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NLWE with a special scale invariant damping in odd space dimension
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). Google Scholar 
[2] 
S. C. Anco and G. Bluman, Direct construction of conservation laws from field equations,, Phys. Rev. Lett., 78 (1987), 2869. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[6] 
G. Bluman, A. Cheviakov and S. Anco, Applications of symmetry methods to partial differential equations,, Springer, (2010). Google Scholar 
[7] 
G. W. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations,, Springer, (2002). Google Scholar 
[8] 
G. W. Bluman and S. Kumei, Symmetries and Differential Equations,, Applied Mathematical Sciences, 81 (1989). Google Scholar 
[9] 
Y. Bozhkov, I. L. Freire and N. H. Ibragimov, Group analysis of the Novikov equation,, Comp. Appl. Math., 33 (2014), 193. Google Scholar 
[10] 
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. Google Scholar 
[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), 195. Google Scholar 
[12] 
P. L. da Silva and I. L. Freire, Strict selfadjointness and shallow water models,, (2013) , (2013). Google Scholar 
[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). Google Scholar 
[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. Google Scholar 
[15] 
A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,, Theor. Math. Phys., 133 (2002), 1463. Google Scholar 
[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). Google Scholar 
[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), 73. Google Scholar 
[18] 
A. N. W. Hone and J. P, Wang, Integrable peak on equations with cubic nonlinearities,, J. Phys. A: Math. Theor., 41 (2008). Google Scholar 
[19] 
C. S. Gardner, Kortewergde Vries equation and generalizations IV. The Kortewegde Vries equation as a Hamiltonian system,, J. Math. Phys., 12 (1971), 1548. Google Scholar 
[20] 
N. H. Ibragimov, Transformation groups applied to mathematical physics,, Translated from the Russian Mathematics and its Applications (Soviet Series), (1985). Google Scholar 
[21] 
N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations,, John Wiley and Sons, (1999). Google Scholar 
[22] 
N. H. Ibragimov, A new conservation theorem,, J. Math. Anal. Appl., 333 (2007), 311. Google Scholar 
[23] 
N. H. Ibragimov, R.S. Khamitova, A. Valenti, Selfadjointness of a generalized CamassaHolm equation,, Appl. Math. Comp., 218 (2011), 2579. Google Scholar 
[24] 
N. H. Ibragimov, Nonlinear selfadjointness and conservation laws,, J. Phys. A: Math. Theor., 44 (2011). Google Scholar 
[25] 
N. H. Ibragimov, Nonlinear selfadjointness in constructing conservation laws,, Archives of ALGA, 7/8 (2011), 1. Google Scholar 
[26] 
N. M. Ivanova and R. O. Popovych, Equivalence of conservation laws and equivalence of potential systems,, Int. J. Theor. Phys., 46 (2007), 2658. Google Scholar 
[27] 
Y. Mi, C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions,, J. Diff. Equ., 254 (2013), 961. Google Scholar 
[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. Google Scholar 
[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), 952. Google Scholar 
[30] 
R. M. Miura, Kortewegde Vries equation and generalizations. I. A remarkable explicit nonlinear transformation,, J. Math. Phys., 9 (1968), 1202. Google Scholar 
[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), 1204. Google Scholar 
[32] 
V. S. Novikov, Generalizations of the CamassaHolm equation,, J. Phys. A: Math. Theor., 42 (2009). Google Scholar 
[33] 
P. J. Olver, Euler operators and conservation laws of the BBM equation,, Math. Proc. Cambridge Phils. Soc., 85 (1979), 143. Google Scholar 
[34] 
P. J. Olver, Conservation laws and null divergences,, Math. Proc. Camb. Phil. Soc., 94 (1983), 529. Google Scholar 
[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. Google Scholar 
[36] 
P. J. Olver, Applications of Lie groups to differential equations,, Springer, (1986). Google Scholar 
[37] 
R. O. Popovych and N. Ivanova, Hierarchy of conservation laws of diffusionconvection equations,, J. Math. Phys., 46 (2005). Google Scholar 
[38] 
R. O. Popovych and A. M. Samoilenko, Local conservation laws of secondorder evolution equations,, J. Phys. A, 41 (2008). Google Scholar 
[39] 
R. O. Popovych and A. Sergyeyev, Conservation laws and normal forms of evolution equations,, Phys. Lett. A, 374 (2010), 2210. Google Scholar 
[40] 
A. M. Vinogradov, Local symmetries and conservation laws,, Acta Appl. Math., 2 (1984), 21. Google Scholar 
