Local Dynamics of Two-Component Parabolic Systems of Schröedinger Type

The local dynamics of a class of two-component nonlinear systems of parabolic equations is considered; this class is important for applications. Under sufficiently natural conditions on the coefficients of the linearized equation, the case of infinite dimension is realized, which is critical in the problem of stationary stability. An algorithm of normalization is proposed, i.e., a reduction to an infinite system of ordinary differential equations for slowly varying amplitudes. The situations are highlighted in which the corresponding systems can be compactly written in the form of boundary value problems with special nonlinearities. These boundary value problems play the role of normal forms for the original parabolic systems. Scalar complex parabolic equations of Schrödinger type are considered as important applications. The role of resonance relations when constructing nonlinear functions entering normal forms are revealed.