Perturbative Analysis of Wave Interaction in Nonlinear Systems

We propose a new way to handle obstacles to asymptotic integrability in perturbed nonlinear PDEs in the method of normal forms (NFs) in the case of multiwave solutions. Instead of including the whole obstacle in the NF, we include only its resonant part (if it exists) in the NF and assign the remainder to the homological equation. This leaves the NF integrable, and its solutions retain the character of the solutions of the unperturbed equation. We use the freedom in the expansion to construct canonical obstacles that are confined to the interaction region of the waves. For soliton solutions (e. g., of the KdV equation), the interaction region is a finite domain around the origin; the canonical obstacles then do not generate secular terms in the homological equation. When the interaction region is infinite (or semi-infinite, e.g., in wave-front solutions of the Burgers equation), the obstacles may contain resonant terms. The obstacles generate waves of a new type that cannot be written as functionals of the solutions of the NF. When the obstacle contributes a resonant term to the NF, this leads to a nonstandard update of the wave velocity.