Dynamics of two interacting kinks for the $\phi^{6}$ model

Overview & Synopsis

We consider the nonlinear wave equation known as the $\phi^{6}$ model in dimension 1+1. We describe the long time behavior of all the solutions of this model 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 long time less o equivalent than $-\ln{(\epsilon)}\epsilon^{-\frac{1}{2}}$ 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 $(g(t),\partial_{t}g(t))$ and we prove that this estimate is achieved during a finite instant $t=T\lesssim -\ln{(\epsilon)}\epsilon^{-\frac{1}{2}}.$