  The perturbed NLS equation is asymptotically integrable if the zero-order
approximation to the solution is a single soliton. It is not integrable for any
other zero-order solution. In the standard application of a Normal Form
expansion, integrability is recovered if, from second order onwards, the
perturbation obeys algebraic constraints. In general, these are not satisfied.
As a result, the Normal Form is not asymptotically integrable; its solution,
the zero-order approximation, loses the simple structure of the solution of the
unperturbed equation. This work exploits the freedom in the perturbative
expansion to propose an alternative, for which the loss of integrability only
affects higher-order terms in the expansion of the solution. The Normal Form
remains asymptotically integrable, and the zero-order approximation, not
affected by the loss of integrability, has the same structure as the solution
of the unperturbed equation. In the case of a multiple-soliton zero-order
approximation, the loss of integrability affects the higher-order corrections
to the solution in a finite domain around the origin in the x-t plane, the
soliton interaction region. The effect decays exponentially in all directions
in the plane as the distance from the origin grows. The computation is carried
through second order.
