Freedom in Expansion and Asymptotic Integrability of Perturbed Evolution Equations

Abstract

It is shown that the emergence of obstacles to asymptotic integrability in the analysis of perturbed evolution equations may, often, be a consequence of the manner, in which the freedom in the ex-pansion is exploited in the derivation of the equations. Algorithms exist, which yield perturbed evolution equations that are devoid of the obstacles for cases, in which, traditionally, obstacles are encountered. The derivation of the perturbed KdV equation for two physical systems (propagation of small amplitude disturbances on a shallow fluid layer, and the ion acoustic wave equations in Plasma Physics), where a second-order obstacle is anticipated, and of the Burgers equation (one-dimensional propagation of weak shock waves in an ideal gas), where a first-order obstacle is an-ticipated, is examined. In all cases, the anticipated obstacles to integrability can be avoided.