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Year 2023, Volume: 6 Issue: 3, 87 - 96, 21.12.2023
https://doi.org/10.33187/jmsm.1241918

Abstract

References

  • [1] B. Kopçasız, A.R. Seadawy, E. Yaşar, Highly dispersive optical soliton molecules to dual-mode nonlinear Schrödinger wave equation in cubic law media, Opt. Quantum Electron., 54(3) (2022), 1-21.
  • [2] B. Kopçasız, E. Yaşar, Novel exact solutions and bifurcation analysis to dual-mode nonlinear Schrödinger equation, J. Ocean Eng. Sci., (2022).
  • [3] B. Kopçasız, E. Yaşar, The investigation of unique optical soliton solutions for dual-mode nonlinear Schrödinger’s equation with new mechanisms, J. Opt., (2022),1-15.
  • [4] E. Yaşar, B. Kopçasız, Novel multi-wave solutions for the fractional order dual-mode nonlinear Schrödinger equation, Annals Math. Computat. Sci., 16 (2023), 100-111.
  • [5] B. Kopçasız, E. Yaşar, Analytical soliton solutions of the fractional order dual-mode nonlinear Schrödinger equation with time-space conformable sense by some procedures, Opt. Quantum Electron., 55(7) (2023), 629.
  • [6] B. Kopçasız, E. Yaşar, Dual-mode nonlinear Schrödinger equation (DMNLSE): Lie group analysis, group invariant solutions, and conservation laws, Internat. J. Modern Phys. B, (2023) 2450020, 26 pages.
  • [7] Y. Zhang, Lie symmetry analysis and exact solutions of the Sawada-Kotera equation, Turkish J. Math., 41(1) (2017), 158-167.
  • [8] D. Kaya, G. Iskandarova, Lie group analysis for initial and boundary value problem of time-fractional nonlinear generalized KdV partial differential equation, Turkish J. Math., 43(3) (2019), 1263-1275.
  • [9] E. Yaşar, T. Özer, On symmetries, conservation laws, and invariant solutions of the foam-drainage equation, Int. J. Nonlinear. Mech., 46(2) (2011), 357-362.
  • [10] Ö. Orhan, M. Torrisi, R. Tracina, Group methods applied to a reaction-diffusion system generalizing Proteus Mirabilis models, Commun. Nonlinear Sci. Numer. Simul., 70 (2019), 223-233.
  • [11] S. Kumar, B. Mohan, R. Kumar, Lump, soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics, Nonlinear Dyn., 110(1) (2022), 693-704.
  • [12] E.H. Zahran, A. Bekir, New unexpected soliton solutions to the generalized (2+1) Schrödinger equation with its four-mixing waves, Int. J. Mod. Phys. B, 36(25) (2022), 2250166.
  • [13] T.S. Moretlo, A.R. Adem, B. Muatjetjeja, A generalized (1+2)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili (BKP) equation: Multiple exp-function algorithm; conservation laws; similarity solutions, Commun. Nonlinear Sci. Numer. Simul., 106 (2022), 106072.
  • [14] M. Islam, F.A. Abdullah, J.F. Gomez-Aguilar, A variety of solitons and other wave solutions of a nonlinear Schrödinger model relating to ultra-short pulses in optical fibers, Opt. Quantum Electron., 54(12) (2022), 1-21.
  • [15] M. Khater, M. Inc, K.U. Tariq, F. Tchier, H. Ilyas, D. Baleanu, On some novel optical solitons to the cubic–quintic nonlinear Helmholtz model, Opt. Quantum Electron., 54(12) (2022), 1-13.
  • [16] K. Hosseini, E. Hincal, S. Salahshour, M. Mirzazadeh, K. Dehigia, B.J. Nath, On the dynamics of soliton waves in a generalized nonlinear Schrödinger equation, Optik, 272 (2022), 170215.
  • [17] Y.S. Bai, J.T. Pei, W.X. Ma, l-symmetry and m-symmetry reductions and invariant solutions of four nonlinear differential equations, Mathematics, 8(7) (2020), 1138.
  • [18] K. Goodarzi, Order reduction, m-symmetry and m-conservation law of the generalized mKdV equation with constant-coefficients and variable-coefficients, Int. J. Ind. Math., 14(4) (2022), 433-444.
  • [19] H. Jafari, K. Goodarzi, M. Khorshidi, V. Parvaneh, Z. Hammouch, Lie symmetry and m-symmetry methods for nonlinear generalized Camassa–Holm equation, Adv. Differ. Equ., 2021(1) (2021), 1-12.
  • [20] Ö. Orhan, T. Özer, On m-symmetries, m-reductions, and m-conservation laws of Gardner equation, J. Math. Phys., 26(1) (2019), 69-90.
  • [21] C. Muriel, J.L. Romero, New methods of reduction for ordinary differential equations, IMA J. App. Math., 66(2) (2001), 111-125.
  • [22] G. Cicogna, G. Gaeta, Noether theorem for m-symmetries, J.Phys. A: Math Theor., 40(39) (2007), 11899–11921.
  • [23] G. Cicogna, G. Gaeta, P. Morando, On the relation between standard and m-symmetries for PDEs. J. Phys. A, 37(40) (2004), 9467–9486.
  • [24] G. Gaeta, P. Morando, On the geometry of lambda-symmetries and PDE reduction, J.Phys. A: Math Gen., 37(27) (2004), 6955-6975.
  • [25] P.J. Olver, Application of Lie Groups to Differential Equations, New York, Springer-Verlag, 1986.
  • [26] K. Khan, M.A. Akbar, S.M. Islam, Exact solutions for (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations, SpringerPlus, 3(1) (2014), 1-8.
  • [27] E. Yusufoğlu, New solitonary solutions for the MBBM equations using Exp-function method, Phys. Lett. A, 372(4) (2008), 442-446.
  • [28] E.M.E Zayed, S. Al-Joudi, Applications of an extended (G0=G)-expansion method to find exact solutions of nonlinear PDEs in mathematical physics, Math. Prob Eng., 2010 (2010), 1-19.
  • [29] M. Khorshidi, M. Nadjafikhah, H. Jafari, M. Al Qurashi, Reductions and conservation laws for BBM and modified BBM equations, Open Maths., 14(1) (2016), 1138-1148.

