We apply the travelling wavelets method to gravitational instability theory for the investigation of large-scale structure formation in cosmology. As the first step of our approach, the method is first applied to the 1D cosmological Euler-Poisson equation system. We test the stability of the linear (evolution) regime in this plane-symmetric case. As a result, our analysis confirms the existence of the linear regime for some configurations of fields describing the evolution of cosmological structures. Moreover, it provides us with estimates for the delay needed for structures of given scale and magnitude to deviate from the linear regime. We also exhibit other configurations for which the linear approximation is not valid at any time. In particular, density defaults (i.e. holes) turn out to be highly non-linear structures that dominate the evolution.