An inverse scattering scheme for the regularized long-wave equation

Abstract

The regularized long-wave (RLW) equation was proposed as an alternative model to the Korteweg–de Vries (KdV) equation to describe small-amplitude long waves in shallow water. However, unlike the KdV equation—which is exactly solvable by the inverse scattering method—the RLW equation is deemed not to be completely integrable. In this paper, an inverse scattering scheme is described for solving the RLW equation which is based on the “dressing method” of Zakharov and Shabat. Significantly, the compatibility of the “dressed” operators leads to two governing equations which, taken together, reduce to the RLW equation. One of these equations links the RLW equation to the (completely integrable) shallow water wave (ASWW) equation due to Ablowitz et al.; the second is a linear wave constraint that signals the nonintegrability of the RLW equation and prohibits the existence of multisoliton solutions. Contrary to results already reported in the literature, the proposed scheme associates the RLW equation with a third-order scattering problem. Moreover, as a special case of the RLW scheme, we obtain an inverse scattering procedure for the integrable ASWW equation; its formulation via the dressing method would appear to be given here for the first time. Explicit solutions are constructed for both equations using the respective inverse scattering schemes.