Integrating factors and conservation laws forsome Camassa-Holm type equations

Marianna Euler; Norbert Euler

doi:10.3934/cpaa.2012.11.1421

Article Contents

2012, Volume 11, Issue 4: 1421-1430. Doi: 10.3934/cpaa.2012.11.1421

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

Marianna Euler^1, and
Norbert Euler^1,

1.
Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå

Received: May 2011

Revised: June 2011

Published: July 2012

Abstract / Introduction Related Papers Cited by

Abstract

We classify all first-order integrating factors and the corresponding conservation laws for a class of Camassa-Holm type equations.

Keywords:

Mathematics Subject Classification: 35Q35, 76B15.

Citation:

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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), 567-585.doi: doi.org/10.1017/S0956792501004661.
[2]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.doi: doi.org/10.1103/PhysRevLett.71.1661.
[3]	A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141 (1997), 218-235.doi: doi.org/10.1006/jdeq.1997.3333.
[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), 475-504.doi: doi.org/10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.
[5]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.doi: doi.org/10.1007/s00205-008-0128-2.
[6]	A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.doi: doi.org/10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.
[7]	A. Degasperis, D. D. Holm and A. N. W. Hone, A New Integrable Equation with Peakon Solutions, Theor. and Math. Phys., 133 (2002), 1463-1474.doi: doi.org/10.1023/A:1021186408422.
[8]	N. Euler and M. Euler, A tree of linearisable second-order evolution equations by generalised hodograph transformations, J. Nonlinear Math. Phys., 8 (2001), 342-362.doi: doi.org/10.2991/jnmp.2001.8.3.3.
[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), 489-504.doi: doi.org/10.1142/S1402925109000509.
[10]	N. Euler and M. Euler, Multipotentialisation and iterating-solution formulae: The Krichever-Novikov equation, J. Nonlinear Math. Phys., 16 Suppl. (2009), 93-106.doi: doi.org/10.1142/S1402925109000340.
[11]	N. Euler and M. Euler, The converse problem for the multipotentialisation of evolution equations and systems, J. Nonlinear Math. Phys., 18 Suppl. (2011), 77-105.doi: doi.org/10.1142/S1402925111001295.
[12]	A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys A, 41 (2008), 372002, 10 pp.
[13]	N. H. Ibragimov, R. S. Khamitova and A. Valenti, Self-adjointness of a generalized Camassa-Holm equation, arXiv: 1102.5719v2 [math-ph], April 1, 2011
[14]	R. S. Johnson and Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82.
[15]	J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163.doi: doi.org/10.2991/jnmp.2004.11.2.2.
[16]	Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.
[17]	V. Novikov, Generalisations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14 pp.
[18]	X. Wu, On the Cauchy problem for the periodic generalized Degasperis-Procesi equation, J. Funct. Anal., 260 (2011), 1428-1445.doi: doi.org/10.1016/j.jfa.2010.10.014.