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August  2018, 11(4): 631-641. doi: 10.3934/dcdss.2018038

Symmetries and conservation laws of a KdV6 equation

Department of Mathematics, University of Cádiz, PO.BOX 40, 11510 Puerto Real, Cádiz, Spain

* Corresponding author: M.S. Bruzón.

Received  December 2016 Revised  May 2017 Published  November 2017

In the present work we make an analysis of the Korteweg-de Vries of sixth order. We apply the classical Lie method of infinitesimals and the nonclassical method, due to Bluman and Cole, to deduce new symmetries of the equation which cannot be obtained by Lie classical method. Moreover, we obtain ten different conservation laws depending on the parameters and we conclude that potential symmetries project on the infinitesimals corresponding to the classical symmetries.

Citation: María Santos Bruzón, Tamara María Garrido. Symmetries and conservation laws of a KdV6 equation. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 631-641. doi: 10.3934/dcdss.2018038
References:
[1]

S. C. Anco and G. W. Bluman, Direct construction of conservation laws from field equations, Physical Review Letters, 78 (1997), 2869-2873.  doi: 10.1103/PhysRevLett.78.2869.

[2]

S. C. Anco and G. W. Bluman, Direct construction method for conservation laws of partial differential equations part 2: General treatment, European Journal of Applied Mathematics, 13 (2002), 567-585.  doi: 10.1017/S0956792501004661.

[3]

S. C. Anco and G. W. Bluman, Direct constrution method for conservation laws of partial differential equations part 1: Examples of conservation law classifications, European Journal of Applied Mathematics, 13 (2002), 545-566.  doi: 10.1017/S0956792501004661.

[4]

S. C. Anco, Generalization of Noether’s theorem in modern form to non-variational partial differential equations, To appear in Fields Institute Communications: Recent progress and Modern Challen ges in Applied Mathematics, Modeling and Computational Science, 79 (2017), 119-182, arXiv: 1605.08734. doi: 10.1007/978-1-4939-6969-2_5.

[5]

G. W. Bluman and J. Cole, General similarity solution of the heat equation, Journal of Mathematics and Mechanics, 18 (1969), 1025-1042. 

[6]

G. W. Bluman and S. Kumei, On the remarkable nonlinear diffusion equation, Journal of Mathematical Physics, 21 (1980), 1019-1023.  doi: 10.1063/1.524550.

[7]

G. W. BlumanS. Kumei and G. J. Reid, New classes of symmetries for partial differential equations, Journal of Mathematics and Mechanics, 29 (1988), 806-811.  doi: 10.1063/1.527974.

[8]

M. S. Bruzón, M. L. Gandarias and J. Ramírez, Symmetry and Perturbation Theory, World Scientific Publising Company, 2005.

[9]

M. S. BruzónT. M. Garrido and R. de la Rosa, Conservation laws and exact solutions of a Generalized Benjamin-Bona-Mahony-Burgers equation, Chaos, Solitons & Fractals, 89 (2016), 578-583.  doi: 10.1016/j.chaos.2016.03.034.

[10]

P. J. CaudreyR. K. Dodd and J. D. Gibbon, New hierarchy of koterweg de vries equations, Proceedings of the Royal Society of London series A-mathematical physical and engineering sciences, 351 (1976), 407-422.  doi: 10.1098/rspa.1976.0149.

[11]

P. A. Clarkson, Nonclassical symmetry reductions of the boussinesq equation, Chaos, Solitons & Fractals, 5 (1995), 2261-2301.  doi: 10.1016/0960-0779(94)E0099-B.

[12]

P. A. Clarkson and T. J. Priestley, Symmetries of a generalised boussinesq equation, Institute of Mathematics and Statistics, University of Kent at Canterbury.

[13]

V. G. Drinfeld and V. V. Sokolov, Nonclassical symmetry reductions of the boussinesq equation, Doklady akademii nauk sssr, 258 (1981), 11-16. 

[14]

A. P. Fordy and J. Gibbons, Some remarkable nonlinear transformations, Physics Letters A, 75 (1980), p325. doi: 10.1016/0375-9601(80)90829-4.

[15]

M. L. Gandarias and M. S. Bruzón, Classical and nonclassical symmetries of a generalized boussinesq equation, Journal of Nonlinear Mathematical Physics, 5 (1998), 8-12.  doi: 10.2991/jnmp.1998.5.1.2.

[16]

T. M. GarridoA. A. KasatkinM. S. Bruzón and R. K. Gazizov, Lie symmetries and equivalence transformations for the barenblatt-gilman model, Journal of Computational and Applied Mathematics, 318 (2017), 253-258.  doi: 10.1016/j.cam.2016.09.023.

[17]

W. Hereman and B. Huard, symmgrp2009. max: A macsyma/maxima program for the calculation of lie point symmetries of large systems of differential equations, http://inside.mines.edu/whereman/software.html (2009).

[18]

A. Karasu-Kalkanli, A. Karasu, A. Sakovich, S. Sakovich and R. Turhan, A new integrable generalization of the korteweg de vries equation, Journal of Mathematical Physics, 49 (2008), 073516, 10 pp. doi: 10.1063/1.2953474.

[19]

D. J. Kaup, On the inverse scattering problem for cubic eigenvalue problems, Studies in Applied Mathematics, 62 (1980), 189-216.  doi: 10.1002/sapm1980623189.

[20]

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986. doi: 10.1007/978-1-4684-0274-2.

[21]

J. Satsuma and R. Hirota, A coupled kdv equation is one case of the four-reduction of the kp hierarchy, Journal of the Physical Society of Japan, 51 (1982), 3390-3397.  doi: 10.1143/JPSJ.51.3390.

