On Symmetry Analysis and Exact Wave Solutions of New Modified Novikov Equation
In this paper, we study a new modified Novikov equation for its classical and nonclassical symmetries and use the symmetries to reduce it to a nonlinear ordinary differential equation (ODE). With the aid of solutions of the nonlinear ODE by using the modified (G/G)-expansion method proposed recently, multiple exact traveling wave solutions are obtained and the traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.
[1] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations
and Inverse Scattering Transform, Cambridge University Press, Cambridge
1991.
[2] A.H. Khater, W. Malfliet, D.K. Callebaut, E.S. Kamel, The tanh method,
a simple transformation and exact analytical solutions for nonlinear reaction-diffusion equation, Chaos, Solitons and Fractals, vol. 14, 2002,
pp. 513-522.
[3] E.G. Fan, Y.C. Hong, Generalized tanh method to special types of nonlinear equations, Z. Naturforsch A, vol. 57, 2002, pp. 692-700.
[4] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, vol. 30, 2006, pp. 700-708.
[5] E.Yusufoglu, A. Bekir, M. Alp, Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using Sine-Cosine method, Chaos, Solitons and Fractals, vol. 37, 2008, pp. 1193-1197.
[6] M. Wang, Solitary wave solutions for variant Boussinesq equations,
Physics Letters A, vol. 199, 1995, pp. 169-172.
[7] M.L. Wang, J.L. Zhang and X.Z. Li, The (G/G)-expansion method and
traveling wave solutions of nonlinear evolution equations in mathematical
physics, Physics Letters A, vol. 372, 2008, pp. 417-423.
[8] R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Physics Review Letters, vol. 77, 1993, pp. 1661-1664.
[9] A. Degasperis, D.D. Holm, A.N.W. Hone, A new integrable equation
with Peakon solutions, Theoretical and Mathematical Physics, vol. 133, 2002, pp. 170-183.
[10] A.M. Wazwaz, Solitary wave solutions for modified forms of Degasperis-
Procesi and Camassa-Holm equations, Physics Letters A, vol. 325,
2006, pp. 500-504.
[11] A.M. Wazwaz, New solitary wave solutions to the modified forms of
Degasperis-Procesi and Camassa-Holm equations, Applied Mathematics
and Computation, vol. 186, 2007, pp. 130-141.
[12] V.S. Novikov, Generalisations of the CamassaHolm equation, Journal of Physics A: Mathematical and Theoretical, vol. 42, 2009, pp. 342002.
[13] A.N.W. Hone, J.P. Wang, Integrable peakon equations with cubic
nonlinearity, Journal of Physics A: Mathematical and Theoretical, vol.
41, 2008, pp. 372002.
[14] A.N.W. Hone, H. Lundmark, J. Szmigielski, Explicit multipeakon solutions
of Novikovs cubically nonlinear integrable CamassaHolm type
equation, Dynamics of Partial Differential Equations, vol. 6, 2009, pp.
253-289.
[15] H.Kheiri, M. R. Moghaddam, V. Vafaei, Application of (G/G)-
expansion method for the Burgers, Burgers-Huxley and modified Burgers-
KdV equations, PRAMANA-journal of physics, vol. 76, 2011, pp.
831-842.
[16] P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate
Texts Math., vol. 107, 1993.
[17] G.W. Bluman, J.D. Cole, Similarity Methods for Differential Equations,
Springer Verlag, 1974.
[18] L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic
Press, 1982.
[19] R.K. Gupta, K. Singh, Symmetry Analysis and Some Exact Solutions of
Cylindrically Symmetric Null Fields in General Relativity, Communictaions
in Nonlinear Science and Numerical Simulation, vol. 16, 2011,
pp. 4189-4196.
[20] R.K. Gupta, Anupma, The Dullin-Gottwald-Holm equation: Classical
Lie approach and exact Solutions, International Journal of Nonlinear
Science, vol. 10, 2010, pp. 146-152.
[21] A. Bansal, R.K. Gupta, Modified (G/G)-expansion method for finding
exact wave solutions of the coupled Klein-Gordon-Schr¨odinger equation,
Mathematical Methods in the Applied Sciences (accepted).
[22] G. W. Bluman, J. D. Cole, The general similarity solution of the heat
equation, Journal of Mathematics and Mechanics, vol. 18, 1969, pp.
1025-1042.
[23] G. W. Bluman, S. Kumei, Symmetries and Differential Equations,
Springer, 1989.
[24] P.A. Clarkson, M.D. Kruskal, New similarity reductions of the Boussinesq
equation, Journal of Mathematical Physics, vol. 30, 1989, pp.
2201-2213.
[25] S.Y. Lou, Localized excitations in (3+1) dimensions: Dromions, Ringshape
and Bubble like solitons, Chinese Physics Letters, vol. 21, 2004,
pp. 1020-1023.
[26] H.C. Ma and S.Y. Lou, Finite symmetry transformation groups and
exact solutions of Lax integrable systems, Communications in Theoretical
Physics, vol. 44, 2005, pp. 193-196.
[1] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations
and Inverse Scattering Transform, Cambridge University Press, Cambridge
1991.
