Single-mode and dual-mode nongomogeneous dissipative structures in the nonlocal model of erosion

We consider a periodic boundary-value problem for a nonlinear equation with the deviating spatial argument in the case when the deviation is small. This equation is called a spatially nonlocal erosion equation. It describes the formation of undulating surface relief under the influence of ion bombardment and can be interpreted as a development of the well-known Bradley–Harper model. It is shown that the nonhomogeneous surface relief can occur when the stability of the homogeneous states of equilibrium changes. In this boundary value problem the loss of stability can occur at the higher modes and a number of such modes. The mode number depends on many factors. For example, it depends on the angle of incidence. It is also shown that the nonlinear boundary value problem can be included into the class of abstract parabolic equations. Solvability of this problem was studied in the works by P.E. Sobolevsky, and this method assumes to use the analytical theory of semigroups of bounded linear operators. In order to solve the occurring bifurcation problems there were used the investigation methods of dynamical systems with an infinite-dimensional phase space (a space of initial conditions) such as: the method of integral manifolds, the method of Poincare–Dulac normal forms and asymptotic methods of analysis. Both possible in the given situation problems were studied: in codimension one and in codimension two. In particular, asymptotic formulas were obtained for solutions which describe nonhomogeneous undulating surface relief. The question about the stability of these solutions was studied. And the analysis of normal form was given. Also the asymptotic formulas for the nonhomogeneous undulating solutions were obtained. In conclusion some possible interpretations of the obtained results are indicated.