American Institute of Mathematical Sciences

October  2020, 13(10): 2789-2802. doi: 10.3934/dcdss.2020169

A study of a generalized first extended (3+1)-dimensional Jimbo-Miwa equation

 1 International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa 2 Department of Mathematics, College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China

* Corresponding author: Chaudry Masood Khalique

Received  January 2019 Published  October 2020 Early access  December 2019

This paper aims to study a generalized first extended (3+1)- dimensional Jimbo-Miwa equation. Symmetry reductions on this equation are performed several times and it is reduced to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is found in terms of the incomplete elliptic integral function. Also exact solutions are constructed using the $({G'}/{G})-$expansion method. Thereafter the conservation laws of the underlying equation are computed by invoking the conservation theorem due to Ibragimov. The conservation laws obtained contain an energy conservation law and three momentum conservation laws.

Citation: Chaudry Masood Khalique, Letlhogonolo Daddy Moleleki. A study of a generalized first extended (3+1)-dimensional Jimbo-Miwa equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2789-2802. doi: 10.3934/dcdss.2020169
References:
 [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511623998. [2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1966 [3] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences, 81. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-4307-4. [4] M. S. Bruzón and T. M. Garrido, Symmetries and conservation laws of a KdV6 equation, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 631-641.  doi: 10.3934/dcdss.2018038. [5] B. T. Cao, Solutions of Jimbo-Miwa equation and Konopelchenko-Dudrovsky equations, Acta Appl. Math., 112 (2010), 181-203.  doi: 10.1007/s10440-009-9559-5. [6] M. T. Darvishi and M. Najafi, Some complexiton type solutions of the $(3+1)$-dimensional Jimbo-Miwa equation, Int. J. Comput. Math. Sci., 6 (2012), 25-27. [7] A. Fatima, F. M. Mahomed and C. M. Khalique, Conditional symmetries of nonlinear third-order ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 655-666.  doi: 10.3934/dcdss.2018040. [8] C. A. Gomez, A. H. Salas and B. Acevedo Frias, Exact solutions to KdV6 equation by using a new approach of the projective Riccati equation method, Math. Prob. Eng., 2010 (2010), 797084, 10 pp. doi: 10.1155/2010/797084. [9] R. Hirota, Exact solution of the KdV equation for multiple collisions of solutions, Phys. Rev. Lett., 27 (1971), 1192-1194. [10] N. H. Ibragimov, R. L. Anderson and et al, CRC Handbook of Lie Group Analysis of Differential Equatios, Vol. 3. New Trends in Theoretical Developments and Computational Methods, CRC Press, Boca Raton, FL, 1996. [11] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley Series in Mathematical Methods in Practice, 4. John Wiley & Sons, Ltd., Chichester, 1999. [12] N. H. Ibragimov, Modern group analysis, Nonlinear Dynam. Springer, Dordrecht, 28 (2002), 103-230. [13] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328.  doi: 10.1016/j.jmaa.2006.10.078. [14] M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. Res. Inst. Math. Sci., 19 (1983), 943-1001.  doi: 10.2977/prims/1195182017. [15] M. T. Kajani and F. T. Kajani, Homotopy analysis method for solving nonlinear system of equations, 13 (2011), 2471–2473. [16] N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Solitons Fract., 24 (2005), 1217-1231.  doi: 10.1016/j.chaos.2004.09.109. [17] N. A. Kudryashov, Exact solitary waves of the Fisher equation, Phys. Lett. A, 342 (2005), 99-106.  