Frontal Asymptotic of Solution of the Klein-Gordon Equation

An asymptotic behavior as t \to \infty of solutions of the equation u_t = u_xx + f(u) for finite initial data is investigated. Using numerical method it was found that 1) the external discontinuity is formed behind the formal boundary of influence of initial data, this shock moves with unit velocity; 2) for large values of t the solution becomes high-oscillatory. Moreover, all oscillations are condensed to the shock. Using this results the special stretching 'self-similar' coordinates were found. The corresponding ordinary equation describes near-shock asymptotic. These solutions are close to Bessel functions. Some results concerning the limit solutions are presented too. In particular, the sufficient condition the limit shock is formulated.