Journal of Applied Nonlinear Dynamics
Lie Symmetry Analysis and Conservation Laws of a TwoWave Mode Equation for the Integrable KadomtsevPetviashvili Equation
Journal of Applied Nonlinear Dynamics 10(1) (2021) 6579  DOI:10.5890/JAND.2021.03.004
T.S. Moretlo$^1$, B. Muatjetjeja$^{1,2}$, A.R. Adem$^3$
$^1$ Department of Mathematical Sciences, NorthWest University, Private Bag X 2046, Mmabatho 2735,
South Africa
$^2$ Department of Mathematics, Faculty of Science, University of Botswana, Private
Bag 22, Gaborone, Botswana
$^3$ Department of Mathematical Sciences, University of South Africa, UNISA 0003,
South Africa
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Abstract
Lie symmetry analysis is performed on a twowave mode equation for the integrable KadomtsevPetviashvili (TKP) equation which describes the propagation of two different wave modes in the same direction simultaneously. The similarity reductions and an exact solution are computed. In addition to this, we derive the conservation laws for the underlying equation.
References

[1] 
Yuan, Y.Q., Tian, B., Liu, L., Wu, X.Y., and Sun, Y. (2018), Solitons for the (2+1)dimensional KonopelchenkoDubrovsky equations,
Journal of Mathematical Analysis and Applications,
460, 476486.


[2] 
Gao, X.Y. (2019), Mathematical view with observational/experimental consideration
on certain (2+1)dimensional waves in the cosmic/laboratory
dusty plasmas, Applied Mathematics Letters, 91, 165172


[3] 
Du, Z., Tian, B., Chai, H.P., Sun, Y., and Zhao, X.H. (2018), Rogue waves for the coupled variablecoefficient fourthorder nonlinear Schr\"{o}dinger equations in an inhomogeneous optical fiber, Chaos, Solitons and Fractals, 109, 9098.


[4] 
Du, X.X., Tian, B., Wu, X.Y., Yin, H.M., and Zhang, C.R. (2018), Lie group analysis, analytic solutions and conservation laws of
the (3+1)dimensional ZakharovKuznetsovBurgers equation
in a collisionless magnetized electronpositronion plasma,
The European Physical Journal Plus, 133, 378.


[5] 
Liu, L., Tian, B., Yuan, Y.Q., and Du, Z. (2018), Darkbright solitons and semirational rogue waves for the coupled SasaSatsuma equations,
Physical Review E,
97, 052217.


[6] 
Liu, L., Tiana, B., Yuan, Y.Q., and Sun, Y. (2018), Bright and dark Nsoliton solutions for the (2+1)dimensional
Maccari system, The European Physical Journal Plus,
133, 72.


[7] 
Zhao, X.H., Tian, B., Xie, X.Y., Wu, X.Y., Sun, Y., and Guo, Y.J. (2018), Solitons, B\"{a}cklund transformation and Lax pair for
a (2+1)dimensional DaveyStewartson system on
surface waves of finite depth, Waves in Random and Complex Media, 28(2), 356366


[8] 
Zhang, C.R., Tian, B., Wu, X.Y., Yuan, Y.Q., and Du, X.X. (2018),
Rogue waves and solitons of the coherentlycoupled
nonlinear Schr\"{o}dinger equations
with the positive coherent coupling, Physica Scripta,
93, 095202, (11pp).


[9] 
Wu, X.Y., Tian, B., Liu, L., and Sun, Y. (2018), Rogue waves for a variablecoefficient KadomtsevPetviashvili equation in fluid mechanics, Computers and Mathematics with Applications,
76, 215223.


[10] 
L\"{u}, X., Wang, J.P., Lin, F.H., and Zhou, X.W. (2018), Lump dynamics of a generalized twodimensional Boussinesq equation in shallow water,
Nonlinear Dynamics, 91, 12491259.


[11] 
Lin, F.H., Chen, S.T., Qu, Q.X., Wang, J.P., Zhou, X.W., and
L\"{u}, X. (2018), Resonant multiple wave solutions to a new (3+1)dimensional generalized KadomtsevPetviashvili equation: Linear superposition principle, Applied Mathematics Letters,
78, 112117.


[12] 
Gao, L.N., Zi, Y.Y., Yin, Y.H., Ma, W.X., and L\"{u}, X. (2017), B\"{a}cklund transformation, multiple wave solutions and lumpsolutions to a
(3+1)dimensional nonlinear evolution equation, Nonlinear Dynamics,
89, 22332240.


[13] 
Yin, Y.H., Ma, W.X., Liu, J.G., and L\"{u}, X. (2018), Diversity of exact solutions to a (3+1)dimensional nonlinear evolution equation and its reduction, Computers and Mathematics with Applications,
76, 12751283.


[14] 
L\"{u}, X., Chen, S.T., and Ma, W.X. (2016), Constructing lump solutions to a generalized KadomtsevPetviashviliBoussinesq equation,
Nonlinear Dynamics, 86, 523534.


[15] 
L\"{u}, X. and Ma, W.X. (2016), Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation, Nonlinear Dynamics, 85, 12171222.


[16] 
Gao, L.N., Zhao, X.Y., Zi, Y.Y., Yu, J., and L\"{u}, X. (2016), Resonant behavior of multiple wave solutions to a Hirota bilinear equation, Computers and Mathematics with Applications,
72, 12251229.


[17] 
L\"{u}, X. and Lin, F. (2016), Soliton excitations and shapechanging collisions in alphahelical proteins with interspine coupling at higher order, Communications in Nonlinear Science and Numerical Simulation,
32, 241261.


[18]  Wazwaz, A.M. (2005), Exact solutions for the ZKMEW equation by using the tanh and sinecosine methods,
International Journal of Computer Mathematics, 82, 699708.


[19]  Wazwaz, A.M. (2006), The tanh and the sinecosine methods for a reliable treatment of the modified equal width equation and its variants, Communications in Nonlinear Science and Numerical Simulation, 11, 148160.


[20]  Wazwaz, A.M. (2006), New travelling wave solutions of different physical structures
to generalized BBM equation, Physics Letters A,
355, 358362.


[21]  Wazwaz, A.M. (2010), A study on KdV and Gardner equations with timedependent coefficients and forcing terms,
Applied Mathematics and Computation, 217, 22772281.


[22]  Wazwaz, A.M. (2010), Completely integrable coupled KdV and coupled KP systems, Communications in Nonlinear
Science and Numerical Simulations, 15, 28282835.


[23]  Wazwaz, A.M. (2011), Integrability of coupled KdV equations,
Central European Journal of Physics, 9, 835840.


[24] 
Korteweg, D.J. and de Vries, G. (1895), On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine, 39, 422443.


[25] 
Kadomtsev, B.B. and Petviashvili, V.I. (1970), On the stability of solitary waves in weakly dispersive media, Soviet Physics  Doklady,
15, 539541.


[26] 
Korsunsky, S.V. (1994), Soliton solutions for a secondorder KdV equation, Physics Letters A, 185, 174176.


[27] 
Wazwaz, A.M. (2017), A study on a twowavemode KadomtsevPetviashvili equation: conditions for multiple soliton solutions to exist, Mathematical Methods in the Applied Sciences, 40, 41284133


[28]  Muatjetjeja, B. and Adem, A.R. (2017), RosenauKdV Equation Coupling with the RosenauRLW Equation: Conservation Laws and Exact Solutions,
International Journal of Nonlinear Sciences and Numerical Simulation, 18, 451456.
