Elementary particles as solutions of the Sivashinsky equation

Abstract

Stationary spherically symmetric solutions of the Sivashinsky equation (1/ c 2) S 2 t −(∇ S) 2)= m 2 c 2+ α ℏ □ S+ ( β ℏ 3/ m 2 c 2)□ 2 S > are looked for. The problem is reduced to finding all admissible solutions y= y( r) of the ODE [ d 2/ dr 2+(2/ r) d/ dr+1]( d/ dr+2/ r) y=- y 2, i.e., solutions which are defined on the half-line 0≤ r≤ ∞ and satisfy y(0)= y(∞)=0. It is proved that for each y'(0) ϵ [-0.34, 0.33] the solution of the initial value problem with y(0)= y″ (0)=0 is admissible. In terms of the action function S these solutions are interpreted as free relativistic particles. The proof of the result employs analytic asymptotic expansion near r=0 and r=∞ and a new technique of rigorous estimates by a computer. The proved interval of existence y'(0) ϵ [-0.34, 0.33] nearly coincides with the one obtained by numerical experiments.