July  2012, 11(4): 1421-1430. doi: 10.3934/cpaa.2012.11.1421

Integrating factors and conservation laws for some Camassa-Holm type equations

1. 

Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden, Sweden

Received  May 2011 Revised  June 2011 Published  January 2012

We classify all first-order integrating factors and the corresponding conservation laws for a class of Camassa-Holm type equations.
Citation: Marianna Euler, Norbert Euler. Integrating factors and conservation laws for some Camassa-Holm type equations. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1421-1430. doi: 10.3934/cpaa.2012.11.1421
References:
[1]

C. S. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations Part II: General treatment,, Euro. Jnl of Applied Mathematics, 13 (2002), 567.  doi: doi.org/10.1017/S0956792501004661.  Google Scholar

[2]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.  doi: doi.org/10.1103/PhysRevLett.71.1661.  Google Scholar

[3]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation,, J. Differential Equations, 141 (1997), 218.  doi: doi.org/10.1006/jdeq.1997.3333.  Google Scholar

[4]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475.  doi: doi.org/10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.  Google Scholar

[5]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165.  doi: doi.org/10.1007/s00205-008-0128-2.  Google Scholar

[6]

A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603.  doi: doi.org/10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.  Google Scholar

[7]

A. Degasperis, D. D. Holm and A. N. W. Hone, A New Integrable Equation with Peakon Solutions,, Theor. and Math. Phys., 133 (2002), 1463.  doi: doi.org/10.1023/A:1021186408422.  Google Scholar

[8]

N. Euler and M. Euler, A tree of linearisable second-order evolution equations by generalised hodograph transformations,, J. Nonlinear Math. Phys., 8 (2001), 342.  doi: doi.org/10.2991/jnmp.2001.8.3.3.  Google Scholar

[9]

N. Euler and M. Euler, On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations: Two linearisable hierarchies,, J. Nonlinear Math. Phys., 16 (2009), 489.  doi: doi.org/10.1142/S1402925109000509.  Google Scholar

[10]

N. Euler and M. Euler, Multipotentialisation and iterating-solution formulae: The Krichever-Novikov equation,, J. Nonlinear Math. Phys., 16 Suppl. (2009), 93.  doi: doi.org/10.1142/S1402925109000340.  Google Scholar

[11]

N. Euler and M. Euler, The converse problem for the multipotentialisation of evolution equations and systems,, J. Nonlinear Math. Phys., 18 Suppl. (2011), 77.  doi: doi.org/10.1142/S1402925111001295.  Google Scholar

[12]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys A, 41 (2008).   Google Scholar

[13]

N. H. Ibragimov, R. S. Khamitova and A. Valenti, Self-adjointness of a generalized Camassa-Holm equation,, arXiv: 1102.5719v2 [math-ph], (1102).   Google Scholar

[14]

R. S. Johnson and Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 457 (2002), 63.   Google Scholar

[15]

J. Lenells, A variational approach to the stability of periodic peakons,, J. Nonlinear Math. Phys., 11 (2004), 151.  doi: doi.org/10.2991/jnmp.2004.11.2.2.  Google Scholar

[16]

Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation,, Comm. Pure Appl. Math., 62 (2009), 125.   Google Scholar

[17]

V. Novikov, Generalisations of the Camassa-Holm equation,, J. Phys. A, 42 (2009).   Google Scholar

[18]

X. Wu, On the Cauchy problem for the periodic generalized Degasperis-Procesi equation,, J. Funct. Anal., 260 (2011), 1428.  doi: doi.org/10.1016/j.jfa.2010.10.014.  Google Scholar

show all references

References:
[1]

C. S. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations Part II: General treatment,, Euro. Jnl of Applied Mathematics, 13 (2002), 567.  doi: doi.org/10.1017/S0956792501004661.  Google Scholar

[2]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.  doi: doi.org/10.1103/PhysRevLett.71.1661.  Google Scholar

[3]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation,, J. Differential Equations, 141 (1997), 218.  doi: doi.org/10.1006/jdeq.1997.3333.  Google Scholar

[4]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475.  doi: doi.org/10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.  Google Scholar

[5]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165.  doi: doi.org/10.1007/s00205-008-0128-2.  Google Scholar

[6]

A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603.  doi: doi.org/10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.  Google Scholar

[7]

A. Degasperis, D. D. Holm and A. N. W. Hone, A New Integrable Equation with Peakon Solutions,, Theor. and Math. Phys., 133 (2002), 1463.  doi: doi.org/10.1023/A:1021186408422.  Google Scholar

[8]

N. Euler and M. Euler, A tree of linearisable second-order evolution equations by generalised hodograph transformations,, J. Nonlinear Math. Phys., 8 (2001), 342.  doi: doi.org/10.2991/jnmp.2001.8.3.3.  Google Scholar

[9]

N. Euler and M. Euler, On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations: Two linearisable hierarchies,, J. Nonlinear Math. Phys., 16 (2009), 489.  doi: doi.org/10.1142/S1402925109000509.  Google Scholar

[10]

N. Euler and M. Euler, Multipotentialisation and iterating-solution formulae: The Krichever-Novikov equation,, J. Nonlinear Math. Phys., 16 Suppl. (2009), 93.  doi: doi.org/10.1142/S1402925109000340.  Google Scholar

[11]

N. Euler and M. Euler, The converse problem for the multipotentialisation of evolution equations and systems,, J. Nonlinear Math. Phys., 18 Suppl. (2011), 77.  doi: doi.org/10.1142/S1402925111001295.  Google Scholar

[12]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys A, 41 (2008).   Google Scholar

[13]

N. H. Ibragimov, R. S. Khamitova and A. Valenti, Self-adjointness of a generalized Camassa-Holm equation,, arXiv: 1102.5719v2 [math-ph], (1102).   Google Scholar

[14]

R. S. Johnson and Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 457 (2002), 63.   Google Scholar

[15]

J. Lenells, A variational approach to the stability of periodic peakons,, J. Nonlinear Math. Phys., 11 (2004), 151.  doi: doi.org/10.2991/jnmp.2004.11.2.2.  Google Scholar

[16]

Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation,, Comm. Pure Appl. Math., 62 (2009), 125.   Google Scholar

[17]

V. Novikov, Generalisations of the Camassa-Holm equation,, J. Phys. A, 42 (2009).   Google Scholar

[18]

X. Wu, On the Cauchy problem for the periodic generalized Degasperis-Procesi equation,, J. Funct. Anal., 260 (2011), 1428.  doi: doi.org/10.1016/j.jfa.2010.10.014.  Google Scholar

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