 5967–5971. 10.1016/j.amc.2010.12.008Search in Google Scholar

[22] R. X. Yao, W. Wang and T. H. Chen, New solutions of three nonlinear space- and time-fractional partial differential equations in mathematical physics, Commun. Theor. Phys. (Beijing) 62 (2014), no. 5, 689–696. 10.1088/0253-6102/62/5/10Search in Google Scholar

[23] S. Zhang, New exact solutions of the KdV–Burgers–Kuramoto equation, Phys. Lett. A 358 (2006), no. 5–6, 414–420. 10.1016/j.physleta.2006.05.071Search in Google Scholar

Received: 2020-02-28
Revised: 2020-11-23
Accepted: 2020-11-23
Published Online: 2021-10-23

© 2021 Walter de Gruyter GmbH, Berlin/Boston