On the exact solutions and conservation laws of a generalized (1+2)-dimensional Jaulent-Miodek equation with a power law nonlinearity

Document Type : Research Paper


1 Department of Mathematical Sciences North-West University Private Bag X 2046, Mmabatho 2735 Republic of South Africa

2 Department of Mathematical Sciences, North-West University Private Bag X 2046 Mmabatho 2735, Republic of South Africa

3 Department of Mathematics Faculty of Science, University of Botswana Private Bag 22, Gaborone, Botswana

4 Department of Mathematical Sciences, University of South Africa, UNISA 0003, Republic of South Africa


In this paper, a generalized (1+2)-dimensional Jaulent-Miodek equation with a power law nonlinearity is examined, which arises in numerous problems in nonlinear science. The computed conservation laws reside in enormously crucial areas both at the foundations of nonlinear science such as biology, physics and other related areas. Exact solutions are acquired using the Lie symmetry method. In addition to exact solutions, we also present conservation laws. The arbitrary functions in the multipliers lead to infinitely many conservation laws.


[1] A.R. Adem, Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry
analysis and extended tanh method, Comput. Math. Appl. 74 (2017) 1897–1902.
[2] A.R Adem, On the solutions and conservation laws of a two-dimensional Korteweg de Vries model: Multiple
exp-function method, J. Appl. Anal. 24 (2018) 27–33.
[3] S.J. Chen, W.X. Ma and X. L¨u, Backlund transformation, exact solutions and interaction behaviour of the (3+1)- ¨
dimensional Hirota-Satsuma-Ito-like equation, Commun. Nonlinear Sci. Numerical Simul. 83 (2020) 105135.
[4] Z. Du, B. Tian, H.P. Chai, Y. Sun and X.H. Zhao, Rogue waves for the coupled variable-coefficient fourth-order
nonlinear Schrodinger equations in an inhomogeneous optical fiber ¨ , Chaos, Solitons Fract. 109 (2018) 90-–98.
[5] X.X. Du, B. Tian, X.Y. Wu, H.M. Yin and C.R. Zhang, Lie group analysis, analytic solutions and conservation
laws of the (3 + 1)-dimensional Zakharov-Kuznetsov-Burgers equation in a collisionless magnetized electronpositron-ion plasma, The European Phys. J. Plus 133 (2018) 378.
[6] X. Guan, W. Liu, Q. Zhou and A. Biswas, Some lump solutions for a generalized (3+1)-dimensional KadomtsevPetviashvili equation, Appl. Math. Comput. 366 (2020) 124757.
[7] X. Guan, W. Liu, Q. Zhou and A. Biswas, Darboux transformation and analytic solutions for a generalized
super-NLS-mKdV equation, Nonlinear Dyn. 98 (2019) 1491–1500.
[8] T.M Garrido, R.D.L Rosa, E. Recio and M.S Bruzon, Symmetries, solutions and conservation laws for the (2+1)
filtration-absorption model, J. Math. Chem. 57 (2019) 1301–1313.
[9] X.Y. Gao, Mathematical view with observational/experimental consideration on certain (2+1)-dimensional waves
in the cosmic/laboratory dusty plasmas, Appl. Math. Lett. 91 (2019) 165-–172.
[10] Y.F. Hua, B.L. Guo, W.X. Ma and X. L¨u, Interaction behavior associated with a generalized (2 + 1)-dimensional
Hirota bilinear equation for nonlinear waves, Appl. Math. Model. 74 (2019) 184–198.
[11] X. Liu, W. Liu, H. Triki, Q. Zhou and A. Biswas, Periodic attenuating oscillation between soliton interactions
for higher-order variable coefficient nonlinear Schrodinger equation ¨ , Nonlinear Dyn. 96 (2019) 801–809.
[12] L. Liu, B. Tian, Y.Q. Yuan and Z. Du, Dark-bright solitons and semirational rogue waves for the coupled SasaSatsuma equations, Phys. Rev. E 97 (2018) 052217.
