Dynamics of two-component parabolic systems of Schr\"{o}dinger type

S. A. Kashchenko

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 26 (2018) no. 5, pp. 81-100.

Résumé

Issue. The paper considers the local dynamics of important for applications class of two-component nonlinear systems of parabolic equations. These systems contain a small parameter appearing in the diffusion coefficients and characterizing «closeness» of the initial system of a parabolic type to a hyperbolic one. On quite natural conditions critical cases in the problem about balance state stability are realized to linearized equation coefficients. Innovation. An important thing here is the fact that these critical cases have an infinite dimension: infinitely many roots of a standard equation go to the imaginary axis when a small parameter vanishes. The specificity of all considered critical cases is typical of Schrödinger type systems and of a classical Schrödinger equation, in particular. These peculiarities are connected with the arrangement of roots of a standard equation. Three most important cases are stood here. Note that they fundamentally differ from each other. This difference is basically determined by the presence of specific resonance relations in the considered cases. It is these relations that define the structure of nonlinear functions included in normal forms. Investigation methods. A normalization algorithm is offered, that is the reduction of the initial system to the infinite system of ordinary differential equations for slowly changing amplitudes. Results. The situations when the corresponding systems can be compactly written as boundary-value problems with special nonlinearities are picked out. These boundary-value problems play the role of normal forms for initial parabolic systems. Their nonlocal dynamics determines the behavior of the solutions of the initial system with the initial conditions from some sufficiently small and not depending on a small parameter balance state neighborhood. Scalar complex parabolic Schr.odinger equations are considered as important applications. Conclusions. The problem about the local dynamics of two-component parabolic systems of Schr.odinger type is reduced to the investigation of nonlocal behavior of the solutions of special nonlinear evolutionary equations.

Détail

Keywords: dynamics, normal forms, Schrödinger equation.