In this paper we study the Cauchy problem for the generalized equation of finite-depth fluids ∂tu- G(∂2xu)-∂x (up/p)=0, where G(·) is a singular integral, and p is an integer larger than 1. We obtain the long time behavior of the fundamental solution of linear problem, and prove that the solutions of the nonlinear problem with small initial data for p>5/2+√21/2 are decay in time and freely asymptotic to solutions of the linear problem. In addition we also study some properties of the singular integral G(·) in Lq(R) with q>1.