The generalized (2 + 1) and (3+1)-dimensional with advanced analytical wave solutions via computational applications

Document Type : Research Paper


1 Mathematics Department, Education for Pure Science Faculty, Basrah University, Basrah, Iraq

2 Department of Basic Science (Mathematics), Deanship of Common First Year, Majmaah University, Riyadh, Saudi Arabia


The analytical solutions for an important generalized Nonlinear evolution equations NLEEs dynamical partial differential equations (DPDEs) that involve independent variables represented by the (2 + 1)-dimensional breaking soliton equation, the (2 + 1)-dimensional Calogero--Bogoyavlenskii--Schiff  (CBS) equation, and the (2 +1)-dimensional Bogoyavlenskii's breaking soliton equation (BE), and some new exact propagating solutions to a generalized (3+1)-dimensional KP equation with variable coefficients are constructed by using a new algorithm of the first integral method (NAFIM) and determined some analytical solutions by appointing special values of the parameters. In addition to that, we showed a new variety and unique travelling wave solutions by graphical illustration with symbolic computations.


[1] M. Abdoon, Programming first integral method general formula for the solving linear and nonlinear equations,
Appl. Math. 6 (2015) 568.
[2] Al-Amr and O. Mohammed, Exact solutions of the generalized (2+1)-dimensional nonlinear evolution equations
via the modified simple equation method, Comput. Math. Appl. 69(5) (2015) 390–397.
[3] M. Al-Amr, Exact solutions of the generalized (2+ 1)-dimensional nonlinear evolution equations via the modified
simple equation method, Comput. Math. Appl. 5 (2015) 390–397.
[4] A. Aleixo and A. Balantekin, Algebraic construction of coherent states for nonlinear quantum deformed systems,
J. Phys. A: Math. Theor. 16 (2012) 165302.
[5] T. Bo. and G. YI-Tian, On the generalized Tanh method the (2+1)-dimensional breaking soliton equations, Mod.
Phys. Lett. A. 38 (1995) 2937–2941.
[6] B. Buti, V. Galinski, V. Shevchenko, G. Lakhina, B. Tsurutani, B. Goldstein, B. Diamond and M. Medvedev,
Evolution of Nonlinear Alfv´en Waves in Streaming Inhomogeneous Plasmas, Astrophysical J. 523 (1999) 849.
[7] O. Bogoyavlenskii, Breaking solitons in 2+ 1-dimensional integrable equations, Russian Math. Surv. 45(4) (1990).
[8] R. Cherniha and S. Kovalenko, Lie symmetries and reductions of multi-dimensional boundary value problems of
the Stefan type, J. Phys. A: Math. Theo. 44 (2011) 485202.
[9] M. Darvishi and M. Najafi, Some exact solutions of the (2+ 1)-dimensional breaking soliton equation using the
three-wave method, Int. J. Comput. Math. Sci. 6 (2012) 13–16.
[10] T. Ding and C. Li, Ordinary differential equations, Theor. Math. Phys. 137 (1996) 1367–1377.
[11] M. Ekici, D. Duran and A. Sonmezoglu, Constructing of exact solutions to the (2+ 1)-dimensional breaking
soliton equations by the multiple (G’/G)-expansion method, J. Adv. Math. Stud. 1 (2014) 27–30.
[12] L. Faeza, First integral method for constructing new exact solutions of the important nonlinear evolution equations
in physics, Phys. Conf. Ser. 2020.
[13] Y. Guldem and D. Durmus, Solution of the (2+1) donation breaking soliton equation by using two different
methods, J. Eng. Tech. App. Sci. 1 (2016) 13–18.
[14] G. Hami and F. Omer, ¨ Benjamin-Bona-Mahony equation by using the sn-ns method and the tanh-coth method,
Math. Moravica, 1 (2017) 95–103.
[15] T. Kobayashi and K. Toda, The Painlev´e test and reducibility to the canonical forms for higher-dimensional
soliton equations with variable-coefficients, Symmet. Integ. Geom. Meth. Appl. 2 (2006) 063.
[16] A. Lisok, A. Trifonov and A. Shapovalov, The evolution operator of the Hartree-type equation with a quadratic
