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Loop groups and equations of KdV type. (English) Zbl 0592.35112
This paper deals with a construction of solutions of KdV-type equations through infinite dimensional Grassmannian constructions initiated by M. Sato. M. Sato and Y. Sato [Nonlinear partial differential equations in applied science, Proc. U.S.-Jap. Semin., Tokyo 1982, North-Holland Math. Stud. 81, 259–271 (1983; Zbl 0528.58020)] gave a method to construct general solutions of KdV equation systematically in 1979. It was soon developed by E. Date, M. Jimbo, M. Kashiwara and T. Miwa [Publ. Res. Inst. Math. Sci. 18, 1111–1119 (1982; Zbl 0571.35101)] in more general situations. The main aims of this paper are to determine what class of solutions is obtained by this method, to illustrate in detail how the geometry of the Grassmannian is reflected in properties of the solutions fit into the picture. Moreover they try to explain the geometric meaning of the ”\(\tau\)-function”. The authors describe the ”Sato”-theory from their view point by using loop groups in order to give a clear and self contained account of the theory, but no new type of solutions has been obtained in their framework.
Reviewer: M.Muro

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
14M15 Grassmannians, Schubert varieties, flag manifolds
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