The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra. (English) Zbl 1420.37084
Summary: M. Mulase solved the Cauchy problem of the Kadomtsev-Petviashvili (KP) hierarchy in an algebraic category in [Invent. Math. 92, No. 1, 1–46 (1988; Zbl 0666.35074)], making use of a delicate factorization of an infinite-dimensional group of formal pseudodifferential operators of infinite order. We prove Mulase’s factorization theorem in a smooth category in the setting of formal pseudo-differential operators with coefficients in a (non-commutative) algebra equipped with a valuation. As an application, we solve the initial value problem for the KP hierarchy using \(r\)-matrix theory.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Citations:
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