[13] L. Liu, B. Tiana, Y.Q. Yuan and Y. Sun, Bright and dark N-soliton solutions for the (2 + 1)-dimensional Maccari
system, European Phys. J. Plus 133 (2018) 72.
[14] W. Liu, Y. Zhang, A.M. Wazwaz and Q. Zhou, Analytic study on triple-S, triple-triangle structure interactions
for solitons in inhomogeneous multi-mode fiber, Appl. Math. Comput. 361 (2019) 325–331.
[15] S. Liu, Q. Zhou, A. Biswas and W. Liu, Phase-shift controlling of three solitons in dispersion-decreasing fibers,
Nonlinear Dyn. 98 (2019) 395–401.
[16] B. Muatjetjeja, Coupled Lane-Emden-Klein-Gordon-Fock system with central symmetry: Symmetries and conservation laws, J. Diff. Equ. 263 (2017) 8322–8328.
[17] B. Muatjetjeja, On the symmetry analysis and conservation laws of the (1+ 1)-dimensional Henon-Lane-Emden
system, Math. Meth. Appl. Sci. 40 (2017) 1531–1537.
[18] E. Recio, T.M Garrido, R.D.L Rosa and M.S Bruzon, Conservation laws and Lie symmetries a (2+1)-dimensional
thin film equation, J. Math. Chem. 57 (2019) 1243–1251.
[19] S. Saez, R.D.L Rosa, E. Recio, T.M Garrido and M.S Bruzon, Lie symmetries and conservation laws for a
generalized (2+1)-dimensional nonlinear evolution equation, J. Math. Chem. 58 (2020) 775–798.
[20] A.M. Wazwaz, Multiple kink solutions and multiple singular kink solutions for (2+1)-dimensional nonlinear models
generated by the Jaulent-Miodek hierarchy, Phys. Lett. A 373 (2009) 1844–1846.
[21] A.M. Wazwaz, Multiple complex soliton solutions for integrable negative-order KdV and integrable negative-order
modified KdV equations, Appl. Math. Lett. 88 (2019) 1–7.
[22] A.M. Wazwaz, Multiple complex soliton solutions for the integrable KdV, fifth-order Lax, modified KdV, Burgers,
and Sharma-Tasso-Olver equations, Chinese J. Phys. 59 (2019) 372–378.
[23] A.M. Wazwaz and M.S. Osman, The combined multi-waves polynomial solutions in a two-layer-liquid medium,
Comput. Math. Appl. 76 (2018) 276–283.
[24] A.M. Wazwaz and, M.S. Osman, An efficient algorithm to construct multi-soliton rational solutions of the (2+1)-
dimensional KdV equation with variable coefficients, Appl. Math. Comput. 321 (2018) 282–289.
[25] X.Y.Wu, B.Tian, L. Liu and Y. Sun, Rogue waves for a variable-coefficient Kadomtsev–Petviashvili equation in
fluid mechanics, Comput. Math. Appl. 76 (2018) 215–223.
[26] H.N. Xu, W.Y. Ruan, Y. Zhang and X. L¨u, Multi-exponential wave solutions to two extended Jimbo-Miwa equations and the resonance behavior, Appl. Math. Lett. 99 (2020) 105976.
[27] Y.Q. Yuan, B. Tian, L. Liu, X.Y. Wu and Y. Sun, Solitons for the (2 +1)-dimensional Konopelchenko-Dubrovsky
equations, J. Math. Anal. Appl. 460 (2018) 476–486.A generalized (1+2)-dimensional Jaulent-Miodek equation 1735
[28] C.R. Zhang, B. Tian, X.Y. Wu, Y.Q. Yuan and X.X. Du, Rogue waves and solitons of the coherentlycoupled
nonlinear Schrodinger equations with the positive coherent coupling ¨ , Phys. Scripta 93 (2018) 095202.
[29] X.H. Zhao, B. Tian, X.Y. Xie, X.Y. Wu, Y. Sun and Y.J. Guo, Solitons, Backlund transformation and Lax pair for ¨
a (2+1)-dimensional Davey-Stewartson system on surface waves of finite depth, Waves in Random and Complex
Media 28 (2018) 356–366.
Volume 13, Issue 1
March 2022
Pages 1721-1735
  • Receive Date: 10 June 2020
  • Revise Date: 14 August 2020
  • Accept Date: 28 September 2020
  • First Publish Date: 11 November 2021