# 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), 2869-2873. [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), 545-566. [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), 566-585. [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), 47-78. [6] G. Bluman, A. Cheviakov and S. Anco, Applications of symmetry methods to partial differential equations, Springer, New York, (2010). [7] G. W. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, (2002). [8] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences, 81, Springer, New York, (1989). [9] Y. Bozhkov, I. L. Freire and N. H. Ibragimov, Group analysis of the Novikov equation, Comp. Appl. Math., 33, (2014), 193-202. [10] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993),1661-1664. [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), 195-212. [12] P. L. da Silva and I. L. Freire, Strict self-adjointness and shallow water models, (2013) arXiv:1312.3992. [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), DOI: 10.5540/03.2015.003.01.0022. [14] P. L. da Silva and I. L. Freire, On the group analysis of a modified Novikov equation, in Interdisciplinary Topics in Applied Mathematics, Modelling and Computational Science. Springer Proceedings in Mathematics and Statistics, 117 (2015), 161-166, DOI: 10.1007/978-3-319-12307-3_23. [15] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys., 133, (2002), 1463-1474. [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), 194501, 4pp. [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), 73-95. [18] A. N. W. Hone and J. P, Wang, Integrable peak on equations with cubic nonlinearities, J. Phys. A: Math. Theor., 41, (2008), 372002, 10 pp. [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), 1548-1551. [20] N. H. Ibragimov, Transformation groups applied to mathematical physics, Translated from the Russian Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, (1985). [21] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley and Sons, Chirchester (1999). [22] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333, (2007), 311-328. [23] N. H. Ibragimov, R.S. Khamitova, A. Valenti, Self-adjointness of a generalized Camassa-Holm equation, Appl. Math. Comp., 218, (2011), 2579-2583. [24] N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor., 44, (2011) 432002, 8 pp. [25] N. H. Ibragimov, Nonlinear self-adjointness in constructing conservation laws, Archives of ALGA, 7/8, (2011), 1-90. [26] N. M. Ivanova and R. O. Popovych, Equivalence of conservation laws and equivalence of potential systems, Int. J. Theor. Phys., 46, (2007), 2658-2668. [27] Y. Mi, C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions, J. Diff. Equ., 254, (2013), 961-982. [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), 422-443. [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),952-960. [30] R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9, (1968), 1202-1204. [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) 1204-1209. [32] V. S. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42, (2009) 342002, 14pp. [33] P. J. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Cambridge Phils. Soc., 85, (1979), 143-160. [34] P. J. Olver, Conservation laws and null divergences, Math. Proc. Camb. Phil. Soc., 94, (1983), 529-540. [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), 353-380. [36] P. J. Olver, Applications of Lie groups to differential equations, Springer, New York, (1986). [37] R. O. Popovych and N. Ivanova, Hierarchy of conservation laws of diffusion-convection equations, J. Math. Phys., 46, (2005), 43502. [38] R. O. Popovych and A. M. Samoilenko, Local conservation laws of second-order evolution equations, J. Phys. A, 41, (2008), 362002. [39] R. O. Popovych and A. Sergyeyev, Conservation laws and normal forms of evolution equations, Phys. Lett. A, 374, (2010), 2210-2217. [40] A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., 2, (1984), 21-78.

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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), 2869-2873. [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), 545-566. [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), 566-585. [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), 47-78. [6] G. Bluman, A. Cheviakov and S. Anco, Applications of symmetry methods to partial differential equations, Springer, New York, (2010). [7] G. W. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, (2002). [8] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences, 81, Springer, New York, (1989). [9] Y. Bozhkov, I. L. Freire and N. H. Ibragimov, Group analysis of the Novikov equation, Comp. Appl. Math., 33, (2014), 193-202. [10] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993),1661-1664. [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), 195-212. [12] P. L. da Silva and I. L. Freire, Strict self-adjointness and shallow water models, (2013) arXiv:1312.3992. [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), DOI: 10.5540/03.2015.003.01.0022. [14] P. L. da Silva and I. L. Freire, On the group analysis of a modified Novikov equation, in Interdisciplinary Topics in Applied Mathematics, Modelling and Computational Science. Springer Proceedings in Mathematics and Statistics, 117 (2015), 161-166, DOI: 10.1007/978-3-319-12307-3_23. [15] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys., 133, (2002), 1463-1474. [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), 194501, 4pp. [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), 73-95. [18] A. N. W. Hone and J. P, Wang, Integrable peak on equations with cubic nonlinearities, J. Phys. A: Math. Theor., 41, (2008), 372002, 10 pp. [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), 1548-1551. [20] N. H. Ibragimov, Transformation groups applied to mathematical physics, Translated from the Russian Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, (1985). [21] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley and Sons, Chirchester (1999). [22] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333, (2007), 311-328. [23] N. H. Ibragimov, R.S. Khamitova, A. Valenti, Self-adjointness of a generalized Camassa-Holm equation, Appl. Math. Comp., 218, (2011), 2579-2583. [24] N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor., 44, (2011) 432002, 8 pp. [25] N. H. Ibragimov, Nonlinear self-adjointness in constructing conservation laws, Archives of ALGA, 7/8, (2011), 1-90. [26] N. M. Ivanova and R. O. Popovych, Equivalence of conservation laws and equivalence of potential systems, Int. J. Theor. Phys., 46, (2007), 2658-2668. [27] Y. Mi, C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions, J. Diff. Equ., 254, (2013), 961-982. [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), 422-443. [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),952-960. [30] R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9, (1968), 1202-1204. [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) 1204-1209. [32] V. S. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42, (2009) 342002, 14pp. [33] P. J. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Cambridge Phils. Soc., 85, (1979), 143-160. [34] P. J. Olver, Conservation laws and null divergences, Math. Proc. Camb. Phil. Soc., 94, (1983), 529-540. [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), 353-380. [36] P. J. Olver, Applications of Lie groups to differential equations, Springer, New York, (1986). [37] R. O. Popovych and N. Ivanova, Hierarchy of conservation laws of diffusion-convection equations, J. Math. Phys., 46, (2005), 43502. [38] R. O. Popovych and A. M. Samoilenko, Local conservation laws of second-order evolution equations, J. Phys. A, 41, (2008), 362002. [39] R. O. Popovych and A. Sergyeyev, Conservation laws and normal forms of evolution equations, Phys. Lett. A, 374, (2010), 2210-2217. [40] A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., 2, (1984), 21-78.
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