We discuss the nonlinear propagation of waves of short wavelength in dispersive systems and we present the simplest nonlinear short-wave dynamics: $u_{tx} = u - 3 u^2$. We present an algorithm to integrate numerically this equation for periodic and non-periodic boundary conditions. We also prove the existence and uniqueness of solutions for the Cauchy problem of the simplest nonlinear short-wave equation for the periodic case.