Abstract
Integrable dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is considered. Addition formula for the τ-function and conformally invariant equations for the dispersionless BKP (dBKP) hierarchy are derived. Symmetry constraints for the dBKP hierarchy are studied.
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Kuperschmidt B A and Manin A Yu, Long wave equations with a free surface. I. Conservation laws and solutions. (Russian) Funk. Anal. Appl. 11(3) (1977), 31–42; Long wave equations with a free surface. II. The Hamiltonian structure and the higher equations, (Russian) Funk. Anal. Appl. 17(1) (1978), 25–37.
Zakharov V E, Benney equations and quasiclassical approximation in the inverse problem method, (Russian) Func. Anal. Priloz. 14 (1980), 89–98; On the Benney equations. Physica 3D (1981), 193–202.
Lax P D and Levermore C D, The small dispersion limit of the Korteweg-de Vries equation. I, II, III, Commun. Pure Appl. Math. 36 (1983), 253–290, 571–593, 809–830.
Krichever I M, The averaging method for two-dimensional “integrable” equations, Func. Anal. Priloz. 22 (1988), 37–52; The τ-function of the universal Whitham hierarchy, matrix models and topological field theories. Commun. Pure Appl. Math. 47 (1994), 437–475; The dispersionless Lax equations and topological minimal models, Comm. Math. Phys. 143 (1992), 415–429.
Kodama Y, A method for solving the dispersionless KP equation and its exact solutions, Phys. Lett. 129A (1988), 223–226; Solutions of the dispersionless Toda equation. Phys. Lett. 147A (1990), 477–482.
Dubrovin B A and Novikov S P, Hydrodynamics of weakly deformed soliton lattices, Differential geometry and Hamiltonian theory. Russian Math. Surveys 44 (1989), 35–124.
Takasaki K and Takebe T, SDiff(2) KP hierarchy, Int. J. Mod. Phys. A, Suppl. 1B (1992), 889–922; Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), 743–808.
Ercolani N M et al, eds. Singular limits of dispersive waves, Nato Adv. Sci. Inst. Ser. B Phys. 320, Plenum, New York, 1994.
Carroll R and Kodama Y, Solution of the dispersionless Hirota equations, J. Phys. A Math. Gen. 28 (1995), 6373–6387.
Dubrovin B A, Hamiltonian formalism of Whitham-type hierarchies and topological Landau-Ginsburg models, Comm. Math. Phys. 145 (1992), 195–207; Dubrovin B A, Integrable systems in topological field theory, Nucl. Phys. B379 (1992), 627–289; Dubrovin B A and Zhang Y, Bi-Hamiltonian hierarchies in 2D topological field theory at one-loop approximation, Comm. Math. Phys. 198 (1998), 311–361.
Aoyama S and Kodama Y, Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy, Commun. Math. Phys. 182 (1996), 185–219.
Gibbons Y and Tsarev S P, Conformal maps and reductions of the Benney equations, Phys. Lett. 258A (1999), 263–271.
Wiegmann P B and Zabrodin A, Conformal maps and integrable hierarchies, Commun. Math. Phys. 213 (2000), 523–538.
Krichever I, Mineev-Weinstein M, Wiegmann P, Zabrodin A, Laplacian Growth and Whitham Equations of Soliton Theory, nlin.SI/0311005
Dunajski M, Mason L J and Tod P, Einstein-Weyl geometry, the dKP equation and twistor theory, J. Geom. Phys. 37 (2001), 63–93.
Ferapontov E V and Khusnutdinova K R, Hydrodynamic reductions of multidimensional dispersionless PDEs: the test for integrability, J. Math. Phys. 45(6) (2004), 2365–2377.
Martínez Alonso L and Mañas M, Additional symmetries and solutions of the dispersionless KP hierarchy. Integrability, topological solitons and beyond, J. Math. Phys 44(8) (2003), 3294–3308.
Konopelchenko B G, Martínez Alonso L and Ragnisco, O, The \(\mathop \partial \limits^ - \)-approach to the dispersionless KP hierarchy, J. Phys. A: Math. Gen. 34 (2001), 10209–10217.
Konopelchenko B and Martínez Alonso L, \(\mathop \partial \limits^ - \)-equations, integrable deformations of quasicon-formal mappings and Whitham hierarchy, Phys. Lett. A 286 (2001), 161–166.
Konopelchenko B and Martínez Alonso L, Dispersionless scalar integrable hierarchies, Whitham hierarchy, and the quasiclassical \(\mathop \partial \limits^ - \)-dressing method, J. Math. Phys. 43(7) (2002), 3807–3823.
Konopelchenko B and Martínez Alonso L, Nonlinear dynamics on the plane and integrable hierarchies of infinitesimal deformations, Stud. Appl. Math. 109(4) (2002), 313–336.
Bogdanov L V, Konopelchenko B G and Martínez Alonso L, The quasiclassical \(\mathop \partial \limits^ - \)-method: generating equations for dispersionless integrable hierarchies, Teor. Mat. Fiz. 134(1) (2003), 46–54.
Bogdanov L V, and Konopelchenko, B G, Nonlinear Beltrami equation and τ-function for dispersionless hierarchies, Phys. Lett. A 322(5–6) (2004), 330–337.
Bogdanov L V and Konopelchenko B G, Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type, Phys. Lett. A 330(6) (2004), 448–459.
Date E, Jimbo M, Kashiwara M, Miwa T, Transformation groups for soliton equations. VI. KP hierarchies of orthogonal and symplectic type, J. Phys. Soc. Japan 50(11) (1981), 3813–3818.
Jimbo M, Miwa T, Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19(3) (1983), 943–1001.
Date E, Jimbo M, Kashiwara M, Miwa, T, Transformation groups for soliton equations, in Nonlinear integrable systems — classical theory and quantum theory (Kyoto, 1981), World Sci. Publishing, Singapore, 1983, 39–119.
Takasaki K, Quasi-classical limit of BKP hierarchy and W-infinity symmetries, Lett. Math. Phys 28 (1993), 177–185.
Konopelchenko B G and Schief W K, Menelaus’ theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy, J. Phys. A 35(29) (2002), 6125–6144.
Konopelchenko B G and Schief W K, Reciprocal figures, graphical statics, and inversive geometry of the Schwarzian BKP hierarchy, Stud. Appl. Math. 109(2) (2002), 89–124.
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This article is part of the special issue published in honour of Francesco Calogero on the occasion of his 70th birthday
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Bogdanov, L.V., Konopelchenko, B.G. On Dispersionless BKP Hierarchy and its Reductions. J Nonlinear Math Phys 12 (Suppl 1), 64–73 (2005). https://doi.org/10.2991/jnmp.2005.12.s1.6
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DOI: https://doi.org/10.2991/jnmp.2005.12.s1.6