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More Solutions of Coupled Equal Width Wave Equations Arising in Plasma and Fluid Dynamics

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Abstract

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 ((Chauhan et al. (2020) Int J Appl Comput Math 6(159):1–17), (Pandir and Ulusoy (2012) J Math 2013(1–6):201276), (EL-Sayed et al. (2014) Int J Adv Appl Math Mech 2(1):19– 25) and Raslan et al (2017) J Egypt Math Soc 25, 350–354) confirm the novelty of solutions. Finally, trigonometric and travelling 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.

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The simulation is to trace the profiles in MATLAB without using any associated data from an authorised agency or laboratory.

References

  1. Peregrine, D.H.: Calculations of the development of an undular bore. J. Fluid. Mech. 25, 321–330 (1966)

    Article  Google Scholar 

  2. Morrison, P.J., Meiss, J.D., Cary, J.R.: Scattering of regularized long wave solitary wave. Phys. D 11, 324–336 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gardner, L.R.T., Gardner, A.: Solitary waves of the equal width wave equation. J. Comput. Phys. 101, 218–223 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Petviashvili, V.I.: Sov. J. Plasma Phys. 3(150) (1977)

  5. Meiss, J.D., Horton, W.: Fluctuation spectra of a drift wave soliton gas. Phys. Fluids 25, 1838 (1982)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  7. Lu, D., Seadawy, A.R., 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)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  9. Munir, M., Athar, M., Sarwar, S., Shatanawi, W.: Lie symmetries of generalized equal width wave equations. AIMS Math 6(1), 12148–12165 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mohyud-Din, S.T., Yildirim, A., Berberler, M.E., Hosseini, M.M.: Numerical solution of modified equal width wave Equation. World Appl. Sci. J. 8(7), 792–798 (2010)

    Google Scholar 

  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)

  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

  13. Ali, A.H.A., Soliman, A.A., Raslan, K.R.: Soliton solution for nonlinear partial differential equations by Cosine-function method. Phys. Lett. A 368, 299–304 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Raslan, K.R., El-Danaf, T.S., Ali, K.K.: New exact solution of coupled general equal width wave equation using Sine-Cosine function method. J. Egypt. Math. Soc. 25, 350–354 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yusufoglu, E., Bekir, A.: Numerical simulation of equal width wave equation. Comput. Math. Appl. 54, 1147–1153 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Banaja, M.A., Bakodah, H.O.: Runge-Kutta integration of the equal width wave equation using the method of lines. Math. Probl. Eng. 274579, 1–10 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bluman, G.W., Cole, J.D.: Similarity Methods. Differ.Eqs. Springer-Verlag, New York (1974)

    Book  Google Scholar 

  18. Olver, P.J.: Appl. Lie Groups Differ. Eqs. Springer-Verlag, New York (1993)

    Google Scholar 

  19. Ovsiannikov, L.V.: Group Anal. Differ. Eqs. Academic Press, New York (1982)

    Google Scholar 

  20. Kumar, R., Kumar, M., Tiwari, A.K.: Dynamics of some more invariant solutions of (3+1)-Burgers’ system. Int. J. Comput. Meth. Eng. Sci. Mech. 22(3), 225–234 (2021)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  22. Kumar, R., Kumar, A.: Dynamical behavior of similarity solutions of CKOEs with conservation law. Appl. Math. Comput. 422, 1–18 (2022)

    MathSciNet  MATH  Google Scholar 

  23. Kumar, R., Verma, R.S.: Dynamics of invariant solutions of mKDV-ZK arising in a homogeneous magnetised plasma. Nonlinear Dyn. (2022). https://doi.org/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, R.S.: Dynamics of some new solutions to the coupled DSW equations traveling horizontally on the seabed. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.04.015

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  27. Kumar, S., Kumar, S.K., 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kumar, M., Tanwar, D.V., Kumar, R.: On Lie symmetries and soliton solutions of \((2+1)\)-dimensional Bogoyavlenskii equations. Nonlinear Dyn. 94(4), 2547–2561 (2018)

    MATH  Google Scholar 

  29. Ray, S.S., Ravi, L.K., Sahoo, S.: New exact solutions of coupled Boussinesq-Burgers equations by exp-function method. J. Ocean Eng. Sci. 2, 34–46 (2017)

    Article  Google Scholar 

  30. Ray, S.S.: Lie symmetries, exact solutions and conservation laws of the Oskolkov-Benjamin-Bona-Mahony-Burgers equation. Mod. Phys. Lett. B 34(1), 2050012 (2020)

    Article  MathSciNet  Google Scholar 

  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), 125202 (2021)

    Article  Google Scholar 

  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)

    Google Scholar 

  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). https://doi.org/10.1016/j.joes.2021.10.002

  34. Kumar, S., Dhiman, S.K.: 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)

    Google Scholar 

  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), 037109 (2022)

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.joes.2021.12.007

    Article  Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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, 110507 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  42. Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333, 311–328 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Faculty of Engineering & Technology, V. B. S. Purvanchal University, Jaunpur, 222003, India

    Raj Kumar & Avneesh Kumar

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  1. Raj Kumar
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Correspondence to Avneesh Kumar.

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Kumar, R., Kumar, A. More Solutions of Coupled Equal Width Wave Equations Arising in Plasma and Fluid Dynamics. Int. J. Appl. Comput. Math 8, 186 (2022). https://doi.org/10.1007/s40819-022-01400-7

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