# American Institute of Mathematical Sciences

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

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

 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.
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. [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. [3] G. W. Bluman and J. Cole, The General Similarity Solution of the Heat Equation, Phys. J. Math. Mech. 18 (1969) 1025-1042. [4] P. A. Clarkson, Nonclassical Symmetry Reductions of the Boussinesq Equation. Chaos, Solitons & Fractal, 5 12 (1995) 2261-2301. [5] P. A. Clarkson and E. L. Mansfield, Algorithms for the nonclassical method of symmetry reductions. SIAM J. Appl. Math., 55 (1994), 1693-1719. [6] P. A. Clarkson and T. J. Priestley, Symmetries of a Generalised Boussinesq equation. Electronic Edition, 1996. [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. [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. [9] N. H. Ibragimov, Transformation groups applied to mathematical physics, Reidel-Dordrecht, (1985). [10] P. Olver, Applications of Lie groups to differential equations, Springer-Verlag, New York, (1993). [11] L. V. Ovsyannikov, Group analysis of differential equations, Academic, New York, (1982). [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.

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##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [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. [3] G. W. Bluman and J. Cole, The General Similarity Solution of the Heat Equation, Phys. J. Math. Mech. 18 (1969) 1025-1042. [4] P. A. Clarkson, Nonclassical Symmetry Reductions of the Boussinesq Equation. Chaos, Solitons & Fractal, 5 12 (1995) 2261-2301. [5] P. A. Clarkson and E. L. Mansfield, Algorithms for the nonclassical method of symmetry reductions. SIAM J. Appl. Math., 55 (1994), 1693-1719. [6] P. A. Clarkson and T. J. Priestley, Symmetries of a Generalised Boussinesq equation. Electronic Edition, 1996. [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. [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. [9] N. H. Ibragimov, Transformation groups applied to mathematical physics, Reidel-Dordrecht, (1985). [10] P. Olver, Applications of Lie groups to differential equations, Springer-Verlag, New York, (1993). [11] L. V. Ovsyannikov, Group analysis of differential equations, Academic, New York, (1982). [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.
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