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An equation unifying both CamassaHolm 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 
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), 28692873. 
[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), 545566. 
[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), 566585. 
[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), 4778. 
[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), 193202. 
[10] 
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993),16611664. 
[11] 
P. A. Clarkson, E. L. Mansfield and T. J. Priestley, Symmetries of a class of nonlinear thirdorder partial differential equations, Math. Comput. Modelling., 25, (1997), 195212. 
[12] 
P. L. da Silva and I. L. Freire, Strict selfadjointness 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 selfadjointness, (work presented in the CNMACBrazil), 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), 161166, DOI: 10.1007/9783319123073_23. 
[15] 
A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys., 133, (2002), 14631474. 
[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, CamassaHolm, Kortewegde Vries5 and other asymptotically equivalent equations for shallow water waves, Fluid Dynamics Research, 333 (2003), 7395. 
[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, Kortewergde Vries equation and generalizations IV. The Kortewegde Vries equation as a Hamiltonian system, J. Math. Phys., 12, (1971), 15481551. 
[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), 311328. 
[23] 
N. H. Ibragimov, R.S. Khamitova, A. Valenti, Selfadjointness of a generalized CamassaHolm equation, Appl. Math. Comp., 218, (2011), 25792583. 
[24] 
N. H. Ibragimov, Nonlinear selfadjointness and conservation laws, J. Phys. A: Math. Theor., 44, (2011) 432002, 8 pp. 
[25] 
N. H. Ibragimov, Nonlinear selfadjointness in constructing conservation laws, Archives of ALGA, 7/8, (2011), 190. 
[26] 
N. M. Ivanova and R. O. Popovych, Equivalence of conservation laws and equivalence of potential systems, Int. J. Theor. Phys., 46, (2007), 26582668. 
[27] 
Y. Mi, C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions, J. Diff. Equ., 254, (2013), 961982. 
[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), 422443. 
[29] 
M. D. Kruskal, R. M. Miura, C. S. Gardner and N. J. Zabusky, Kortewegde Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws, J. Math. Phys., 11, (1970),952960. 
[30] 
R. M. Miura, Kortewegde Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9, (1968), 12021204. 
[31] 
R. M. Miura, C. S. Gardner and M. D. Kruskal, Kortewegde Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys., 9, (1968) 12041209. 
[32] 
V. S. Novikov, Generalizations of the CamassaHolm 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), 143160. 
[34] 
P. J. Olver, Conservation laws and null divergences, Math. Proc. Camb. Phil. Soc., 94, (1983), 529540. 
[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), 353380. 
[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 diffusionconvection equations, J. Math. Phys., 46, (2005), 43502. 
[38] 
R. O. Popovych and A. M. Samoilenko, Local conservation laws of secondorder 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), 22102217. 
[40] 
A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., 2, (1984), 2178. 
show all references
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), 28692873. 
[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), 545566. 
[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), 566585. 
[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), 4778. 
[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), 193202. 
[10] 
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993),16611664. 
[11] 
P. A. Clarkson, E. L. Mansfield and T. J. Priestley, Symmetries of a class of nonlinear thirdorder partial differential equations, Math. Comput. Modelling., 25, (1997), 195212. 
[12] 
P. L. da Silva and I. L. Freire, Strict selfadjointness 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 selfadjointness, (work presented in the CNMACBrazil), 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), 161166, DOI: 10.1007/9783319123073_23. 
[15] 
A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys., 133, (2002), 14631474. 
[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, CamassaHolm, Kortewegde Vries5 and other asymptotically equivalent equations for shallow water waves, Fluid Dynamics Research, 333 (2003), 7395. 
[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, Kortewergde Vries equation and generalizations IV. The Kortewegde Vries equation as a Hamiltonian system, J. Math. Phys., 12, (1971), 15481551. 
[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), 311328. 
[23] 
N. H. Ibragimov, R.S. Khamitova, A. Valenti, Selfadjointness of a generalized CamassaHolm equation, Appl. Math. Comp., 218, (2011), 25792583. 
[24] 
N. H. Ibragimov, Nonlinear selfadjointness and conservation laws, J. Phys. A: Math. Theor., 44, (2011) 432002, 8 pp. 
[25] 
N. H. Ibragimov, Nonlinear selfadjointness in constructing conservation laws, Archives of ALGA, 7/8, (2011), 190. 
[26] 
N. M. Ivanova and R. O. Popovych, Equivalence of conservation laws and equivalence of potential systems, Int. J. Theor. Phys., 46, (2007), 26582668. 
[27] 
Y. Mi, C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions, J. Diff. Equ., 254, (2013), 961982. 
[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), 422443. 
[29] 
M. D. Kruskal, R. M. Miura, C. S. Gardner and N. J. Zabusky, Kortewegde Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws, J. Math. Phys., 11, (1970),952960. 
[30] 
R. M. Miura, Kortewegde Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9, (1968), 12021204. 
[31] 
R. M. Miura, C. S. Gardner and M. D. Kruskal, Kortewegde Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys., 9, (1968) 12041209. 
[32] 
V. S. Novikov, Generalizations of the CamassaHolm 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), 143160. 
[34] 
P. J. Olver, Conservation laws and null divergences, Math. Proc. Camb. Phil. Soc., 94, (1983), 529540. 
[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), 353380. 
[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 diffusionconvection equations, J. Math. Phys., 46, (2005), 43502. 
[38] 
R. O. Popovych and A. M. Samoilenko, Local conservation laws of secondorder 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), 22102217. 
[40] 
A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., 2, (1984), 2178. 
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