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 TMF, 2003, Volume 137, Number 1, Pages 121–136 (Mi tmf250)

Nonintegrability of a Fifth-Order Equation with Integrable Two-Body Dynamics

D. D. Holma, A. Honeb

a Los Alamos National Laboratory
b University of Kent

Abstract: We consider a fifth-order partial differential equation (PDE) that is a generalization of the integrable Camassa–Holm equation. This fifth-order PDE has exact solutions in terms of an arbitrary number of superposed pulsons with a geodesic Hamiltonian dynamics that is known to be integrable in the two-body case $N=2$. Numerical simulations show that the pulsons are stable, dominate the initial value problem, and scatter elastically. These characteristics are reminiscent of solitons in integrable systems. But after demonstrating the nonexistence of a suitable Lagrangian or bi-Hamiltonian structure and obtaining negative results from Painlevé analysis and the Wahlquist–Estabrook method, we assert that this fifth-order PDE is not integrable.

Keywords: Hamiltonian dynamics, nonintegrability, elastic scattering, pulsons

DOI: https://doi.org/10.4213/tmf250

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English version:
Theoretical and Mathematical Physics, 2003, 137:1, 1459–1471

Bibliographic databases:

Citation: D. D. Holm, A. Hone, “Nonintegrability of a Fifth-Order Equation with Integrable Two-Body Dynamics”, TMF, 137:1 (2003), 121–136; Theoret. and Math. Phys., 137:1 (2003), 1459–1471

Citation in format AMSBIB
\Bibitem{HolHon03} \by D.~D.~Holm, A.~Hone \paper Nonintegrability of a Fifth-Order Equation with Integrable Two-Body Dynamics \jour TMF \yr 2003 \vol 137 \issue 1 \pages 121--136 \mathnet{http://mi.mathnet.ru/tmf250} \crossref{https://doi.org/10.4213/tmf250} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2048095} \zmath{https://zbmath.org/?q=an:1178.37104} \elib{http://elibrary.ru/item.asp?id=14421838} \transl \jour Theoret. and Math. Phys. \yr 2003 \vol 137 \issue 1 \pages 1459--1471 \crossref{https://doi.org/10.1023/A:1026060924520} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000186557700012} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Holm DD, Hone ANW, “A class of equations with peakon and pulson solutions (with an appendix by Harry Braden and John Byatt-Smith)”, Journal of Nonlinear Mathematical Physics, 12 (2005), 380–394, Suppl. 1
2. Ivanov, RI, “Water waves and integrability”, Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 365:1858 (2007), 2267
3. S. P. Popov, “Application of the quasi-spectral fourier method to soliton equations”, Comput. Math. Math. Phys., 50:12 (2010), 2064–2070
4. S. P. Popov, “Numerical study of Peakons and $k$-Solitons of the Camassa–Holm and Holm–Hone equations”, Comput. Math. Math. Phys., 51:7 (2011), 1231–1238
5. Han L. Cui W., “Infinite Propagation Speed and Asymptotic Behavior For a Generalized Fifth-Order Camassa-Holm Equation”, Appl. Anal., 98:3 (2019), 536–552
6. Wang G., Yong X., Huang Y., Tian J., “Symmetry, Pulson Solution, and Conservation Laws of the Holm-Hone Equation”, Adv. Math. Phys., 2019, 4364108
7. Zhang Yu., Liu Q., Qiao Zh., “Fifth-Order B-Family Novikov (Fobfn) Model With Pseudo-Peakons and Multi-Peakons”, Mod. Phys. Lett. B, 33:18 (2019), 1950205
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