Generalization of the method of Krylov-Bogolyubov-Mitropol'skij (Q1824103)

The method of Krylov-Bogolyubov-Mitropol'skij is one of most famous methods for solving differential equations by a small parameter [see \textit{N. N. Bogolyubov} and \textit{Yu. A. Mitropol'skij}, Asymptotical methods in theory of nonlinear oscillations, Moscow, Nauka (1974; Zbl 0303.34043), p. 504]. In order to solve the weakly nonlinear evolution equation \[ u_ t+Lu=\epsilon f(u),\quad t>0,\quad u(0)=u_ 0, \] where L is a linear operator, f is a nonlinear operator, and u is an unknown solution of t and spatial variables \(X=(x_ 0,...,x_ m)\). The asymptotic method presented here is close to the classical KBM's method. There are also two other asymptotic methods and possibilities to apply such method. Examples describing weakly nonlinear waves in gas dynamics are presented.