Abstract

Localized Solutions of the Non-Linear Klein-Gordon Equation in Many Dimensions

M.V.Perel, I.V.Fialkovsky (Dept. of Mathematical Physics, Physics Faculty, St. Petersburg University)

e-mail: perel@mph.phys.spbu.ru

We construct the non-stationary localized solution of the non-linear Klein-Gordon equation with constant coefficients. To do it we reduce this partial differential equation in many dimensions to the ordinary differential equation with one complex variable depending on the spatial coordinates, time and free parameters. We investigate this more simple equation. We find the asymptotics of the solution of this equation outside some moving in the space sphere for fixed times and large distance from its center. This asymptotics coincides with the simple exact solution of the corresponded linear Klein-Gordon equation [Perel M.V., Fialkovsky I.V. "Exponentially localized solutions to the Klein-Gordon equation", Zapiski nauch. sem.POMI, 245, p.187-198, 2001]. It decreases exponentially in all directions away from this sphere. The radius of the sphere, the speed of its movement, the scales of time and distance can be ruled with free parameters.

Section : 1