The Cauchy problem for the equation \((\alpha >0\), \(\lambda >1\in {\mathbb{N}}):\) \[ \partial u/\partial t+(\partial /\partial x)(u^{\lambda}/\lambda)+(\partial /\partial x)(-\partial^ 2/\partial x^ 2)^{\alpha}u=0 \] which describes the propagation of non-linear waves in a dispersive medium is considered. This equation reduces to classical equations (Korteweg-de Vries, modified Korteweg-de Vries, Benjamin-Ono) for particular values of \(\lambda\) and \(\alpha\). The detailed large distance behaviour of the fundamental solution of the linear problem is obtained and it is shown that for \(\alpha\geq 1/2\) and \(\lambda >\alpha +3/2+(\alpha^ 2+3\alpha +5/4)^{1/2}\), solutions of the non-linear equation with small initial conditions are smooth in the large and asymptotic when \(t\to \pm \infty\) to solutions of the linear problem.