Abstract
In this letter we discuss the nonlinear propagation of waves of
short wavelenght in dispersive systems. We propose a family of
equations that is likely to described the asymptotic behaviour
of a large class of systems. We then restrict our attention to
the analysis of the simplest nonlinear short-wave dynamics
given by $U_{0\xi\tau}=U_{0}-3(U_0)^2$.
We integrate numerically this equation for periodic and
non-periodic boundary conditions, and we find that short-waves
may exist only if the amplitude of the initial profile is not
too large.