Existence of two-solitary waves with logarithmic distance for the nonlinear Klein-Gordon equation

We consider the focusing nonlinear Klein-Cordon (NLKG) equation partial derivative(tt)u - Delta(u) + u -vertical bar u vertical bar(p-1) u = 0, (t, x) is an element of R x R-d for 1 <= d <= 5 and p > 2 subcritical for the (H)over dot(1) norm. In this paper, we show the existence of a solution u (t) of the equation such that parallel to u(t) - Sigma(k=1,2) Qk(t)parallel to(H1) + parallel to partial derivative(t)u(t)parallel to(L2) -> 0 as t -> +infinity, where Q(k) (t, x) are two solitary waves of the equation with translations z(k) : R -> R-d satisfying vertical bar z(1)(t) - z(2) (t)vertical bar similar to 2 log(t) as t -> +infinity. This behavior is due to the strong interactions between solitary waves which is in contrast with the previous work (R. Cote and Y. Martel, Multi-travelling waves for the nonlinear Klein-Gordon equation, Trans. Amer. Math. Soc. 370(10) (2018) 7461-7487] on multisolitary waves of the (NLKG), devoted to the case of solitary waves with different speeds. This work is motivated by previous similar existence results for the nonlinear Schrodinger and generalized Korteweg-de Vries equations.