Abstract
We show how to construct the Hamiltonian structures of any reduction of the Benney chain (dispersionless KP). The construction follows the scheme suggested by Ferapontov, leading in general to nonlocal Hamiltonian structures. In some special cases these reduce to local structures. All the geometric objects which define the Poisson bracket, the metric, connection and Riemann curvature, are obtained explicitly, in terms of the n-parameter family of conformal maps associated with the reduction.
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Baldwin S., Gibbons J.: Hyperelliptic reduction of the Benney moment equations. J. Phys. A 36(31), 8393–8417 (2003)
Baldwin S., Gibbons J.: Higher genus hyperelliptic reductions of the Benney equations. J. Phys. A 37(20), 5341–5354 (2004)
Baldwin S., Gibbons J.: Genus 4 trigonal reduction of the Benney equations. J. Phys. A 39(14), 3607–3639 (2006)
Benney D.J.: Some properties of long nonlinear waves. Stud. Appl. Math. 52, 45–50 (1973)
Dubrovin, B., Liu, S.Q., Zhang, Y.: Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures. http://arXiv.org/abs/0710.3115v1[math.DG], 2007
Dubrovin B.A., Novikov S.P.: Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method. Dokl. Akad. Nauk SSSR 270(4), 781–785 (1983)
Dubrovin B.A., Novikov S.P.: Poisson brackets of hydrodynamic type. Dokl. Akad. Nauk SSSR 279(2), 294–297 (1984)
Duren P.L.: Univalent functions. Springer-Verlag, New York (1983)
Ferapontov E.V.: Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type. Funkts. Anal. i Pril. 25(3), 37–49 (1991) 95
Gibbons, J., Kodama, Y.: Solving dispersionless Lax equations. In: Singular limits of dispersive waves (Lyon, 1991), vol. 320 of NATO Adv. Sci. Inst. Ser. B Phys., New York: Plenum 1994, pp. 61–66
Gibbons J., Tsarev S.P.: Reductions of the Benney equations. Phys. Lett. A 211(1), 19–24 (1996)
Gibbons J., Tsarev S.P.: Conformal maps and reductions of the Benney equations. Phys. Lett. A 258(4–6), 263–271 (1999)
Kodama, Y., Gibbons, J.: Integrability of the dispersionless KP hierarchy. In: Nonlinear world, vol. 1 (Kiev, 1989), River Edge, NJ: World Sci. Publ., 1990, pp. 166–180
Kokotov, A., Korotkin, D.: A new hierarchy of integrable systems associated to Hurwitz spaces. http://arXiv.org/math-ph/0112051v3, 2001
Konopel′chenko B., Martines Alonso L., Medina E.: Quasiconformal mappings and solutions of the dispersionless KP hierarchy. Teoret. Mat. Fiz. 133(2), 247–258 (2002)
Kupershmidt B.A., Manin Yu.I.: Long wave equations with a free surface. I. Conservation laws and solutions. Funkt. Anal. i Pril. 11(3), 31–42 (1977)
Kupershmidt B.A., Manin Yu.I.: Long wave equations with a free surface. II. The Hamiltonian structure and the higher equations. Funkt. Anal. i Pril. 12(1), 25–37 (1978)
Lebedev D.R., Manin Yu.I.: Conservation laws and Lax representation of Benney’s long wave equations. Phys. Lett. A 74(3–4), 154–156 (1979)
Mokhov O.I., Ferapontov E.V.: Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature. Usp Mat. Nauk 45(3), 191–192 (1990)
Tsarëv S.P.: Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Dokl. Akad. Nauk SSSR 282(3), 534–537 (1985)
Yu L., Gibbons J.: The initial value problem for reductions of the Benney equations. Inverse Problems 16(3), 605–618 (2000)
Zakharov V.E.: Benney equations and quasiclassical approximation in the inverse problem. Funkt. Anal. i Pril. 14, 15–24 (1980)
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Communicated by L. Takhtajan
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Gibbons, J., Lorenzoni, P. & Raimondo, A. Hamiltonian Structures of Reductions of the Benney System. Commun. Math. Phys. 287, 291–322 (2009). https://doi.org/10.1007/s00220-008-0686-z
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DOI: https://doi.org/10.1007/s00220-008-0686-z