[2] A.H. Khater, W. Malfliet, D.K. Callebaut, E.S. Kamel, The tanh method,
a simple transformation and exact analytical solutions for nonlinear reaction-diffusion equation, Chaos, Solitons and Fractals, vol. 14, 2002,
pp. 513-522.
[3] E.G. Fan, Y.C. Hong, Generalized tanh method to special types of nonlinear equations, Z. Naturforsch A, vol. 57, 2002, pp. 692-700.
[4] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, vol. 30, 2006, pp. 700-708.
[5] E.Yusufoglu, A. Bekir, M. Alp, Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using Sine-Cosine method, Chaos, Solitons and Fractals, vol. 37, 2008, pp. 1193-1197.
[6] M. Wang, Solitary wave solutions for variant Boussinesq equations,
Physics Letters A, vol. 199, 1995, pp. 169-172.
[7] M.L. Wang, J.L. Zhang and X.Z. Li, The (G/G)-expansion method and
traveling wave solutions of nonlinear evolution equations in mathematical
physics, Physics Letters A, vol. 372, 2008, pp. 417-423.
[8] R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Physics Review Letters, vol. 77, 1993, pp. 1661-1664.
[9] A. Degasperis, D.D. Holm, A.N.W. Hone, A new integrable equation
with Peakon solutions, Theoretical and Mathematical Physics, vol. 133, 2002, pp. 170-183.
[10] A.M. Wazwaz, Solitary wave solutions for modified forms of Degasperis-
Procesi and Camassa-Holm equations, Physics Letters A, vol. 325,
2006, pp. 500-504.
[11] A.M. Wazwaz, New solitary wave solutions to the modified forms of
Degasperis-Procesi and Camassa-Holm equations, Applied Mathematics
and Computation, vol. 186, 2007, pp. 130-141.
[12] V.S. Novikov, Generalisations of the CamassaHolm equation, Journal of Physics A: Mathematical and Theoretical, vol. 42, 2009, pp. 342002.
[13] A.N.W. Hone, J.P. Wang, Integrable peakon equations with cubic
nonlinearity, Journal of Physics A: Mathematical and Theoretical, vol.
41, 2008, pp. 372002.
[14] A.N.W. Hone, H. Lundmark, J. Szmigielski, Explicit multipeakon solutions
of Novikovs cubically nonlinear integrable CamassaHolm type
equation, Dynamics of Partial Differential Equations, vol. 6, 2009, pp.
253-289.
[15] H.Kheiri, M. R. Moghaddam, V. Vafaei, Application of (G/G)-
expansion method for the Burgers, Burgers-Huxley and modified Burgers-
KdV equations, PRAMANA-journal of physics, vol. 76, 2011, pp.
831-842.
[16] P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate
Texts Math., vol. 107, 1993.
[17] G.W. Bluman, J.D. Cole, Similarity Methods for Differential Equations,
Springer Verlag, 1974.
[18] L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic
Press, 1982.
[19] R.K. Gupta, K. Singh, Symmetry Analysis and Some Exact Solutions of
Cylindrically Symmetric Null Fields in General Relativity, Communictaions
in Nonlinear Science and Numerical Simulation, vol. 16, 2011,
pp. 4189-4196.
[20] R.K. Gupta, Anupma, The Dullin-Gottwald-Holm equation: Classical
Lie approach and exact Solutions, International Journal of Nonlinear
Science, vol. 10, 2010, pp. 146-152.
[21] A. Bansal, R.K. Gupta, Modified (G/G)-expansion method for finding
exact wave solutions of the coupled Klein-Gordon-Schr¨odinger equation,
Mathematical Methods in the Applied Sciences (accepted).
[22] G. W. Bluman, J. D. Cole, The general similarity solution of the heat
equation, Journal of Mathematics and Mechanics, vol. 18, 1969, pp.
1025-1042.
[23] G. W. Bluman, S. Kumei, Symmetries and Differential Equations,
Springer, 1989.
[24] P.A. Clarkson, M.D. Kruskal, New similarity reductions of the Boussinesq
equation, Journal of Mathematical Physics, vol. 30, 1989, pp.
2201-2213.
[25] S.Y. Lou, Localized excitations in (3+1) dimensions: Dromions, Ringshape
and Bubble like solitons, Chinese Physics Letters, vol. 21, 2004,
pp. 1020-1023.
[26] H.C. Ma and S.Y. Lou, Finite symmetry transformation groups and
exact solutions of Lax integrable systems, Communications in Theoretical
Physics, vol. 44, 2005, pp. 193-196.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:62326", author = "Anupma Bansal and R. K. Gupta", title = "On Symmetry Analysis and Exact Wave Solutions of New Modified Novikov Equation", abstract = "In this paper, we study a new modified Novikov equation for its classical and nonclassical symmetries and use the symmetries to reduce it to a nonlinear ordinary differential equation (ODE). With the aid of solutions of the nonlinear ODE by using the modified (G/G)-expansion method proposed recently, multiple exact traveling wave solutions are obtained and the traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.
", keywords = "New Modified Novikov Equation, Lie Classical Method, Nonclassical Method, Modified (G'/G)-Expansion Method, Traveling Wave Solutions.", volume = "6", number = "8", pages = "1161-8", }
