American Institute of Mathematical Sciences

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2015, 2015(special): 151-158. doi: 10.3934/proc.2015.0151

Classical and nonclassical symmetries and exact solutions for a generalized Benjamin equation

M. S. Bruzón 1, , M. L. Gandarias 1, and  J. C. Camacho 1,

1. 

Departamento de Matemáticas, Universidad de Cádiz, PO.BOX 40, 11510 Puerto Real, Cádiz, Spain, Spain, Spain

Received  September 2014 Revised  December 2014 Published  November 2015

We apply the Lie-group formalism to deduce symmetries of a generalized Benjamin equation. We make an analysis of the symmetry reductions of the equation. In order to obtain travelling wave solutions we apply an indirect F-function method. We obtained in an unified way simultaneously many periodic wave solutions expressed by various single and combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions. We compare these solutions with the solutions derived by other authors by using different methods and we observe that we have obtained new solutions for this equation.
Keywords: nonclassical symmetries., Exact solutions, classical symmetries.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 76M6.
Citation: M. S. Bruzón, M. L. Gandarias, J. C. Camacho. Classical and nonclassical symmetries and exact solutions for a generalized Benjamin equation. Conference Publications, 2015, 2015 (special) : 151-158. doi: 10.3934/proc.2015.0151
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. Google Scholar

[2]

F. M. Belgacem, H. Bulut, H. M. Baskonus and T. Akturk, Mathematical analysis of the Generalized Benjamin and Burger-Kdv Equations via the Extended Trial Equation Method, J. Assoc. of Arab Univ. Basic and Appl. Sc., (2013), preprint. Google Scholar

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G. W. Bluman and J. Cole, The General Similarity Solution of the Heat Equation, Phys. J. Math. Mech. 18 (1969) 1025-1042. Google Scholar

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P. A. Clarkson, Nonclassical Symmetry Reductions of the Boussinesq Equation. Chaos, Solitons & Fractal, 5 12 (1995) 2261-2301. Google Scholar

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P. A. Clarkson and E. L. Mansfield, Algorithms for the nonclassical method of symmetry reductions. SIAM J. Appl. Math., 55 (1994), 1693-1719. Google Scholar

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P. A. Clarkson and T. J. Priestley, Symmetries of a Generalised Boussinesq equation. Electronic Edition, 1996. Google Scholar

[7]

M.L. Gandarias and M.S. Bruzón, Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation, J. Nonlin. Math. Phys. 5 (1998) 8-12. Google Scholar

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W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. Van Immerzeele and A. Meerpole, Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method, J. Phys. A. Math. Gen., 19 (5) (1986), 607-628. Google Scholar

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N. H. Ibragimov, Transformation groups applied to mathematical physics, Reidel-Dordrecht, (1985). Google Scholar

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P. Olver, Applications of Lie groups to differential equations, Springer-Verlag, New York, (1993). Google Scholar

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L. V. Ovsyannikov, Group analysis of differential equations, Academic, New York, (1982). Google Scholar

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N. Taghizadeh, M. Mirzazadeh and S. R. Moosavi Noori, Exact Solutions of the Generalized Benjamin Equation and (3 + 1)- Dimensional Gkp Equation by the Extended Tanh Method, Appl. Appl. Math., 7 (2012), 175-187. Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. Google Scholar

[2]

F. M. Belgacem, H. Bulut, H. M. Baskonus and T. Akturk, Mathematical analysis of the Generalized Benjamin and Burger-Kdv Equations via the Extended Trial Equation Method, J. Assoc. of Arab Univ. Basic and Appl. Sc., (2013), preprint. Google Scholar

[3]

G. W. Bluman and J. Cole, The General Similarity Solution of the Heat Equation, Phys. J. Math. Mech. 18 (1969) 1025-1042. Google Scholar

[4]

P. A. Clarkson, Nonclassical Symmetry Reductions of the Boussinesq Equation. Chaos, Solitons & Fractal, 5 12 (1995) 2261-2301. Google Scholar

[5]

P. A. Clarkson and E. L. Mansfield, Algorithms for the nonclassical method of symmetry reductions. SIAM J. Appl. Math., 55 (1994), 1693-1719. Google Scholar

[6]

P. A. Clarkson and T. J. Priestley, Symmetries of a Generalised Boussinesq equation. Electronic Edition, 1996. Google Scholar

[7]

M.L. Gandarias and M.S. Bruzón, Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation, J. Nonlin. Math. Phys. 5 (1998) 8-12. Google Scholar

[8]

W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. Van Immerzeele and A. Meerpole, Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method, J. Phys. A. Math. Gen., 19 (5) (1986), 607-628. Google Scholar

[9]

N. H. Ibragimov, Transformation groups applied to mathematical physics, Reidel-Dordrecht, (1985). Google Scholar

[10]

P. Olver, Applications of Lie groups to differential equations, Springer-Verlag, New York, (1993). Google Scholar

[11]

L. V. Ovsyannikov, Group analysis of differential equations, Academic, New York, (1982). Google Scholar

[12]

N. Taghizadeh, M. Mirzazadeh and S. R. Moosavi Noori, Exact Solutions of the Generalized Benjamin Equation and (3 + 1)- Dimensional Gkp Equation by the Extended Tanh Method, Appl. Appl. Math., 7 (2012), 175-187. Google Scholar

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