
Previous Article
Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape
 PROC Home
 This Issue

Next Article
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). 
[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 thirdorder partial differential equations,, Math. Comput. Modelling., 25 (1997), 195. 
[12] 
P. L. da Silva and I. L. Freire, Strict selfadjointness and shallow water models,, (2013) , (2013). 
[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). 
[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, CamassaHolm, Kortewegde Vries5 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, Kortewergde Vries equation and generalizations IV. The Kortewegde 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, Selfadjointness of a generalized CamassaHolm equation,, Appl. Math. Comp., 218 (2011), 2579. 
[24] 
N. H. Ibragimov, Nonlinear selfadjointness and conservation laws,, J. Phys. A: Math. Theor., 44 (2011). 
[25] 
N. H. Ibragimov, Nonlinear selfadjointness 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, Kortewegde Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws,, J. Math. Phys., 11 (1970), 952. 
[30] 
R. M. Miura, Kortewegde 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, Kortewegde 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 CamassaHolm 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 diffusionconvection equations,, J. Math. Phys., 46 (2005). 
[38] 
R. O. Popovych and A. M. Samoilenko, Local conservation laws of secondorder 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. 
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), 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 thirdorder partial differential equations,, Math. Comput. Modelling., 25 (1997), 195. 
[12] 
P. L. da Silva and I. L. Freire, Strict selfadjointness and shallow water models,, (2013) , (2013). 
[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). 
[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, CamassaHolm, Kortewegde Vries5 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, Kortewergde Vries equation and generalizations IV. The Kortewegde 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, Selfadjointness of a generalized CamassaHolm equation,, Appl. Math. Comp., 218 (2011), 2579. 
[24] 
N. H. Ibragimov, Nonlinear selfadjointness and conservation laws,, J. Phys. A: Math. Theor., 44 (2011). 
[25] 
N. H. Ibragimov, Nonlinear selfadjointness 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, Kortewegde Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws,, J. Math. Phys., 11 (1970), 952. 
[30] 
R. M. Miura, Kortewegde 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, Kortewegde 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 CamassaHolm 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 diffusionconvection equations,, J. Math. Phys., 46 (2005). 
[38] 
R. O. Popovych and A. M. Samoilenko, Local conservation laws of secondorder 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. 
[1] 
Jae Min Lee, Stephen C. Preston. Local wellposedness of the CamassaHolm equation on the real line. Discrete & Continuous Dynamical Systems  A, 2017, 37 (6) : 32853299. doi: 10.3934/dcds.2017139 
[2] 
Marianna Euler, Norbert Euler. Integrating factors and conservation laws for some CamassaHolm type equations. Communications on Pure & Applied Analysis, 2012, 11 (4) : 14211430. doi: 10.3934/cpaa.2012.11.1421 
[3] 
Giuseppe Maria Coclite, Lorenzo Di Ruvo. A note on the convergence of the solution of the high order CamassaHolm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems  A, 2017, 37 (3) : 12471282. doi: 10.3934/dcds.2017052 
[4] 
Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solutions of the CamassaHolm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems  A, 2016, 36 (6) : 29812990. doi: 10.3934/dcds.2016.36.2981 
[5] 
Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$Component CamassaHolm equation with peakons. Discrete & Continuous Dynamical Systems  A, 2017, 37 (3) : 15751601. doi: 10.3934/dcds.2017065 
[6] 
Helge Holden, Xavier Raynaud. Dissipative solutions for the CamassaHolm equation. Discrete & Continuous Dynamical Systems  A, 2009, 24 (4) : 10471112. doi: 10.3934/dcds.2009.24.1047 
[7] 
Zhenhua Guo, Mina Jiang, Zhian Wang, GaoFeng Zheng. Global weak solutions to the CamassaHolm equation. Discrete & Continuous Dynamical Systems  A, 2008, 21 (3) : 883906. doi: 10.3934/dcds.2008.21.883 
[8] 
Milena Stanislavova, Atanas Stefanov. Attractors for the viscous CamassaHolm equation. Discrete & Continuous Dynamical Systems  A, 2007, 18 (1) : 159186. doi: 10.3934/dcds.2007.18.159 
[9] 
Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized CamassaHolm equation. Discrete & Continuous Dynamical Systems  A, 2015, 35 (3) : 871889. doi: 10.3934/dcds.2015.35.871 
[10] 
Yu Gao, JianGuo Liu. The modified CamassaHolm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems  B, 2018, 23 (6) : 25452592. doi: 10.3934/dcdsb.2018067 
[11] 
Xi Tu, Zhaoyang Yin. Local wellposedness and blowup phenomena for a generalized CamassaHolm equation with peakon solutions. Discrete & Continuous Dynamical Systems  A, 2016, 36 (5) : 27812801. doi: 10.3934/dcds.2016.36.2781 
[12] 
Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the CamassaHolm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 2937. doi: 10.3934/proc.2015.0029 
[13] 
Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized CamassaHolm equation via characteristics. Discrete & Continuous Dynamical Systems  A, 2018, 38 (10) : 52055220. doi: 10.3934/dcds.2018230 
[14] 
Yongsheng Mi, Chunlai Mu. On a threeComponent CamassaHolm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305339. doi: 10.3934/krm.2014.7.305 
[15] 
Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized CamassaHolm equation. Discrete & Continuous Dynamical Systems  A, 2013, 33 (4) : 17131739. doi: 10.3934/dcds.2013.33.1713 
[16] 
Shihui Zhu. Existence and uniqueness of global weak solutions of the CamassaHolm equation with a forcing. Discrete & Continuous Dynamical Systems  A, 2016, 36 (9) : 52015221. doi: 10.3934/dcds.2016026 
[17] 
Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higherorder μCamassaHolm equation. Discrete & Continuous Dynamical Systems  A, 2018, 38 (8) : 41634187. doi: 10.3934/dcds.2018181 
[18] 
Danping Ding, Lixin Tian, Gang Xu. The study on solutions to CamassaHolm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483492. doi: 10.3934/cpaa.2006.5.483 
[19] 
Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified CamassaHolm equation. Discrete & Continuous Dynamical Systems  A, 2018, 38 (9) : 44494465. doi: 10.3934/dcds.2018194 
[20] 
David F. Parker. Higherorder shallow water equations and the CamassaHolm equation. Discrete & Continuous Dynamical Systems  B, 2007, 7 (3) : 629641. doi: 10.3934/dcdsb.2007.7.629 
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]