doi: 10.1016/j.physleta.2005.05.025. [18] N. A. Kudryashov, One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2248-2253.  doi: 10.1016/j.cnsns.2011.10.016. [19] N. A. Kudryashov, Analitical Theory of Nonlinear Differential Equations, Institute of Computer Investigations, Moskow-Igevsk, 2004. [20] N. A. Kudryashov, On new travelling wave solutions" of the KdV and the KdV-Burgers equations, Commun. Nonlinear Sci. Numer. Simulat., 14 (2009), 1891-1900.  doi: 10.1016/j.cnsns.2008.09.020. [21] J. Liu and Y. Zhang, Construction of lump soliton and mixed lump stripe solutions of $(3+1)$-dimensional soliton equation, Results in Physics, 10 (2018), 94-98. [22] S. K. Liu, Z. T. Fu and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions on nonlinear wave equations, Phys. Lett. A., 289 (2001), 69-74.  doi: 10.1016/S0375-9601(01)00580-1. [23] W.-X. Ma and J.-M. Lee, A transformed rational function method and exact solutions to the $(3+1)$-dimensional Jimbo-Miwa equation, Chaos Solitons Fractals, 42 (2009), 1356-1363.  doi: 10.1016/j.chaos.2009.03.043. [24] F. Mahomed, Recent trends in symmetry analysis of differential equations, Not. S. Afr. Math. Soc., 33 (2002), 11-40.  doi: 10.1198/106186002317375721. [25] W. Malfiet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650-654.  doi: 10.1119/1.17120. [26] P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition, Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993. [27] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, Inc., New York-London, 1982. [28] M. Saifur Rahman and A. S. M. Z. Hasan, Modified harmonic balance method for the solution of nonlinear jerk equations, Results in Physics, 8 (2018), 893-897.  doi: 10.1016/j.rinp.2018.01.030. [29] I. Simbanefayi and C. M. Khalique, Travelling wave solutions and conservation laws for the Korteweg-de Vries-Bejamin-Bona-Mahony equation, Results in Physics, 8 (2018), 57-63. [30] H.-Q. Sun and A. H. Chen, Lump and limp-kink solutions of the $(3+1)$-dimensional Jimbo-Miwa and extended Jimbo-Miwa equations, Appl. Math. Lett., 68 (2017), 55-61.  doi: 10.1016/j.aml.2016.12.008. [31] C.-Y. Wang, The analytic solutions of Schrödinger equation with Cubic-Quintic nonlinearities, Results in Physics, 10 (2018), 150-154.  doi: 10.1016/j.rinp.2018.05.017. [32] M. L. Wang, X. Z. Li and J. L. Zhang, The $(G'/G)-$expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A., 372 (2008), 417-423.  doi: 10.1016/j.physleta.2007.07.051. [33] A.-M. Wazwaz, The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations, Appl. Math. and Comput., 188 (2007), 1467-1475.  doi: 10.1016/j.amc.2006.11.013. [34] A.-M. Wazwaz, Multiple-soliton solutions for the Calogero-Bogoyavlenskii-Schiff, Jimbo-Miwa and YTSF equations, Appl. Math. Comput., 203 (2008), 592-597.  doi: 10.1016/j.amc.2008.05.004. [35] A.-M. Wazwaz, Multiple-soliton solutions for the extended $(3+1)$-dimensional Jimbo-Miwa equation, Appl. Math. Lett., 64 (2017), 21-26.  doi: 10.1016/j.aml.2016.08.005. [36] G. Q. Xu, The soliton solutions, dromions of the Kadomtsev-Petviashvili and Jimbo-Miwa equations in $(3+1)$-dimensionals, Chaos Solitons Fractals, 30 (2006), 71-76.  doi: 10.1016/j.chaos.2005.08.089. [37] L. J. Zhang and C. M. Khalique, Classification and bifurcation of a class of second-order odes and its application to nonlinear PDEs, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 759-772.  doi: 10.3934/dcdss.2018048. [38] Y. B. Zhou, M. L. Wang and Y. M. Wang, Periodic wave solutions to coupled KdV equations with variable coefficients, Phys. Lett. A, 308 (2003), 31-36.  doi: 10.1016/S0375-9601(02)01775-9.

