Dynamics of Two Interacting Kinks for the $$\phi ^{6}$$ Model

Abstract

We consider the nonlinear wave equation known as the $$\phi ^{6}$$ model in dimension 1+1. We describe the long-time behavior of this model’s solutions close to a sum of two kinks with energy slightly larger than twice the minimum energy of non-constant stationary solutions. We prove orbital stability of two moving kinks. We show for low energy excess $$\epsilon $$ that these solutions can be described for a long time of order $$-\ln {(\epsilon )}\epsilon ^{-\frac{1}{2}}$$ as the sum of two moving kinks such that each kink’s center is close to an explicit function which is a solution of an ordinary differential system. We give an optimal estimate in the energy norm of the remainder and we prove that this estimate is achieved during a finite instant t of order $$-\ln {(\epsilon )}\epsilon ^{-\frac{1}{2}}.$$