Painleve integrability of two sets of nonlinear evolution equations with nonlinear dispersions

It is proven that the nonlinear evolution equations (K(m,n) equations), u(1) + (u(m))(x) + (u(n))(xxx) = 0 are Painleve integrable for n = m - 2 and n = m - 1 with positive integer n. Especially, the solutions of the K(3,2) and K(4,2) models are single valued not only about a movable singularity manifold but also about a movable zero manifold. By using the general hodograph transformation, we know that there are five integrable K(m,n) models for negative n, K(- 1/2, - 1/2),K(3/2, - 1/2), K(1/2,- 1/2), K(- 1, - 2) and K(- 2, - 2), which are equivalent to the potential KdV, mKdV and CDF equations. However,the K(m, n) models for positive n are not equivalent to the known third order semilinear integrable ones. (C) 1999 Published by Elsevier Science B.V. All rights reserved.