Some Remarks on the KP System of the Camassa–Holm Hierarchy
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

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We study a Kadomtsev–Petviashvili system for the local Camassa–Holm hierarchy obtaining a candidate to the Baker–Akhiezer function for its first reduction generalizing the local Camassa–Holm. We focus our attention on the differences with the standard KdV-KP case.
Keywords: KP hierarchy; CH hierarchy; Sato Grassmannian.
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a46/}
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Giovanni Ortenzi. Some Remarks on the KP System of the Camassa–Holm Hierarchy. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a46/

[1] Aratyn H., van de Leur J., “Clifford algebra derivations of tau functions for two-dimensional integrable models with positive and negative flows”, SIGMA, 3 (2007), 020, 29 pp., ages ; nlin.SI/0605027 | MR | Zbl

[2] Camassa R., Holm D. D., “An integrable shallow water equation with peaked solitons”, Phys. Rev. Lett., 71 (1993), 1661–1664 ; patt-sol/9305002 | DOI | MR | Zbl

[3] Casati P., Lorenzoni P., Ortenzi G., Pedroni M., “On the local and nonlocal Camassa–Holm hierarchies”, J. Math. Phys., 46 (2005), 042704, 8 pp., ages | DOI | MR | Zbl

[4] Chen M., Liu S., Zhang Y., “A two-component generalization of the Camassa–Holm equation and its solutions”, Lett. Math. Phys., 75 (2006), 1–15 ; nlin.SI/0501028 | DOI | MR | Zbl

[5] Constantin A., “On the inverse spectral problem for the Camassa–Holm equation”, J. Funct. Anal., 155 (1998), 352–363 | DOI | MR | Zbl

[6] Clarkson P. A., Gordoa P. R., Pickering A., “Multicomponent equations associated to non-isospectral scattering problems”, Inverse Problems, 13 (1997), 1463–1476 | DOI | MR | Zbl

[7] Constantin A., McKean H. P., “A shallow water equation on the circle”, Comm. Pure Appl. Math., 52 (1999), 949–982 | 3.0.CO;2-D class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[8] Date E., Jimbo M., Kashiwara M., Miwa T., “Transformation groups for soliton equations”, Nonlinear Integrable Systems, eds. M. Jimbo and T. Miwa, World Scientific, Singapore, 1983, 39–119 | MR

[9] Russian Math. Surveys, 44:6 (1989), 35–124 | DOI | MR | Zbl

[10] Dubrovin B., Zhang Y., Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants, math.DG/0108160

[11] Estevez P. G., Prada J., “Hodograph transformations for a Camassa–Holm hierarchy in $2+1$ dimensions”, J. Phys. A: Math. Gen., 38 (2005), 1287–1297 ; nlin.SI/0412019 | DOI | MR | Zbl

[12] Falqui G., “On a Camassa–Holm type equation with two dependent variables”, J. Phys. A: Math. Gen., 39 (2006), 327–342 ; nlin.SI/0505059 | DOI | MR | Zbl

[13] Falqui G., Magri F., Pedroni M., “Bihamiltonian geometry, Darboux coverings and linearization of the KP hierarchy”, Comm. Math. Phys., 197 (1998), 303–324 ; solv-int/9806002 | DOI | MR | Zbl

[14] Fontanelli L., Lorenzoni P., Pedroni M., A 3-component extension of the Camassa–Holm hierarchy

[15] Ivanov R., “Extended Camassa–Holm hierarchy and conserved quantities”, Z. Naturforschung A, 61 (2006), 133–138; nlin.SI/0601066

[16] Khesin B., Misiolek G., “Euler equations on homogeneous spaces and Virasoro orbits”, Adv. Math., 176 (2003), 116–144 ; math.SG/0210397 | DOI | MR | Zbl

[17] Konopelchenko B., Martinez Alonso L., Medina E., “Singular sector of the Kadomtsev–Petviashvili hierarchy, $\overline\partial$ operators of nonzero index, and associated integrable systems”, J. Math. Phys., 41 (2000), 385–413 ; solv-int/9806001 | DOI | MR | Zbl

[18] Kraenkel R. A., Senthilvelan M., Zenchuk A. I., “Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa–Holm equation”, Phys. Lett. A, 273 (2000), 183–193 | DOI | MR | Zbl

[19] Lenells J., “Conservation laws of the Camassa–Holm equation”, J. Phys. A: Math. Gen., 38 (2005), 869–880 | DOI | MR | Zbl

[20] Lorenzoni P., Pedroni M., “On the bi-Hamiltonian structures of the Camassa–Holm and Harry Dym equations”, Int. Math. Res. Not., 75 (2004), 4019–4029 ; nlin.SI/0407057 | DOI | MR | Zbl

[21] Martinez Alonso L., Shabat A. B., “On the prolongation of a hierarchy of hydrodynamic chains”, New trends in integrability and partial solvability, NATO Sci. Ser. II Math. Phys. Chem., 132, Kluwer Acad. Publ., Dordrecht, 2004, 263–280 | MR | Zbl

[22] Theoret. and Math. Phys., 140 (2004), 1073–1085 ; nlin.SI/0312043 | DOI | MR

[23] Magri F., Casati P., Falqui G., Pedroni M., “Eight lectures on integrable systems”, Integrability of Nonlinear Systems, Lecture Notes in Physics, 638, eds. Y. Kosmann-Schwarzbach et al., 2004, 209–250

[24] Casati P., Falqui G., Magri F., Pedroni M., Soliton equations, bi-Hamiltonian manifolds and integrability, 21$^{\rm o}$ Coloquio Brasileiro de Matemetica (21st Brazilian Mathematics Colloquium), Instituto de Matemetica Pura e Aplicada (IMPA), Rio de Janeiro, 1997 | MR

[25] Reyes E. G., “Geometric integrability of the Camassa–Holm equation”, Lett. Math. Phys., 59 (2002), 117–131 | DOI | MR | Zbl

[26] Shabat A., “Universal solitonic hierarchy”, J. Nonlinear Math. Phys., 12, suppl. 1 (2005), 614–624 | DOI | MR

[27] Sato M., Sato Y., “Soliton equations as dynamical systems on infinite–dimensional Grassmann manifold”, Nonlinear PDEs in Applied Sciences (US–Japan Seminar, Tokyo), eds. P. Lax and H. Fujita, North-Holland, Amsterdam, 1982, 259–271 | MR

[28] Segal G., Wilson G., “Loop groups and equations of the KdV type”, Publ. Math. IHES, 61 (1985), 5–65 | MR | Zbl