# American Institute of Mathematical Sciences

2015, 2015(special): 304-311. doi: 10.3934/proc.2015.0304

## An equation unifying both Camassa-Holm and Novikov equations

 1 Centro de Matemática, Computação e Cognição, Universidade Federal do ABC - UFABC, Rua Santa Adélia, 166, Bairro Bangu, 09.210 -- 170, Santo André, SP, Brazil, Brazil

Received  September 2014 Revised  January 2015 Published  November 2015

In this paper we derive a new equation unifying the Camassa-Holm and Novikov equations invariant under the scaling transformation $(x,t,u)\mapsto(x,\lambda^{-b}t,\lambda u)$ and admitting a certain multiplier.
Citation: Priscila Leal da Silva, Igor Leite Freire. An equation unifying both Camassa-Holm and Novikov equations. Conference Publications, 2015, 2015 (special) : 304-311. doi: 10.3934/proc.2015.0304
##### References:
 [1] M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering,, Cambridge University Press, (1991). [2] S. C. Anco and G. Bluman, Direct construction of conservation laws from field equations,, Phys. Rev. Lett., 78 (1987), 2869. [3] S. C. Anco, and G. Bluman, Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications,, European J. Appl. Math., 13 (2002), 545. [4] S. C. Anco, and G. Bluman, Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications,, European J. Appl. Math., 13 (2002), 566. [5] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Roy. Soc. London, 272 (1972), 47. [6] G. Bluman, A. Cheviakov and S. Anco, Applications of symmetry methods to partial differential equations,, Springer, (2010). [7] G. W. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations,, Springer, (2002). [8] G. W. Bluman and S. Kumei, Symmetries and Differential Equations,, Applied Mathematical Sciences, 81 (1989). [9] Y. Bozhkov, I. L. Freire and N. H. Ibragimov, Group analysis of the Novikov equation,, Comp. Appl. Math., 33 (2014), 193. [10] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. [11] P. A. Clarkson, E. L. Mansfield and T. J. Priestley, Symmetries of a class of nonlinear third-order partial differential equations,, Math. Comput. Modelling., 25 (1997), 195. [12] P. L. da Silva and I. L. Freire, Strict self-adjointness and shallow water models,, (2013) , (2013). [13] P. L. da Silva and I. L. Freire, On certain shallow water models, scaling invariance and strict self-adjointness, (work presented in the CNMAC-Brazil),, Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, (2015). [14] P. L. da Silva and I. L. Freire, On the group analysis of a modified Novikov equation,, in Interdisciplinary Topics in Applied Mathematics, 117 (2015), 161. [15] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,, Theor. Math. Phys., 133 (2002), 1463. [16] R. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion,, Phys. Rev. Lett., 87 (2001). [17] H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid Dynamics Research, 333 (2003), 73. [18] A. N. W. Hone and J. P, Wang, Integrable peak on equations with cubic nonlinearities,, J. Phys. A: Math. Theor., 41 (2008). [19] C. S. Gardner, Kortewerg-de Vries equation and generalizations IV. The Korteweg-de Vries equation as a Hamiltonian system,, J. Math. Phys., 12 (1971), 1548. [20] N. H. Ibragimov, Transformation groups applied to mathematical physics,, Translated from the Russian Mathematics and its Applications (Soviet Series), (1985). [21] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations,, John Wiley and Sons, (1999). [22] N. H. Ibragimov, A new conservation theorem,, J. Math. Anal. Appl., 333 (2007), 311. [23] N. H. Ibragimov, R.S. Khamitova, A. Valenti, Self-adjointness of a generalized Camassa-Holm equation,, Appl. Math. Comp., 218 (2011), 2579. [24] N. H. Ibragimov, Nonlinear self-adjointness and conservation laws,, J. Phys. A: Math. Theor., 44 (2011). [25] N. H. Ibragimov, Nonlinear self-adjointness in constructing conservation laws,, Archives of ALGA, 7/8 (2011), 1. [26] N. M. Ivanova and R. O. Popovych, Equivalence of conservation laws and equivalence of potential systems,, Int. J. Theor. Phys., 46 (2007), 2658. [27] Y. Mi, C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions,, J. Diff. Equ., 254 (2013), 961. [28] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,, Phil. Mag., 39 (1895), 422. [29] M. D. Kruskal, R. M. Miura, C. S. Gardner and N. J. Zabusky, Korteweg-de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws,, J. Math. Phys., 11 (1970), 952. [30] R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation,, J. Math. Phys., 9 (1968), 1202. [31] R. M. Miura, C. S. Gardner and M. D. Kruskal, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion,, J. Math. Phys., 9 (1968), 1204. [32] V. S. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A: Math. Theor., 42 (2009). [33] P. J. Olver, Euler operators and conservation laws of the BBM equation,, Math. Proc. Cambridge Phils. Soc., 85 (1979), 143. [34] P. J. Olver, Conservation laws and null divergences,, Math. Proc. Camb. Phil. Soc., 94 (1983), 529. [35] P. J. Olver, Conservation laws of free boundary problems and the classification of conservation laws for water waves,, Trans. Amer. Math. Soc., 277 (1983), 353. [36] P. J. Olver, Applications of Lie groups to differential equations,, Springer, (1986). [37] R. O. Popovych and N. Ivanova, Hierarchy of conservation laws of diffusion-convection equations,, J. Math. Phys., 46 (2005). [38] R. O. Popovych and A. M. Samoilenko, Local conservation laws of second-order evolution equations,, J. Phys. A, 41 (2008). [39] R. O. Popovych and A. Sergyeyev, Conservation laws and normal forms of evolution equations,, Phys. Lett. A, 374 (2010), 2210. [40] A. M. Vinogradov, Local symmetries and conservation laws,, Acta Appl. Math., 2 (1984), 21.

