Extremals of conserved quantities for an equation are frequently used as first approximations to solutions of the perturbed equation. The subsequent evolution of such an approximation is the subject of this paper. The main idea is as follows. For an evolution equation and in the case of extremal conditions which do not depend on time explicitly, the extremal condition is a differential equation in spatial independent variables. Its solution can be used as an initial datum to the perturbed equation. Then, the approximate solution may be obtained by solving a simplified perturbed equation. In fact, this approach is applicable in wider circumstances, for equations which are first-order in the time variable, as demonstrated by the examples of KdV-B, BBM, and the shallow water equations.
A. Samokhin (Sat,) studied this question.