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References:
 [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511623998. [2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1966 [3] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences, 81. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-4307-4. [4] M. S. Bruzón and T. M. Garrido, Symmetries and conservation laws of a KdV6 equation, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 631-641.  doi: 10.3934/dcdss.2018038. [5] B. T. Cao, Solutions of Jimbo-Miwa equation and Konopelchenko-Dudrovsky equations, Acta Appl. Math., 112 (2010), 181-203.  doi: 10.1007/s10440-009-9559-5. [6] M. T. Darvishi and M. Najafi, Some complexiton type solutions of the $(3+1)$-dimensional Jimbo-Miwa equation, Int. J. Comput. Math. Sci., 6 (2012), 25-27. [7] A. Fatima, F. M. Mahomed and C. M. Khalique, Conditional symmetries of nonlinear third-order ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 655-666.  doi: 10.3934/dcdss.2018040. [8] C. A. Gomez, A. H. Salas and B. Acevedo Frias, Exact solutions to KdV6 equation by using a new approach of the projective Riccati equation method, Math. Prob. Eng., 2010 (2010), 797084, 10 pp. doi: 10.1155/2010/797084. [9] R. Hirota, Exact solution of the KdV equation for multiple collisions of solutions, Phys. Rev. Lett., 27 (1971), 1192-1194. [10] N. H. Ibragimov, R. L. Anderson and et al, CRC Handbook of Lie Group Analysis of Differential Equatios, Vol. 3. New Trends in Theoretical Developments and Computational Methods, CRC Press, Boca Raton, FL, 1996. [11] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley Series in Mathematical Methods in Practice, 4. John Wiley & Sons, Ltd., Chichester, 1999. [12] N. H. Ibragimov, Modern group analysis, Nonlinear Dynam. Springer, Dordrecht, 28 (2002), 103-230. [13] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328.  doi: 10.1016/j.jmaa.2006.10.078. [14] M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. Res. Inst. Math. Sci., 19 (1983), 943-1001.  doi: 10.2977/prims/1195182017. [15] M. T. Kajani and F. T. Kajani, Homotopy analysis method for solving nonlinear system of equations, 13 (2011), 2471–2473. [16] N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Solitons Fract., 24 (2005), 1217-1231.  doi: 10.1016/j.chaos.2004.09.109. [17] N. A. Kudryashov, Exact solitary waves of the Fisher equation, Phys. Lett. A, 342 (2005), 99-106.  doi: 10.1016/j.physleta.2005.05.025. [18] N. A. Kudryashov, One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2248-2253.  doi: 10.1016/j.cnsns.2011.10.016. [19] N. A. Kudryashov, Analitical Theory of Nonlinear Differential Equations, Institute of Computer Investigations, Moskow-Igevsk, 2004. [20] N. A. Kudryashov, On new travelling wave solutions" of the KdV and the KdV-Burgers equations, Commun. Nonlinear Sci. Numer. Simulat., 14 (2009), 1891-1900.  doi: 10.1016/j.cnsns.2008.09.020. [21] J. Liu and Y. Zhang, Construction of lump soliton and mixed lump stripe solutions of $(3+1)$-dimensional soliton equation, Results in Physics, 10 (2018), 94-98. [22] S. K. Liu, Z. T. Fu and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions on nonlinear wave equations, Phys. Lett. A., 289 (2001), 69-74.  doi: 10.1016/S0375-9601(01)00580-1. [23] W.-X. Ma and J.-M. Lee, A transformed rational function method and exact solutions to the $(3+1)$-dimensional Jimbo-Miwa equation, Chaos Solitons Fractals, 42 (2009), 1356-1363.  doi: 10.1016/j.chaos.2009.03.043. [24] F. Mahomed, Recent trends in symmetry analysis of differential equations, Not. S. Afr. Math. Soc., 33 (2002), 11-40.  doi: 10.1198/106186002317375721. [25] W. Malfiet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650-654.  doi: 10.1119/1.17120. [26] P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition, Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993. [27] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, Inc., New York-London, 1982. [28] M. Saifur Rahman and A. S. M. Z. Hasan, Modified harmonic balance method for the solution of nonlinear jerk equations, Results in Physics, 8 (2018), 893-897.  doi: 10.1016/j.rinp.2018.01.030. [29] I. Simbanefayi and C. M. Khalique, Travelling wave solutions and conservation laws for the Korteweg-de Vries-Bejamin-Bona-Mahony equation, Results in Physics, 8 (2018), 57-63. [30] H.-Q. Sun and A. H. Chen, Lump and limp-kink solutions of the $(3+1)$-dimensional Jimbo-Miwa and extended Jimbo-Miwa equations, Appl. Math. Lett., 68 (2017), 55-61.  doi: 10.1016/j.aml.2016.12.008. [31] C.-Y. Wang, The analytic solutions of Schrödinger equation with Cubic-Quintic nonlinearities, Results in Physics, 10 (2018), 150-154.  doi: 10.1016/j.rinp.2018.05.017. [32] M. L. Wang, X. Z. Li and J. L. Zhang, The $(G'/G)-$expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A., 372 (2008), 417-423.  doi: 10.1016/j.physleta.2007.07.051. [33] A.-M. Wazwaz, The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations, Appl. Math. and Comput., 188 (2007), 1467-1475.  doi: 10.1016/j.amc.2006.11.013. [34] A.-M. Wazwaz, Multiple-soliton solutions for the Calogero-Bogoyavlenskii-Schiff, Jimbo-Miwa and YTSF equations, Appl. Math. Comput., 203 (2008), 592-597.  doi: 10.1016/j.amc.2008.05.004. [35] A.-M. Wazwaz, Multiple-soliton solutions for the extended $(3+1)$-dimensional Jimbo-Miwa equation, Appl. Math. Lett., 64 (2017), 21-26.  doi: 10.1016/j.aml.2016.08.005. [36] G. Q. Xu, The soliton solutions, dromions of the Kadomtsev-Petviashvili and Jimbo-Miwa equations in $(3+1)$-dimensionals, Chaos Solitons Fractals, 30 (2006), 71-76.  doi: 10.1016/j.chaos.2005.08.089. [37] L. J. Zhang and C. M. Khalique, Classification and bifurcation of a class of second-order odes and its application to nonlinear PDEs, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 759-772.  doi: 10.3934/dcdss.2018048. [38] Y. B. Zhou, M. L. Wang and Y. M. Wang, Periodic wave solutions to coupled KdV equations with variable coefficients, Phys. Lett. A, 308 (2003), 31-36.  doi: 10.1016/S0375-9601(02)01775-9.
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