A.V. Porubov ANALYTICAL SOLUTIONS AND UNSTEADY PROCESSES GOVERNED BY NON-LINEAR NON-INTEGRABLE EQUATIONS

The benefit of finding particular exact and asymptotic solutions of non-integrable non-linear partial differential equations is considered. It is shown how explicit analytical solutions predict important features of the waves behavior even outside their formal applicability. In particular, an arbitrary initial pulse splits in a numerical solution into the train of localized waves each described by the analytical travelling wave solution. This happens both for the bell-shaped and kink-shaped localized waves. Also numerical simulations demonstrate an incident wave amplification/attenuation to a stable wave with the amplitude and velocity defined by the relationships obtained via an asymptotic solution. Physically reasonable equations are used to illustrate above mentioned statements.