tailieunhanh - Construction of asymptotic in krylov bogoliubov-mitropolsky-sense solution of wave equations

The paper in question gives consideration to two examples of 1 | Vietnam Journa l of Mechanics, VAST , Vol. 29 , No. 3 (2007), pp . 221 - 243 Special Iss ue Dedicated to the Memory of Prof. Nguyen Van Dao CONSTRUCTION OF ASYMPTOTIC IN KRYLOV-BOGOLIUBOV-MITROPOLSKY SENSE SOLUTION OF WAVE EQUATIONS Yu. A. MITROPOLSKY Institute of Mathematics , National Academy of Sciences of Ukraine This article is dedicated to my colleague and intimate friend, who died prematurely being in the prime of life and scientific activity Abstract. The paper in question gives consideration to two examples of 1. 2 ( 8u 8u) u =cf vt, u, 8t' 8x ; (2 .1) for E: = 0, this is the Klein- Gordon equation and, for >. = 0, it turns into the classical wave equation. Equation (2 .1) was studied by many scientists in the course of investigation of nonlinear wave processes in different branches of natural sciences. Below , we dwell on principal aspects of the application of asymptotic methods of nonlinear mechanics t o the construction of approximate solutions of Eq. ( 2 .1). This may be useful for studying special problems of natural sciences that require the investigation of wave processes subject to the action of nonlinear perturbation forces and described by equations of type (). This can also be useful for the analysis of the obtained results. The development and detailed application of the asymptotic method to the solution of a special problem that leads to Eq. (), provided that there is no periodic perturbation with period v, was first realized in [4]. Thus, for E: = 0, Eq. () is the classical Klein- Gordon wave equation -C2 82u 8x 2 + >.2u = 0 (2 .2) admitting a solution of the form u = a cos(kx - wot+ 0 is a small parameter and a function f(vt, u, Ut, ux) is periodic (or almost periodic) in () = vt and has sufficiently many derivatives with respect to the other 2 . . . OU u OU cJ2u variables for all their fimte values. Denote Ut = ot , Utt = ot 2 , Ux = ox, and Uxx = ox 2 · Then, according to the well-known statements of the asymptotic .