Modern mathematical physics is almost exclusively a mathematical theory of
nonlinear partial differential equations describing various physical processes.
Since only a few partial differential equations have succeeded in being solved
explicitly, different qualitative methods play a very important role. One of
the most effective ways of qualitative analysis of differential equations are
asymptotic methods, which enable us to obtain an explicit approximate representation for solutions with respect to a large parameter time. Asymptotic
formulas allow us to know such basic properties of solutions as how solutions grow or decay in different regions, where solutions are monotonous and
where they oscillate, which information about initial data is preserved in the
asymptotic representation of the solution after large time, and so on. It is
interesting to study the influence of the nonlinear term in the asymptotic
behavior of solutions. For example, compared with the corresponding linear
case, the solutions of the nonlinear problem can obtain rapid oscillations, can
converge to a self-similar profile, can grow or decay faster, and so on. It is very
difficult to obtain this information via numerical experiments. Thus asymptotic methods are important not only from the theoretical point of view, but
also they are widely used in practice as a complement to numerical methods.
It is worth mentioning that in practice large time could be a rather bounded
value, which is sufficient for all the transitional processes caused by the initial
perturbations in the system to happen