show all references
References:
[1] 
M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering,, Cambridge University Press, (1991). Google Scholar 
[2] 
S. C. Anco and G. Bluman, Direct construction of conservation laws from field equations,, Phys. Rev. Lett., 78 (1987), 2869. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[6] 
G. Bluman, A. Cheviakov and S. Anco, Applications of symmetry methods to partial differential equations,, Springer, (2010). Google Scholar 
[7] 
G. W. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations,, Springer, (2002). Google Scholar 
[8] 
G. W. Bluman and S. Kumei, Symmetries and Differential Equations,, Applied Mathematical Sciences, 81 (1989). Google Scholar 
[9] 
Y. Bozhkov, I. L. Freire and N. H. Ibragimov, Group analysis of the Novikov equation,, Comp. Appl. Math., 33 (2014), 193. Google Scholar 
[10] 
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. Google Scholar 
[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), 195. Google Scholar 
[12] 
P. L. da Silva and I. L. Freire, Strict selfadjointness and shallow water models,, (2013) , (2013). Google Scholar 
[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). Google Scholar 
[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. Google Scholar 
[15] 
A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,, Theor. Math. Phys., 133 (2002), 1463. Google Scholar 
[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). Google Scholar 
[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), 73. Google Scholar 
[18] 
A. N. W. Hone and J. P, Wang, Integrable peak on equations with cubic nonlinearities,, J. Phys. A: Math. Theor., 41 (2008). Google Scholar 
[19] 
C. S. Gardner, Kortewergde Vries equation and generalizations IV. The Kortewegde Vries equation as a Hamiltonian system,, J. Math. Phys., 12 (1971), 1548. Google Scholar 
[20] 
N. H. Ibragimov, Transformation groups applied to mathematical physics,, Translated from the Russian Mathematics and its Applications (Soviet Series), (1985). Google Scholar 
[21] 
N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations,, John Wiley and Sons, (1999). Google Scholar 
[22] 
N. H. Ibragimov, A new conservation theorem,, J. Math. Anal. Appl., 333 (2007), 311. Google Scholar 
[23] 
N. H. Ibragimov, R.S. Khamitova, A. Valenti, Selfadjointness of a generalized CamassaHolm equation,, Appl. Math. Comp., 218 (2011), 2579. Google Scholar 
[24] 
N. H. Ibragimov, Nonlinear selfadjointness and conservation laws,, J. Phys. A: Math. Theor., 44 (2011). Google Scholar 
[25] 
N. H. Ibragimov, Nonlinear selfadjointness in constructing conservation laws,, Archives of ALGA, 7/8 (2011), 1. Google Scholar 
[26] 
N. M. Ivanova and R. O. Popovych, Equivalence of conservation laws and equivalence of potential systems,, Int. J. Theor. Phys., 46 (2007), 2658. Google Scholar 
[27] 
Y. Mi, C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions,, J. Diff. Equ., 254 (2013), 961. Google Scholar 
[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. Google Scholar 
[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), 952. Google Scholar 
[30] 
R. M. Miura, Kortewegde Vries equation and generalizations. I. A remarkable explicit nonlinear transformation,, J. Math. Phys., 9 (1968), 1202. Google Scholar 
[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), 1204. Google Scholar 
[32] 
V. S. Novikov, Generalizations of the CamassaHolm equation,, J. Phys. A: Math. Theor., 42 (2009). Google Scholar 
[33] 
P. J. Olver, Euler operators and conservation laws of the BBM equation,, Math. Proc. Cambridge Phils. Soc., 85 (1979), 143. Google Scholar 
[34] 
P. J. Olver, Conservation laws and null divergences,, Math. Proc. Camb. Phil. Soc., 94 (1983), 529. Google Scholar 
[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. Google Scholar 
[36] 
P. J. Olver, Applications of Lie groups to differential equations,, Springer, (1986). Google Scholar 
[37] 
R. O. Popovych and N. Ivanova, Hierarchy of conservation laws of diffusionconvection equations,, J. Math. Phys., 46 (2005). Google Scholar 
[38] 
R. O. Popovych and A. M. Samoilenko, Local conservation laws of secondorder evolution equations,, J. Phys. A, 41 (2008). Google Scholar 
[39] 
R. O. Popovych and A. Sergyeyev, Conservation laws and normal forms of evolution equations,, Phys. Lett. A, 374 (2010), 2210. Google Scholar 
[40] 
A. M. Vinogradov, Local symmetries and conservation laws,, Acta Appl. Math., 2 (1984), 21. Google Scholar 
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