Abstract
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.
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).
References
[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/CBO9780511623998Suche 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.2019.163546Suche 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)00725-8Suche 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. 10.1080/00207160.2018.1487555Suche 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.2019.07.004Suche 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. 10.1088/0266-5611/17/2/311Suche in Google Scholar
[7] I. A. Kunin, Elastic Media with Microstructure. I: One-Dimensional Models, Springer Ser. Solid-State Sci. 26, Springer, Berlin, 1982. 10.1007/978-3-642-81748-9Suche 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)00559-3Suche 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.2007.06.093Suche in Google Scholar
[10]
Z. Y. Long, L. Y. Ping and L. Z. Bin,
A connection between the (
[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.2011.12.085Suche in Google Scholar
[12] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Amer. J. Phys. 60 (1992), no. 7, 650–654. 10.1119/1.17120Suche 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. 10.1080/00207160.2019.1634262Suche in Google Scholar
[14] V. B. Matveev and V. B. Matveev, Darboux Transformations and Solitons, Springer, Berlin, 1991. 10.1007/978-3-662-00922-2Suche 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)90120-2Suche 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. 10.1080/00207160.2017.1293820Suche 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)00909-5Suche 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
[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.46.1965Suche 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.2007.11.001Suche 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.2010.12.008Suche 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. 10.1088/0253-6102/62/5/10Suche 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.2006.05.071Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative
- On tangential approximations of the solution set of set-valued inclusions
- Stabilization of the wave equation with a nonlinear delay term in the boundary conditions
- Fixed point to fixed circle and activation function in partial metric space
- A note on the validity of the Schrödinger approximation for the Helmholtz equation
- Certain classes of analytic functions defined by Hurwitz–Lerch zeta function
- A new factor theorem on absolute matrix summability method
- On a solution to a functional equation
- Hydromagnetic effects on non-Newtonian Hiemenz flow
- Stability of a class of entropies based on fractional calculus
- Asymptotic behavior of solution of Whitham–Broer–Kaup type equations with negative dispersion
- Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts
- Weak solutions to the time-fractional g-Navier–Stokes equations and optimal control
- On the asymptotic formulas for perturbations in the eigenvalues of the Stokes equations due to the presence of small deformable inclusions
- Extended homogeneous balance conditions in the sub-equation method
Artikel in diesem Heft
- Frontmatter
- L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative
- On tangential approximations of the solution set of set-valued inclusions
- Stabilization of the wave equation with a nonlinear delay term in the boundary conditions
- Fixed point to fixed circle and activation function in partial metric space
- A note on the validity of the Schrödinger approximation for the Helmholtz equation
- Certain classes of analytic functions defined by Hurwitz–Lerch zeta function
- A new factor theorem on absolute matrix summability method
- On a solution to a functional equation
- Hydromagnetic effects on non-Newtonian Hiemenz flow
- Stability of a class of entropies based on fractional calculus
- Asymptotic behavior of solution of Whitham–Broer–Kaup type equations with negative dispersion
- Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts
- Weak solutions to the time-fractional g-Navier–Stokes equations and optimal control
- On the asymptotic formulas for perturbations in the eigenvalues of the Stokes equations due to the presence of small deformable inclusions
- Extended homogeneous balance conditions in the sub-equation method