More solutions of coupled equal width wave equations arising in plasma and fluid dynamics. (English) Zbl 1513.35036

Summary: The goal of this study is to get new analytical solutions for the system of \((1+1)\)-coupled equal width wave equations, which describe physical processes in turbulence and other unstable coupled systems. The system is reduced to an equivalent system of ordinary differential equations using the Lie symmetries. The reduced system after a suitable choice of arbitrary constants is solved by the Sine-Cosine method for getting final solutions. Observations and comparison with reported results [S. Chauhan et al., Int. J. Appl. Comput. Math. 6, No. 6, Paper No. 159, 17 p. (2020; Zbl 1465.35096); Y. Pandir and H. Ulusoy, J. Math. 2013, Article ID 201276, 5 p. (2013; Zbl 1273.35018); M. F. El-Sayed et al., Int. J. Adv. Appl. Math. Mech. 2, No. 1, 19–25 (2014; Zbl 1359.35017) and K. R. Raslan et al., J. Egypt. Math. Soc. 25, No. 3, 350–354 (2017; Zbl 1377.35052)] confirm the novelty of solutions. Finally, trigonometric and traveling wave solutions are obtained. The resulting solutions include elastic multi solitons, multi solitons kink waves, and undular bore types, which differ from reported works. Additionally, conserved vectors are calculated for the first time in this study, revealing that the system is completely integrable.


35B06 Symmetries, invariants, etc. in context of PDEs
22E70 Applications of Lie groups to the sciences; explicit representations
35C07 Traveling wave solutions
Full Text: DOI


