Front solution

A front is a spatial profile connecting different asymptotic states on its two sides. A travelling wave front moves with a constant speed; a stationary front has zero speed. In a one-dimensional stationary reduction it is a heteroclinic orbit between the corresponding homogeneous equilibria. Equal first-integral values are necessary for a stationary front in a scalar conservative spatial equation.

A stationary front solution of $u_t=u_{xx}+f(u)$ connecting equilibria $u_{-}$ and $u_{+}$ requires ${\int }_{u_{-}}^{u_{+}}f(s)ds=0$. Multiply the stationary equation by $u_x$ and integrate: the derivative term has zero endpoint contribution. This equal-potential condition is the Maxwell balance. For $f(u)=\mu u+\alpha u^3-u^5$, a front from zero to a nonzero constant requires $u_{+}^2=3\alpha /4$ and $\mu =-3{\alpha }^2/16$.

A mesa is a localized broad plateau joined to its surrounding state by two separated fronts. Its leading spatial average equals the plateau fraction times the plateau value; the transition layers give corrections proportional to their width divided by the domain length. A conserved mean can determine the fraction occupied by the plateau, as in the coexistence fraction of a conserved-field amplitude mesa.