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##### References:
 [1] M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering,, Cambridge University Press, (1991). [2] S. C. Anco and G. Bluman, Direct construction of conservation laws from field equations,, Phys. Rev. Lett., 78 (1987), 2869. [3] S. C. Anco, and G. Bluman, Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications,, European J. Appl. Math., 13 (2002), 545. [4] S. C. Anco, and G. Bluman, Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications,, European J. Appl. Math., 13 (2002), 566. [5] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Roy. Soc. London, 272 (1972), 47. [6] G. Bluman, A. Cheviakov and S. Anco, Applications of symmetry methods to partial differential equations,, Springer, (2010). [7] G. W. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations,, Springer, (2002). [8] G. W. Bluman and S. Kumei, Symmetries and Differential Equations,, Applied Mathematical Sciences, 81 (1989). [9] Y. Bozhkov, I. L. Freire and N. H. Ibragimov, Group analysis of the Novikov equation,, Comp. Appl. Math., 33 (2014), 193. [10] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. [11] P. A. Clarkson, E. L. Mansfield and T. J. Priestley, Symmetries of a class of nonlinear third-order partial differential equations,, Math. Comput. Modelling., 25 (1997), 195. [12] P. L. da Silva and I. L. Freire, Strict self-adjointness and shallow water models,, (2013) , (2013). [13] P. L. da Silva and I. L. Freire, On certain shallow water models, scaling invariance and strict self-adjointness, (work presented in the CNMAC-Brazil),, Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, (2015). [14] P. L. da Silva and I. L. Freire, On the group analysis of a modified Novikov equation,, in Interdisciplinary Topics in Applied Mathematics, 117 (2015), 161. [15] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,, Theor. Math. Phys., 133 (2002), 1463. [16] R. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion,, Phys. Rev. Lett., 87 (2001). [17] H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid Dynamics Research, 333 (2003), 73. [18] A. N. W. Hone and J. P, Wang, Integrable peak on equations with cubic nonlinearities,, J. Phys. A: Math. Theor., 41 (2008). [19] C. S. Gardner, Kortewerg-de Vries equation and generalizations IV. The Korteweg-de Vries equation as a Hamiltonian system,, J. Math. Phys., 12 (1971), 1548. [20] N. H. Ibragimov, Transformation groups applied to mathematical physics,, Translated from the Russian Mathematics and its Applications (Soviet Series), (1985). [21] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations,, John Wiley and Sons, (1999). [22] N. H. Ibragimov, A new conservation theorem,, J. Math. Anal. Appl., 333 (2007), 311. [23] N. H. Ibragimov, R.S. Khamitova, A. Valenti, Self-adjointness of a generalized Camassa-Holm equation,, Appl. Math. Comp., 218 (2011), 2579. [24] N. H. Ibragimov, Nonlinear self-adjointness and conservation laws,, J. Phys. A: Math. Theor., 44 (2011). [25] N. H. Ibragimov, Nonlinear self-adjointness in constructing conservation laws,, Archives of ALGA, 7/8 (2011), 1. [26] N. M. Ivanova and R. O. Popovych, Equivalence of conservation laws and equivalence of potential systems,, Int. J. Theor. Phys., 46 (2007), 2658. [27] Y. Mi, C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions,, J. Diff. Equ., 254 (2013), 961. [28] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,, Phil. Mag., 39 (1895), 422. [29] M. D. Kruskal, R. M. Miura, C. S. Gardner and N. J. Zabusky, Korteweg-de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws,, J. Math. Phys., 11 (1970), 952. [30] R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation,, J. Math. Phys., 9 (1968), 1202. [31] R. M. Miura, C. S. Gardner and M. D. Kruskal, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion,, J. Math. Phys., 9 (1968), 1204. [32] V. S. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A: Math. Theor., 42 (2009). [33] P. J. Olver, Euler operators and conservation laws of the BBM equation,, Math. Proc. Cambridge Phils. Soc., 85 (1979), 143. [34] P. J. Olver, Conservation laws and null divergences,, Math. Proc. Camb. Phil. Soc., 94 (1983), 529. [35] P. J. Olver, Conservation laws of free boundary problems and the classification of conservation laws for water waves,, Trans. Amer. Math. Soc., 277 (1983), 353. [36] P. J. Olver, Applications of Lie groups to differential equations,, Springer, (1986). [37] R. O. Popovych and N. Ivanova, Hierarchy of conservation laws of diffusion-convection equations,, J. Math. Phys., 46 (2005). [38] R. O. Popovych and A. M. Samoilenko, Local conservation laws of second-order evolution equations,, J. Phys. A, 41 (2008). [39] R. O. Popovych and A. Sergyeyev, Conservation laws and normal forms of evolution equations,, Phys. Lett. A, 374 (2010), 2210. [40] A. M. Vinogradov, Local symmetries and conservation laws,, Acta Appl. Math., 2 (1984), 21.
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