$\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation

Year 2023, Volume: 6 Issue: 3, 87 - 96, 21.12.2023
https://doi.org/10.33187/jmsm.1241918

Abstract

This work discusses the $%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
$-symmetry and conservation law of $%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
$ procedure for the nonlinear dispersive modified Benjamin-Bona-Mahony equation (NDMBBME). This equation models an approximation for surface long waves in nonlinear dispersive media. It can also describe the hydromagnetic waves in a cold plasma, acoustic waves in inharmonic crystals, and acoustic gravity waves in compressible fluids. First and foremost, we offer some essential pieces of information about the $% %TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
$-symmetry and the conservation law of $%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
$ concepts. In light of such information, $%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
$-symmetries are found. Using characteristic equations, the NDMBBME is reduced to ordinary differential equations (ODEs). We obtained the exact invariant solutions by solving the nonlinear ODEs. Furthermore, employing the variational problem procedure, we get the Lagrangian and the $%
%TCIMACRO{\U{3bc} }%
%BeginExpansion
\mu
%EndExpansion
$-conservation laws. The exact solutions and conservation laws are new for the NDMBBME that are not reported by the other studies. We also demonstrate the properties with figures for these solutions.

References

  • [1] B. Kopçasız, A.R. Seadawy, E. Yaşar, Highly dispersive optical soliton molecules to dual-mode nonlinear Schrödinger wave equation in cubic law media, Opt. Quantum Electron., 54(3) (2022), 1-21.
  • [2] B. Kopçasız, E. Yaşar, Novel exact solutions and bifurcation analysis to dual-mode nonlinear Schrödinger equation, J. Ocean Eng. Sci., (2022).
  • [3] B. Kopçasız, E. Yaşar, The investigation of unique optical soliton solutions for dual-mode nonlinear Schrödinger’s equation with new mechanisms, J. Opt., (2022),1-15.
  • [4] E. Yaşar, B. Kopçasız, Novel multi-wave solutions for the fractional order dual-mode nonlinear Schrödinger equation, Annals Math. Computat. Sci., 16 (2023), 100-111.
  • [5] B. Kopçasız, E. Yaşar, Analytical soliton solutions of the fractional order dual-mode nonlinear Schrödinger equation with time-space conformable sense by some procedures, Opt. Quantum Electron., 55(7) (2023), 629.
  • [6] B. Kopçasız, E. Yaşar, Dual-mode nonlinear Schrödinger equation (DMNLSE): Lie group analysis, group invariant solutions, and conservation laws, Internat. J. Modern Phys. B, (2023) 2450020, 26 pages.
  • [7] Y. Zhang, Lie symmetry analysis and exact solutions of the Sawada-Kotera equation, Turkish J. Math., 41(1) (2017), 158-167.
  • [8] D. Kaya, G. Iskandarova, Lie group analysis for initial and boundary value problem of time-fractional nonlinear generalized KdV partial differential equation, Turkish J. Math., 43(3) (2019), 1263-1275.
  • [9] E. Yaşar, T. Özer, On symmetries, conservation laws, and invariant solutions of the foam-drainage equation, Int. J. Nonlinear. Mech., 46(2) (2011), 357-362.
  • [10] Ö. Orhan, M. Torrisi, R. Tracina, Group methods applied to a reaction-diffusion system generalizing Proteus Mirabilis models, Commun. Nonlinear Sci. Numer. Simul., 70 (2019), 223-233.
