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

This article is cited in 7 scientific papers (total in 7 papers)

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

Full text: PDF file (1045 kB)
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English version:
Theoretical and Mathematical Physics, 2003, 137:1, 1459–1471

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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
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    2. Ivanov, RI, “Water waves and integrability”, Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 365:1858 (2007), 2267  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    3. S. P. Popov, “Application of the quasi-spectral fourier method to soliton equations”, Comput. Math. Math. Phys., 50:12 (2010), 2064–2070  mathnet  crossref  adsnasa
    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  mathnet  crossref  isi
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  isi  scopus
    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  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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