Convective Cahn–Hilliard–Oono Equation

A nonlinear evolutionary partial differential equation is considered that is obtained as a natural (from the physical point of view) generalization of the well-known Cahn–Hilliard equation. The generalized version is supplemented with terms responsible for convection and dissipation. The new version of the equation is considered together with homogeneous Neumann boundary conditions. For such a boundary value problem, local bifurcations of codimensions one and two are studied. In both cases, the existence, stability, and asymptotic representation of spatially inhomogeneous equilibrium states, as well as invariant manifolds formed by such solutions of the boundary value problem, are analyzed. To substantiate the results, methods of the modern theory of infinite-dimensional dynamical systems are used, including the method of integral manifolds and the apparatus of the Poincaré theory of normal forms. Differences between the results of the analysis of bifurcations in the Neumann boundary value problem and the conclusions of the analysis of the periodic boundary value problem studied by the authors of this paper in previous publications are indicated.