  • [11] S. Kumar, B. Mohan, R. Kumar, Lump, soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics, Nonlinear Dyn., 110(1) (2022), 693-704.
  • [12] E.H. Zahran, A. Bekir, New unexpected soliton solutions to the generalized (2+1) Schrödinger equation with its four-mixing waves, Int. J. Mod. Phys. B, 36(25) (2022), 2250166.
  • [13] T.S. Moretlo, A.R. Adem, B. Muatjetjeja, A generalized (1+2)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili (BKP) equation: Multiple exp-function algorithm; conservation laws; similarity solutions, Commun. Nonlinear Sci. Numer. Simul., 106 (2022), 106072.
  • [14] M. Islam, F.A. Abdullah, J.F. Gomez-Aguilar, A variety of solitons and other wave solutions of a nonlinear Schrödinger model relating to ultra-short pulses in optical fibers, Opt. Quantum Electron., 54(12) (2022), 1-21.
  • [15] M. Khater, M. Inc, K.U. Tariq, F. Tchier, H. Ilyas, D. Baleanu, On some novel optical solitons to the cubic–quintic nonlinear Helmholtz model, Opt. Quantum Electron., 54(12) (2022), 1-13.
  • [16] K. Hosseini, E. Hincal, S. Salahshour, M. Mirzazadeh, K. Dehigia, B.J. Nath, On the dynamics of soliton waves in a generalized nonlinear Schrödinger equation, Optik, 272 (2022), 170215.
  • [17] Y.S. Bai, J.T. Pei, W.X. Ma, l-symmetry and m-symmetry reductions and invariant solutions of four nonlinear differential equations, Mathematics, 8(7) (2020), 1138.
  • [18] K. Goodarzi, Order reduction, m-symmetry and m-conservation law of the generalized mKdV equation with constant-coefficients and variable-coefficients, Int. J. Ind. Math., 14(4) (2022), 433-444.
  • [19] H. Jafari, K. Goodarzi, M. Khorshidi, V. Parvaneh, Z. Hammouch, Lie symmetry and m-symmetry methods for nonlinear generalized Camassa–Holm equation, Adv. Differ. Equ., 2021(1) (2021), 1-12.
  • [20] Ö. Orhan, T. Özer, On m-symmetries, m-reductions, and m-conservation laws of Gardner equation, J. Math. Phys., 26(1) (2019), 69-90.
  • [21] C. Muriel, J.L. Romero, New methods of reduction for ordinary differential equations, IMA J. App. Math., 66(2) (2001), 111-125.
  • [22] G. Cicogna, G. Gaeta, Noether theorem for m-symmetries, J.Phys. A: Math Theor., 40(39) (2007), 11899–11921.
  • [23] G. Cicogna, G. Gaeta, P. Morando, On the relation between standard and m-symmetries for PDEs. J. Phys. A, 37(40) (2004), 9467–9486.
  • [24] G. Gaeta, P. Morando, On the geometry of lambda-symmetries and PDE reduction, J.Phys. A: Math Gen., 37(27) (2004), 6955-6975.
  • [25] P.J. Olver, Application of Lie Groups to Differential Equations, New York, Springer-Verlag, 1986.
  • [26] K. Khan, M.A. Akbar, S.M. Islam, Exact solutions for (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations, SpringerPlus, 3(1) (2014), 1-8.
  • [27] E. Yusufoğlu, New solitonary solutions for the MBBM equations using Exp-function method, Phys. Lett. A, 372(4) (2008), 442-446.
  • [28] E.M.E Zayed, S. Al-Joudi, Applications of an extended (G0=G)-expansion method to find exact solutions of nonlinear PDEs in mathematical physics, Math. Prob Eng., 2010 (2010), 1-19.
  • [29] M. Khorshidi, M. Nadjafikhah, H. Jafari, M. Al Qurashi, Reductions and conservation laws for BBM and modified BBM equations, Open Maths., 14(1) (2016), 1138-1148.
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Bahadır Kopçasız 0000-0002-6364-3631

