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A nonlinearly dispersive fifth order integrable equation and its hierarchy. (English) Zbl 1067.35118

Summary: We study the properties of a nonlinearly dispersive integrable system of fifth order and its associated hierarchy. We describe a Lax representation for such a system which leads to two infinite series of conserved charges and two hierarchies of equations that share the same conserved charges. We construct two compatible Hamiltonian structures as well as their Casimir functionals. One of the structures has a single Casimir functional while the other has two. This allows us to extend the flows into negative order and clarifies the meaning of two different hierarchies of positive flows. We study the behavior of these systems under a hodograph transformation and show that they are related to the Kaup-Kupershmidt and the Sawada-Kotera equations under appropriate Miura transformations. We also discuss briefly some properties associated with the generalization of second, third and fourth order Lax operators.

MSC:

35Q58 Other completely integrable PDE (MSC2000)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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References:

[1] Kruskal M. D.Lect. Notes Phys.38(1975) 310 (Springer-Verlag)
[2] Konopelchenko B. G., Publ. Rims Kyoto Univ 29 pp 581– (1993) · Zbl 0798.58037 · doi:10.2977/prims/1195166743
[3] Brunelli J. C., J. Math. Phys. 43 pp 6116– (2002) · Zbl 1060.37053 · doi:10.1063/1.1512974
[4] Olver P. J., Applications of Lie Groups to Differential Equations (1993) · Zbl 0785.58003 · doi:10.1007/978-1-4612-4350-2
[5] Brunelli J. C., J. Math. Phys 45 pp 2633– (2004) · Zbl 1071.37041 · doi:10.1063/1.1756699
[6] Fordy A. P., Phys. Lett. 75 pp 325– (1980) · doi:10.1016/0375-9601(80)90829-4
[7] Kaup D. J., Stud. Appl. Math 62 pp 189– (1980)
[8] Fordy A. P., J. Math. Phys 21 pp 2508– (1980) · Zbl 0456.35079 · doi:10.1063/1.524357
[9] Sawada K., Prog. Theo. Phys 51 pp 1355– (1974) · Zbl 1125.35400 · doi:10.1143/PTP.51.1355
[10] Kawamoto S., J. Phys. Soc. Japan 54 (5) pp 2055– (1985) · doi:10.1143/JPSJ.54.2055
[11] Euler M., Stud. Appl. Math. 111 pp 315– (2003) · Zbl 1141.37351 · doi:10.1111/1467-9590.t01-1-00236
[12] Euler N., J. Nonlinear Math. Phys. 8 pp 342– (2001) · Zbl 0997.35093 · doi:10.2991/jnmp.2001.8.3.3
[13] Gürses M., J. Math. Phys. 40 pp 6473– (1999) · Zbl 0977.37038 · doi:10.1063/1.533102
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