Spatially inhomogeneous dissipative structures in a periodic boundary-value problem for a nonlocal erosion equation (Q2516543)

The author studies spatially periodic solutions of the nonlinear problem \[ u_t = a u_{xx}-c w_x+u-w+b_1 (u-w) w_x+b_2 w_x^2+b_3 (u-w) w_x^2 \] where \(w=w(t,x)=u(t,x-h)\) for some \(h\in \mathbb R^+\) and the coefficients are given as part of the model. This equation is used to model the formation of nanopatterns when a flat surface is bombarded by a flow of ions. The author proposes a mechanism for the formation of ripple nanopatterns due to the loss of stability of the flat surface topography. The ripple topography is found as the solution of bifurcation problems.