Reductions of Lower Triangular Toda Hierarchies
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Abstract
Deforming commutative algebras in the lower triangular (ℤ×ℤ)-matrices yields lower triangular Toda hierarchies and their associated nonlinear equations. Like for their counterpart in the ring of pseudodifferential operators, the KP-hierarchy, one also has for these hierarchies a geometric picture: certain infinite chains of subspaces in an separable Hilbert space provide solutions of lower triangular Toda hierarchies. The KP-hierarchy and its multi-component version contain many interesting subsystems, like e.g. the nth Gelfand–Dickey hierarchy and the AKNS-hierarchy. In this paper one considers analogues of these two subsystems in the context of the lower triangular Toda hierarchies and a geometric description of solutions to both type reductions is given.
Keywords
Lower triangular Toda hierarchy Lax equations Linearization Reduction KP-hierarchy nth Gelfand–Dickey hierarchy AKNS-hierarchy Finite band deformation Commuting flows Banach Lie group Big cellMathematics Subject Classification (2000)
20G15 20G20 22E15 22E46Preview
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References
- 1.Date, E., Kashiwara, M., Jimbo, M., Miwa, T.: Transformation groups for soliton equations. In: Nonlinear Integrable Systems—Classical Theory and Quantum Theory, Kyoto, 1981, pp. 39–119. World Scientific, Singapore (1983) Google Scholar
- 2.Dijkgraaf, R.: Integrable hierarchies and quantum gravity. In: Geometric and Quantum Aspects of Integrable Systems, Scheveningen, 1992. Lecture Notes in Physics, vol. 424, pp. 67–89. Springer, Berlin (1993) CrossRefGoogle Scholar
- 3.Flaschka, H., Newell, A.C., Ratiu, T.: Kac Moody Lie algebras and soliton equations II; Lax equations associated with A 1(1). Physica D 9, 300–323 (1983) MATHCrossRefMathSciNetGoogle Scholar
- 4.Helminck, G.F., Post, G.F.: A convergent framework for the multicomponent KP-hierarchy. Trans. Am. Math. Soc. 324(1), 271–292 (1991) MATHCrossRefMathSciNetGoogle Scholar
- 5.Helminck, G.F., Post, G.F.: The geometry of differential-difference equations. Indag. Math. (N.S.) 5(4), 411–438 (1994) MATHCrossRefMathSciNetGoogle Scholar
- 6.Helminck, G.F., van de Leur, J.W.: An analytic description of the vector constrained KP hierarchy. Commun. Math. Phys. 193(3), 627–641 (1998) MATHCrossRefGoogle Scholar
- 7.Mulase, M.: Cohomological structure in soliton equations and Jacobian varieties. J. Differ. Geom. 19(2), 403–430 (1984) MATHMathSciNetGoogle Scholar
- 8.Sato, M., Sato, Y.: Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold. In: Nonlinear Partial Differential Equations in Applied Science, Tokyo, 1982. North-Holland Mathematics Studies, vol. 81, pp. 259–271. North-Holland, Amsterdam (1983) Google Scholar
- 9.Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES 63, 1–64 (1985) Google Scholar
- 10.Ueno, K., Takasaki, K.: Toda lattice hierarchy. In: Group Representations and Systems of Differential Equations, Tokyo, 1982. Advanced Studies in Pure Mathematics, vol. 4, pp. 1–95. North-Holland, Amsterdam/New York (1984) Google Scholar
- 11.van Moerbeke, P.: Integrable foundations of string theory. In: Proceedings of the CIMPA-School, pp. 163–267. World Scientific, Singapore (1994) Google Scholar