[22]

K. Sawada and T. Kotera, A method for finding n-soliton solutions of the k.d.v. equation and k.d.v. like equation, Progress of theoretical physics, 51 (1974), 1355-1367.  doi: 10.1143/PTP.51.1355.

[23]

J. WeissM. Tabor and G. Carnevale, The painlevé property for partial differential equations, Journal of Mathematical Physics, 24 (1983), 522-526.  doi: 10.1063/1.525721.

show all references

References:
[1]

S. C. Anco and G. W. Bluman, Direct construction of conservation laws from field equations, Physical Review Letters, 78 (1997), 2869-2873.  doi: 10.1103/PhysRevLett.78.2869.

[2]

S. C. Anco and G. W. Bluman, Direct construction method for conservation laws of partial differential equations part 2: General treatment, European Journal of Applied Mathematics, 13 (2002), 567-585.  doi: 10.1017/S0956792501004661.

[3]

S. C. Anco and G. W. Bluman, Direct constrution method for conservation laws of partial differential equations part 1: Examples of conservation law classifications, European Journal of Applied Mathematics, 13 (2002), 545-566.  doi: 10.1017/S0956792501004661.

[4]

S. C. Anco, Generalization of Noether’s theorem in modern form to non-variational partial differential equations, To appear in Fields Institute Communications: Recent progress and Modern Challen ges in Applied Mathematics, Modeling and Computational Science, 79 (2017), 119-182, arXiv: 1605.08734. doi: 10.1007/978-1-4939-6969-2_5.

[5]

G. W. Bluman and J. Cole, General similarity solution of the heat equation, Journal of Mathematics and Mechanics, 18 (1969), 1025-1042. 

[6]

G. W. Bluman and S. Kumei, On the remarkable nonlinear diffusion equation, Journal of Mathematical Physics, 21 (1980), 1019-1023.  doi: 10.1063/1.524550.

[7]

G. W. BlumanS. Kumei and G. J. Reid, New classes of symmetries for partial differential equations, Journal of Mathematics and Mechanics, 29 (1988), 806-811.  doi: 10.1063/1.527974.

[8]

M. S. Bruzón, M. L. Gandarias and J. Ramírez, Symmetry and Perturbation Theory, World Scientific Publising Company, 2005.

[9]

M. S. BruzónT. M. Garrido and R. de la Rosa, Conservation laws and exact solutions of a Generalized Benjamin-Bona-Mahony-Burgers equation, Chaos, Solitons & Fractals, 89 (2016), 578-583.  doi: 10.1016/j.chaos.2016.03.034.

[10]

P. J. CaudreyR. K. Dodd and J. D. Gibbon, New hierarchy of koterweg de vries equations, Proceedings of the Royal Society of London series A-mathematical physical and engineering sciences, 351 (1976), 407-422.  doi: 10.1098/rspa.1976.0149.

[11]

P. A. Clarkson, Nonclassical symmetry reductions of the boussinesq equation, Chaos, Solitons & Fractals, 5 (1995), 2261-2301.  doi: 10.1016/0960-0779(94)E0099-B.

[12]

P. A. Clarkson and T. J. Priestley, Symmetries of a generalised boussinesq equation, Institute of Mathematics and Statistics, University of Kent at Canterbury.

[13]

V. G. Drinfeld and V. V. Sokolov, Nonclassical symmetry reductions of the boussinesq equation, Doklady akademii nauk sssr, 258 (1981), 11-16. 

[14]

A. P. Fordy and J. Gibbons, Some remarkable nonlinear transformations, Physics Letters A, 75 (1980), p325. doi: 10.1016/0375-9601(80)90829-4.

[15]

M. L. Gandarias and M. S. Bruzón, Classical and nonclassical symmetries of a generalized boussinesq equation, Journal of Nonlinear Mathematical Physics, 5 (1998), 8-12.  doi: 10.2991/jnmp.1998.5.1.2.

[16]

T. M. GarridoA. A. KasatkinM. S. Bruzón and R. K. Gazizov, Lie symmetries and equivalence transformations for the barenblatt-gilman model, Journal of Computational and Applied Mathematics, 318 (2017), 253-258.  doi: 10.1016/j.cam.2016.09.023.

[17]

W. Hereman and B. Huard, symmgrp2009. max: A macsyma/maxima program for the calculation of lie point symmetries of large systems of differential equations, http://inside.mines.edu/whereman/software.html (2009).

[18]

A. Karasu-Kalkanli, A. Karasu, A. Sakovich, S. Sakovich and R. Turhan, A new integrable generalization of the korteweg de vries equation, Journal of Mathematical Physics, 49 (2008), 073516, 10 pp. doi: 10.1063/1.2953474.

[19]

D. J. Kaup, On the inverse scattering problem for cubic eigenvalue problems, Studies in Applied Mathematics, 62 (1980), 189-216.  doi: 10.1002/sapm1980623189.

[20]

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986. doi: 10.1007/978-1-4684-0274-2.

[21]

J. Satsuma and R. Hirota, A coupled kdv equation is one case of the four-reduction of the kp hierarchy, Journal of the Physical Society of Japan, 51 (1982), 3390-3397.  doi: 10.1143/JPSJ.51.3390.

[22]

K. Sawada and T. Kotera, A method for finding n-soliton solutions of the k.d.v. equation and k.d.v. like equation, Progress of theoretical physics, 51 (1974), 1355-1367.  doi: 10.1143/PTP.51.1355.

[23]

J. WeissM. Tabor and G. Carnevale, The painlevé property for partial differential equations, Journal of Mathematical Physics, 24 (1983), 522-526.  doi: 10.1063/1.525721.

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