Wave solutions of semilinear parabolic equations (Q1179694)

The paper is concerned with solutions of the type \[ u(x,t)=\chi(\tau)=\chi(x+pt+p_ 0), p,p_ 0 \text{constants} \] of the equation \(u_ t-u_{xx}-F(u)=0\). Interactions of nonlinear waves (kinks), described by semilinear parabolic equations are investigated. Solutions generalizing the Newell solutions are obtained for cubic nonlinearities, and an asymptotic solution is found, describing interactions of kinks which move in a strip between the two roots of the nonlinearity \(F(u)=u(1-u)\) for the Kolmogorov-Petrovsky-Piskunov (KPP)- Fisher equation.