Emrullah Yaşar 0000-0003-4732-5753

Early Pub Date July 25, 2023
Publication Date December 21, 2023
Submission Date January 24, 2023
Acceptance Date April 5, 2023
Published in Issue Year 2023 Volume: 6 Issue: 3

Cite

APA Kopçasız, B., & Yaşar, E. (2023). $\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation. Journal of Mathematical Sciences and Modelling, 6(3), 87-96. https://doi.org/10.33187/jmsm.1241918
AMA Kopçasız B, Yaşar E. $\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation. Journal of Mathematical Sciences and Modelling. December 2023;6(3):87-96. doi:10.33187/jmsm.1241918
Chicago Kopçasız, Bahadır, and Emrullah Yaşar. “$\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation”. Journal of Mathematical Sciences and Modelling 6, no. 3 (December 2023): 87-96. https://doi.org/10.33187/jmsm.1241918.
EndNote Kopçasız B, Yaşar E (December 1, 2023) $\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation. Journal of Mathematical Sciences and Modelling 6 3 87–96.
IEEE B. Kopçasız and E. Yaşar, “$\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation”, Journal of Mathematical Sciences and Modelling, vol. 6, no. 3, pp. 87–96, 2023, doi: 10.33187/jmsm.1241918.
ISNAD Kopçasız, Bahadır - Yaşar, Emrullah. “$\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation”. Journal of Mathematical Sciences and Modelling 6/3 (December 2023), 87-96. https://doi.org/10.33187/jmsm.1241918.
JAMA Kopçasız B, Yaşar E. $\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation. Journal of Mathematical Sciences and Modelling. 2023;6:87–96.
MLA Kopçasız, Bahadır and Emrullah Yaşar. “$\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation”. Journal of Mathematical Sciences and Modelling, vol. 6, no. 3, 2023, pp. 87-96, doi:10.33187/jmsm.1241918.
Vancouver Kopçasız B, Yaşar E. $\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation. Journal of Mathematical Sciences and Modelling. 2023;6(3):87-96.

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