Freedom in the Normal Form Expansion and Obstacles to Asymptotic Integrability: The Perturbed KdV Equation

Integrable evolution equations, to which a perturbation is added, often lose their integrability when
multiple-wave solutions are considered. In the Normal Form expansion, this leads to the emergence of terms that the formalism cannot account for. In the standard application of the method, these terms spoil the integrability of the normal form (hence, “obstacles to asymptotic integrability”), leading to distortion of the multiple-wave structure of the zero-order approximation, even asymptotically far from the interaction region of the waves. While obstacles do not emerge in the case of single-wave solutions, this is not borne out by the analysis of the general case. Exploiting the freedom inherent in the Normal Form method, an alternative expansion algorithm is proposed, resulting in obstacles that are expressed in terms of symmetries of the unperturbed equation, and which vanish explicitly for single-wave zero-order solutions. The normal form remains integrable, generating multiple-wave solutions of the same structure as generated by the unperturbed equation, because the effect of the obstacles is shifted from the normal form to the higher-order terms in the expansion of the solution. The obstacles affect the solution only in the interaction region of the waves, which is a finite domain around the origin in the x-t plane in the case of solitons. Results are presented for the perturbed KdV equation.