Freedom in the expansion and obstacles to integrability in multiple-soliton solutions of the perturbed KdV equation

The construction of solutions of the perturbed KdV equation encounters obstacles to asymptotic integrability beyond the first order, when the zero-order approximation is not a single-soliton wave. In the standard analysis, the obstacles lead to the loss of integrability of the Normal Form, resulting in a zero-order term, which does not have the simple structure of the solution of the unperturbed equation. Exploiting the freedom in the perturbative expansion, an algorithm is proposed that shifts the effect of the obstacles from the Normal Form to the higher-order terms. The expansion has been carried out through third order. Through this order, the Normal Form remains integrable, and the zero-order approximation retains the structure of the unperturbed solution. For multiple-soliton solutions, the resulting obstacles decay exponentially away from the soliton-interaction region, which is a finite domain around the origin in the x-t plane. The effect is demonstrated in detail through second order for the two-soliton case, where the obstacles generate a second-order radiative tail that emanates from the origin, and decays exponentially away from the origin. These results suggest a new meaning to “asymptotic” integrability: The zero-order term is determined by an integrable Normal Form, and the higher-order terms in the expansion of the full solution tend to those of the integrable case (when no obstacles exist) asymptotically away from the origin.