Geodesic
Dispersionless Hirota Equations of Two-Component BKP Hierarchy
Symmetry, integrability and geometry: methods and applications, Tome 2 (2006) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The BKP hierarchy has a two-component analogue (the 2-BKP hierarchy). Dispersionless limit of this multi-component hierarchy is considered on the level of the $\tau$-function. The so called dispersionless Hirota equations are obtained from the Hirota equations of the $\tau$-function. These dispersionless Hirota equations turn out to be equivalent to a system of Hamilton–Jacobi equations. Other relevant equations, in particular, dispersionless Lax equations, can be derived from these fundamental equations. For comparison, another approach based on auxiliary linear equations is also presented.
Keywords: BKP hierarchy; Hirota equation; dispersionless limit. 
@article{SIGMA_2006_2_a56,
     author = {Kanehisa Takasaki},
     title = {Dispersionless {Hirota} {Equations} of {Two-Component} {BKP} {Hierarchy}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2006},
     volume = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a56/}
}
TY  - JOUR
AU  - Kanehisa Takasaki
TI  - Dispersionless Hirota Equations of Two-Component BKP Hierarchy
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2006
VL  - 2
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a56/
LA  - en
ID  - SIGMA_2006_2_a56
ER  - 
%0 Journal Article
%A Kanehisa Takasaki
%T Dispersionless Hirota Equations of Two-Component BKP Hierarchy
%J Symmetry, integrability and geometry: methods and applications
%D 2006
%V 2
%U http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a56/
%G en
%F SIGMA_2006_2_a56
Kanehisa Takasaki. Dispersionless Hirota Equations of Two-Component BKP Hierarchy. Symmetry, integrability and geometry: methods and applications, Tome 2 (2006). http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a56/

[1] Sato M., “Soliton equations as dynamical systems on an infinite dimensional Grassmannian manifold”, Surikaiseki Kenkyusho Kokyuroku (RIMS, Kyoto University), 439 (1981), 30–46 | Zbl

[2] Sato M., Sato Y., “Soliton equations as dynamical systems on an infinite dimensional Grassmannian manifold”, Nonlinear Partial Differential Equations in Applied Science (1982, Tokyo), North-Holland Math. Stud., 81, eds. H. Fujita, P. D. Lax and G. Strang, North-Holland, Amsterdam, 1983, 259–271 | MR | Zbl

[3] Date E., Kashiwara M., Miwa T., “Transformation groups for soliton equations. II. Vertex operators and $\tau$ functions”, Proc. Japan Acad. Ser. A Math. Sci., 57 (1981), 387–392 | DOI | MR | Zbl

[4] Date E., Jimbo M., Kashiwara M., Miwa T., “Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type”, Phys. D, 4 (1981/82), 343–365 | DOI | MR

[5] Date E., Jimbo M., Kashiwara M., Miwa T., “Transformation groups for soliton equations. V. Quasiperiodic solutions of the orthogonal KP equation”, Publ. Res. Inst. Math. Sci., 18 (1982), 1111–1119 | DOI | MR | Zbl

[6] 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 (1981), 3813–3818 | DOI | MR | Zbl

[7] Shiota T., “Prym varieties and soliton equations”, Infinite Dimensional Lie Algebras and Groups, Advanced Series in Math. Phys., 7, ed. V. G. Kac, World Scientific, Singapore, 1989, 407–448 | MR

[8] Krichever I., A characterization of Prym varieties, arXiv:math.AG/0506238 | MR

[9] Takasaki K., “Quasi-classical limit of BKP hierarchy and $W$-infinity symmetries”, Lett. Math. Phys., 28 (1993), 177–185 ; arXiv:hep-th/9301090 | DOI | MR | Zbl

[10] Takasaki K., Takebe T.,, “Integrable hierarchies and dispersionless limit”, Rev. Math. Phys., 7 (1995), 743–808 ; arXiv:hep-th/9405096 | DOI | MR | Zbl

[11] Bogdanov L. V., Konopelchenko B. G., “On dispersionless BKP hierarchy and its reductions”, J. Nonlinear Math. Phys., 12, suppl. 1 (2005), 64–75 ; arXiv:nlin.SI/0411046 | DOI | MR