[1] Peregrine, DH, Calculations of the development of an undular bore, J. Fluid. Mech., 25, 321-330 (1966)
[2] Morrison, PJ; Meiss, JD; Cary, JR, Scattering of regularized long wave solitary wave, Phys. D, 11, 324-336 (1984) · Zbl 0599.76028
[3] Gardner, LRT; Gardner, A., Solitary waves of the equal width wave equation, J. Comput. Phys., 101, 218-223 (1992) · Zbl 0759.65086
[4] Petviashvili, V.I.: Sov. J. Plasma Phys.3(150) (1977)
[5] Meiss, JD; Horton, W., Fluctuation spectra of a drift wave soliton gas, Phys. Fluids, 25, 1838 (1982) · Zbl 0497.76114
[6] Chauhan, S.; Arora, R.; Chauhan, A., Lie symmetry reductions and wave solutions of coupled equal width wave equation, Int. J. Appl. Comput. Math., 6, 159, 1-17 (2020) · Zbl 1465.35096
[7] Lu, D.; Seadawy, AR; Ali, A., Dispersive traveling wave solutions of the equal width and modified equal width equations via mathematical methods and its applications, Results Phys., 9, 313-320 (2018)
[8] Arora, R.; Chauhan, A., Lie symmetry reductions and solitary wave solutions of modified equal width wave equation, Int. J. Appl. Comput. Math., 4, 122, 1-13 (2018) · Zbl 1402.35020
[9] Munir, M.; Athar, M.; Sarwar, S.; Shatanawi, W., Lie symmetries of generalized equal width wave equations, AIMS Math, 6, 1, 12148-12165 (2021) · Zbl 1508.35130
[10] Mohyud-Din, ST; Yildirim, A.; Berberler, ME; Hosseini, MM, Numerical solution of modified equal width wave Equation, World Appl. Sci. J., 8, 7, 792-798 (2010)
[11] Pandir, Y., Ulusoy, H.: New generalized hyperbolic functions to find new exact solutions of the nonlinear partial differential equations, J. Math., 2013, 201276 (1-6) (2012) · Zbl 1273.35018
[12] EL-Sayed, M.F., Moatimid, G.M., Moussa, M.H.M., El-Shiekh, R.M., Al-Khawlani, M.A.: New exact solutions for coupled equal width wave equation and \((2+1)\)-dimensional Nizhnik-Novikov-Veselov system using modified Kudryashov method, Int. J. Adv. Appl. Math. Mech., 2(1) (2014), 19- 25 · Zbl 1359.35017
[13] Ali, AHA; Soliman, AA; Raslan, KR, Soliton solution for nonlinear partial differential equations by Cosine-function method, Phys. Lett. A, 368, 299-304 (2007) · Zbl 1209.35107
[14] Raslan, KR; El-Danaf, TS; Ali, KK, New exact solution of coupled general equal width wave equation using Sine-Cosine function method, J. Egypt. Math. Soc., 25, 350-354 (2017) · Zbl 1377.35052
[15] Yusufoglu, E.; Bekir, A., Numerical simulation of equal width wave equation, Comput. Math. Appl., 54, 1147-1153 (2007) · Zbl 1141.65389
[16] Banaja, MA; Bakodah, HO, Runge-Kutta integration of the equal width wave equation using the method of lines, Math. Probl. Eng., 274579, 1-10 (2015) · Zbl 1394.65088
[17] Bluman, GW; Cole, JD, Similarity Methods. Differ.Eqs. (1974), New York: Springer-Verlag, New York
[18] Olver, PJ, Appl. Lie Groups Differ. Eqs. (1993), New York: Springer-Verlag, New York
[19] Ovsiannikov, LV, Group Anal. Differ. Eqs. (1982), New York: Academic Press, New York
[20] Kumar, R.; Kumar, M.; Tiwari, AK, Dynamics of some more invariant solutions of (3+1)-Burgers’ system, Int. J. Comput. Meth. Eng. Sci. Mech., 22, 3, 225-234 (2021)
[21] Kumar, R.; Kumar, A., Optimal subalgebra of GKP by using Killing form, conservation law and some more solutions, Int. J. Appl. Comput. Math., 8, 11, 1-22 (2021) · Zbl 1499.76084
[22] Kumar, R.; Kumar, A., Dynamical behavior of similarity solutions of CKOEs with conservation law, Appl. Math. Comput., 422, 1-18 (2022) · Zbl 1510.35189
[23] Kumar, R., Verma, R.S.: Dynamics of invariant solutions of mKDV-ZK arising in a homogeneous magnetised plasma. Nonlinear Dyn. (2022). doi:10.21203/rs.3.rs-1411278/v1
[24] Kumar, R., Verma, R.S., Tiwari, A.K.: On similarity solutions to (2+1)-dispersive long-wave equations, J. Ocean Eng. Sci. 1-18 (2021)
[25] Kumar, R.; Verma, RS, Dynamics of some new solutions to the coupled DSW equations traveling horizontally on the seabed, J. Ocean Eng. Sci. (2022)
[26] Kumar, M.; Kumar, R.; Kumar, A., Some more invariant solutions of (2 + 1)-water waves Int, J. Appl. Comput. Math., 7, 18, 1-17 (2021) · Zbl 1499.35159
[27] Kumar, S.; Kumar, SK; Chauhan, A., Symmetry reductions, generalized solutions and dynamics of wave profiles for the (2+1)-dimensional system of Broer-Kaup-Kupershmidt (BKK) equations, Math. Comput. Simul., 196, 319-335 (2022) · Zbl 07487732
[28] Kumar, M.; Tanwar, DV; Kumar, R., On Lie symmetries and soliton solutions of \((2+1)\)-dimensional Bogoyavlenskii equations, Nonlinear Dyn., 94, 4, 2547-2561 (2018) · Zbl 1448.35447
[29] Ray, SS; Ravi, LK; Sahoo, S., New exact solutions of coupled Boussinesq-Burgers equations by exp-function method, J. Ocean Eng. Sci., 2, 34-46 (2017)
[30] Ray, SS, Lie symmetries, exact solutions and conservation laws of the Oskolkov-Benjamin-Bona-Mahony-Burgers equation, Mod. Phys. Lett. B, 34, 1, 2050012 (2020)
[31] Kumar, S.; Rani, S., Invariance analysis, optimal system, closed-form solutions and dynamical wave structures of a (2+1)-dimensional dissipative long wave system, Phys. Scri., 96, 12 (2021)
[32] Kumar, S.; Rani, S., Lie symmetry analysis, group-invariant solutions and dynamics of solitons to the (2+1)-dimensional Bogoyavlenskii-Schieff equation, Pramana - J. Phys., 95, 51, 1-14 (2021)
[33] Kumar, S., Rani, S.: Study of exact analytical solutions and various wave profiles of a new extended (2+1)-dimensional Boussinesq equation using symmetry analysis. J. Ocean Eng. Sci. (2021). doi:10.1016/j.joes.2021.10.002
[34] Kumar, S.; Dhiman, SK, Lie symmetry analysis, optimal system, exact solutions and dynamics of solitons of a (3+1)-dimensional generalised BKP-Boussinesq equation, Pramana-J. Phys., 96, 31, 1-20 (2022)
[35] Kumar, S.; Rani, S., Symmetries of optimal system, various closed-form solutions, and propagation of different wave profiles for the Boussinesq-Burgers system in ocean waves, Phys. Fluids, 34, 3 (2022)
[36] Rani, S.; Kumar, S.; Kumar, R., Invariance analysis for determining the closed-form solutions, optimal system, and various wave profiles for a (2+1)-dimensional weakly coupled B-Type Kadomtsev-Petviashvili equations, J. Ocean Eng. Sci. (2021)
[37] Yadav, S.; Chauhan, A.; Arora, R., Invariance analysis, optimal system and conservation laws of (2+1)-dimensional non-linear Vakhnenko equation, Pramana J. Phys., 95, 8, 1-13 (2021)
[38] Devi, M.; Yadav, S.; Arora, R., Optimal system, invariance analysis of fourth-Order nonlinear ablowitz-Kaup-Newell-Segur water wave dynamical equation using lie symmetry approach, Appl. Math. Comput., 404, 1-15 (2021) · Zbl 1510.35296
[39] Kumar, S.; Kumar, D.; Kumar, A., Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher-dimensional Fokas equation, Chaos Solit. Fractals., 142 (2021) · Zbl 1496.35152
[40] Kumar, S.; Kumar, D.; Kharbanda, H., Lie symmetry analysis, abundant exact solutions and dynamics of multisolitons to the (2+1)-dimensional KP-BBM equation, Pramana-J. Phys., 95, 33, 1-19 (2021)
[41] Yadav, S.; Arora, R., Lie symmetry analysis, optimal system and invariant solutions of (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles, Eur. Phys. J. Plus, 136, 172, 1-25 (2021)
[42] Ibragimov, NH, A new conservation theorem, J. Math. Anal. Appl., 333, 311-328 (2007) · Zbl 1160.35008
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