Overcoming Obstacles to Integrability I: Perturbed Diffusive Systems

Abstract

Obstacles to integrability sometimes hamper the standard Normal Form analysis of perturbed integrable evolution equations. One is then forced to account for them by the Normal Form, which is the dynamical equation obeyed by the zero-order term. This spoils the integrability of the normal form, and the simple wave-nature of the zero-order approximation. To avoid both undesired results, one must require that the obstacles be accounted for by the higher-order terms in the expansion of the solution (the Near-Identity Transformation). We show that this goal can be achieved if the higher-order terms are allowed to depend explicitly on the independent variables,t and x (an option that is not considered in the standard NF analysis). In addition, a particular algorithm for the construction of the NIT leads to a canonical form for the obstacles; they are expressed in terms of the symmetries of the unperturbed equation. The canonical form ensures the explicit vanishing of the obstacles in the case of a single-wave zero-order solution. In the case of a multiple-wave solution, the effect of the canonical obstacles on the higher-order terms in the NIT is confined to the region of interaction among the waves. This often facilitates the derivation of closed-form expressions for the asymptotic effect of the obstacles. These ideas are demonstrated for the cases of the perturbed Burgers and heat diffusion equations.