[12] Konopelchenko B. G., Martinez Alonso L., Ragnisco O., “$\bar\partial$-approach to the dispersionless KP hierarchy”, J. Phys. A: Math. Gen., 34 (2001), 10209–10217 ; arXiv:nlin.SI/0103023 | DOI | MR | Zbl

[13] Konopelchenko B., Martinez Alonso L., “$\bar\partial$-equations, integrable deformations of quasi-conformal mappings and Whitham hierarchy”, Phys. Lett. A, 286 (2001), 161–166 ; arXiv:nlin.SI/0103015 | DOI | MR | Zbl

[14] Konopelchenko B., Martinez Alonso L., “Dispersionless scalar integrable hierarchies, Whitham hierarchy and the quasi-classical $\bar\partial$-dressing method”, J. Math. Phys., 43 (2001), 3807–3823 | DOI | MR

[15] Theoret. and Math. Phys., 134:1 (2003), 39–46 | DOI | MR | Zbl

[16] Krichever I. M., “The $\tau$-function of the universal Whitham hierarchy, matrix models and topological field theories”, Comm. Pure. Appl. Math., 47 (1994), 437–475 ; arXiv:hep-th/9205110 | DOI | MR | Zbl

[17] Carroll R., Kodama Y., “Solutions of the dispersionless Hirota equations”, J. Phys. A: Math. Gen., 28 (1995), 6373–6378 ; arXiv:hep-th/9506007 | DOI | MR

[18] Kostov I. K., Krichever I., Mineev-Weinstein M., Wiegmann P. B., Zabrodin A., “$\tau$-function for analytic curve”, Random Matrices and Their Applications, Math. Sci. Res. Inst. Publ., 40, eds. P. Bleher and A. Its, Cambridge University Press, Cambridge, 2001, 285–299 ; arXiv:hep-th/0005259 | MR | Zbl

[19] Theoret. and Math. Phys., 129:2 (2001), 1511–1525 ; arXiv:math.CV/0104169 | DOI | MR | Zbl

[20] Boyarsky A., Marshakov A., Ruchayskiy O., Wiegmann P., Zabrodin A., “Associativity equations in dispersionless integrable hierarchies”, Phys. Lett. B, 515 (2001), 483–492 ; arXiv:hep-th/0105260 | DOI | MR | Zbl

[21] Teo L.-P.,, “Analytic functions and integrable hierarchies – characterization of tau functions”, Lett. Math. Phys., 64 (2003), 75–92 ; arXiv:hep-th/0305005 | DOI | MR | Zbl

[22] Kac V., van de Leur J., “The geometry of spinors and the multicomponent BKP and DKP hierarchies”, CRM Proc. Lect. Notes, 14, American Mathematical Society, Providence, 1998, 159–202 | MR | Zbl

[23] Date E., Jimbo M., Kashiwara M., Miwa T., “Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy”, Publ. Res. Inst. Math. Sci., 18 (1982), 1077–1110 | DOI | MR | Zbl

[24] Novikov S., Veselov A., “Finite-gap two-dimensional potential Schrödinger operators Explicit formulas and evolution equations”, Soviet Math. Dokl., 30 (1984), 588–591 | MR | Zbl

[25] Ueno K., Takasaki K., “Toda lattice hierarchy”, Group Representations and Systems of Differential Equations (1982, Tokyo), Adv. Stud. Pure Math., 4, North-Holland, Amsterdam, 1984, 1–94 | MR

[26] Konopelchenko B., Moro A., “Integrable equations in nonlinear geometrical optics”, Stud. Appl. Math., 113 (2004), 325–352 ; arXiv:nlin.SI/0403051 | DOI | MR | Zbl

[27] Jimbo M., Miwa T., “Solitons and infinite dimensional Lie algebras”, Publ. Res. Inst. Math. Sci., 19 (1983), 943–1001 | DOI | MR | Zbl

[28] Takebe T., “Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy. I”, Lett. Math. Phys., 21 (1991), 77–84 | DOI | MR | Zbl

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

[30] Dickey L., Soliton equations and Hamiltonian systems, World Scientific, Singapore, 1991 | MR | Zbl