potential, J. Phys. A: Math. Gen. 37 (2004) 4535.
[17] W. Ma, A. Abdeljabbar and M. GamilAsaad, Wronskian and Grammian solutions to a (3+ 1)-dimensional
generalized KP equation, Appl. Math. Comput. 24 (2011) 10016–10023.
[18] W. Ma and A. Abdeljabbar, A bilinear B¨acklund transformation of a (3 + 1)-dimensional generalized KP equation,
Appl. Math. Lett. 25(10) (2012) 1500–1504.
[19] K. Melike, A. Arzu and B. Ahmet, (2016) Solving space-time fractional differential equations by using modified
simple equation method, Commun. Theor. Phys. 5 (2016) 563–568.[20] M. Mirzazadeh, A couple of solutions to a (3+ 1)-dimensional generalized KP equation with variable coefficients
by extended transformed rational function method, Elect. J. Math. Anal. Appl. 1 (2015) 188–194.
[21] A. Mohamed, First integral method: A general formula for nonlinear fractional Klein-Gordon equation using
advanced computing language, Amer. J. Comput. Math. 5 (2015) 127–134.
[22] S. Mohyud-Din, A. Irshad, N. Ahmed and U. Khan, Exact solutions of (3+ 1)-dimensional generalized KP
equation arising in physics, Results Phys. 7 (2017) 3901–3909.
[23] M. Najafi, S. Arbabi and M. Najafi, New application of sine-cosine method for the generalized (2 + 1)-dimensional
nonlinear evolution equations, Int. J. Adv. Math. Sci. 1 (2013) 45–49.
[24] M. Najafi, M. Najafi and S. Arbabi, New exact solutions for the generalized (2+ 1)-dimensional nonlinear evolution
equations by Tanh-Coth method, Int. J. Modern Theo. Phys. 2 (2013) 79–85.
[25] M. Najafi, M. Najafi and S. Arbabi, New application of (G
/G)-expansion method for generalized (2 + 1)-
dimensional nonlinear evolution equations, Int. J. Eng. Math. 5 (2013) Article ID 746910.
[26] V. Orlov and O. Kovalchuk, Research of one class of nonlinear differential equations of third order for mathematical modelling the complex structures, IOP Conf. Ser.: Mater. Sci. Eng. 365 (2018) 042045.
[27] Y. Peng, New types of localized coherent structures in the Bogoyavlenskii-Schiff equation, Int. J. Theor. Phys. 45
(2006) 1779–1783.
[28] A. Pullen, A. Benson and L. Moustakas, Nonlinear evolution of dark matter subhalos and applications to warm
dark matter, Astrophysical J. 792(1) (2014) 24.
[29] A. Seadawy, N. Cheemaa and A. Biswas, Optical dromions and domain walls in (2+ 1)-dimensional coupled
system, Optik, 227 (2020) 165669.
[30] Y. Tang, S. Tao, M. Zhou and Q. Guan, Interaction solutions between lump and other solitons of two classes of
nonlinear evolution equations, Nonlinear Dyn. 89(1) (2017) 429–442 .
[31] M. Tarig and B. Jafar, (2011) Homotopy perturbation method and Elzaki transform for solving system of nonlinear
partial differential equations, Worl. Appl. Sci. J. 7 (2011) 944–948.
[32] A. Wazwaz, Integrable (2+ 1)-dimensional and (3+ 1)-dimensional breaking soliton equations, Physica Scripta,
81 (2010) 035005.
[33] W. X. Ma, Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs, J. Geom. Phys.
133 (2018) 10–16 .
[34] Y. Yıldırım and E. Ya¸sar, A (2+ 1)-dimensional breaking soliton equation: solutions and conservation laws,
Chaos, Solitons Fract. 107 (2018) 146–155.
[35] G. Yildiz and D. Daghan, Solution of the (2+ 1) dimensional breaking Soliton equation by using two different
methods, J. Engin. Tech. Appl. Sci. 1 (2016) 13–18.
[36] V. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP. 5 (1972) 908–914.
[37] Y. Zhou, M. Wang and Y. Wang, Periodic wave solutions to a coupled KdV equations with variable coefficients,
Phys. Lett. A. 1 (2003) 31–36.
Volume 12, Issue 2
November 2021
Pages 1213-1241
  • Receive Date: 18 March 2021
  • Revise Date: 21 April 2021
  • Accept Date: 27 May 2021