Proceedings of the Royal Society of Edinburgh, 128A, 725–743, 1998 On equations of KP-type Rafael Jose ´ Io ´rio Jr.* Instituto de Matema ´tica Pura e Aplicada (IMPA/CNPq), Estrada Dona Castorina 110, Jardim Bota ˆ nico, 22460-320 Rio de Janeiro RJ, Brazil Wagner Vieira Leite Nunes Instituto de Cie ˆncias Matema ´ ticas de Sa ˜o Carlos (ICMSC/USP), Caixa Postal 668, 13560-270 Sa ˜ o Carlos SP, Brazil (MS received 13 September 1996. Revised MS received 23 June 1997) We discuss the local Cauchy problem for the generalised Kadomtsev–Petviashvili equation, namely (∂ t u −iP(D)u +a(u) ∂ x u) x +c ∂2 y u =0, in the periodic and nonperiodic settings. 1. Introduction In this work, we consider the Cauchy problem (∂ t u −iP(D)u +a(u) ∂ x u) x +c ∂2 y u =0, (1.1) u( 0; x, y) =w(x, y), (1.2) with periodic and nonperiodic boundary conditions. In the periodic case, we assume that (P 3 (D) y) (m, n) =p(m) y ˆ (m, n), (m, n)µZ2, (1.3) where p =p(m), mµZ, is real valued, p(0) =0 and there are C l "0 and lµ{2, 3, . . . } such that | p(m) | !C l | m |l, Y | m | sufficiently large. (1.4) In the nonperiodic case, we make a similar assumption, namely (P 3 (D) y)(j, g) =p(j) y ˆ (j, g), (j, g)µR2, (1.5) where p =p(j), jµR is a real odd continuous function and there are C l "0 and lµ{2, 3, . . . } such that | p(j) | !C l | j |l, Y | j | sufficiently large. (1.6) Our purpose in this paper is to extend recent results of Isaza, Mejı ´a and Stallbohm [ 7 ] in the periodic case and of Ukai [ 18 ] and Saut [ 16 ] in the nonperiodic situation. We will be more precise further on. * Part of this work was done while the author was visiting IMECC-Unicamp. Partially supported by FAPESP.