Interaction of kinks in KPP-Fisher equation (Q1178170)

The KPP-Fischer nonlinear equation \[ \partial u/\partial t-\partial^ 2u/\partial x^ 2-u(1-u)=0, \] is known in the physics of nonstationary processes in gaseous and liquid media. This equation has a solution in the form of kink, \[ u(x,t)=[1+\exp(-\tau/\sqrt 6)]^{-2}, \tag{*} \] where \(\tau=\pm x+bt\) and \(b=5/\sqrt 6\), which describes a wave of transition between two stationary states (stable and unstable). The present paper analyses interactions between the kinks of different polarities [different signs in eq. \((*)]\) and it gives the main term of the asymptotic expansion valid for large distances of the interaction, i.e. up to order \(O(e^{-L})\), \(L\) being the distance between the kinks.