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J. Phys. A: Math. Gen. 19, 1889–1898 (1986) 14. : Modified KP and discrete KP. Lett. Math. Phys. 48, 277–289 (1999) 15. : Soliton Equations and Hamiltonian Systems. World Scientific, Singapore (2003) 16. : Explorations of the extended ncKP hierarchy. J. Phys. A: Math. Gen. 37, 10,899–10,930 (2004) 17. : An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions. J. Phys. A: Math. Gen. 38, 5453–5505 (2005) 18. : From nonassociativity to solutions of the KP hierarchy.

N = 2, 3, . , where (m) (m+1) (m) ∂ (un ) = un and ∂ (vw) = ∂ (v) w + v ∂ (w) for v, w ∈ A. We demand that un , n = 2, 3, . , m = 0, 1, 2, . 62) 12 Recall that ϕ = tr(Qφ ) (cf. 21) determines a homomorphism if Q has rank 1. 46), then ϕ solves the scalar pDKP hierarchy. 14) requires q = p for all i, j = 1, . . , N. i j 2 Weakly Nonassociative Algebras, Riccati and KP Hierarchies 21 where V≥0 is the projection of a pseudo-differential operator V to its differential operator part, and L = ∂ + u2 ∂ −1 + u3 ∂ −2 + · · · .

SIAM J. Appl. Math. 32, 707–710 (1977) 29. : On KP generators and the geometry of the HBDE. J. Geom. Phys. 56, 282–309 (2006) 30. : The Kadomtsev-Petviashvili equations and fundamental string theory. Commun. Math. Phys. 117, 331–348 (1988) 31. : Higher order Riccati equations as Bäcklund transformations. preprint CRM-2469 (1997) 26 A. Dimakis and F. Müller-Hoissen 32. : Negative flows of the potential KP-hierarchy. Trans. Am. Math. Soc. 348, 375–390 (1996) 33. : Commuting rings of difference operators and an adelic flag manifold.