An inverse scattering study of the radiation solution of the Korteweg–de Vries equation: the generic and nongeneric cases

Abstract

Although the asymptotic (t → ∞) behaviour of radiation solutions of the Korteweg–de Vries (KdV) equation has been investigated in the literature using the inverse scattering transform method (ISTM), the complete temporal (and spatial) evolution has not been studied in detail using this method. In this paper we discuss the application of the inverse scattering expansion method, a method that we have successfully applied to a wide variety of nonlinear evolution equations of physical interest, to both the generic and nongeneric cases. Using model inputs as illustrative examples, we find that (a) unlike all other problems studied to date, the "natural" expansion parameter is not the (dimensionless) area of the input potential in the direct (eigenvalue) problem, and (b) the value of the reflection coefficient R(k) at zero eigenvalue, i.e., k = 0, plays a crucial role in the success or failure of what we will call the "standard" expansion method. The standard expansion method works for input potentials belonging to the "nongeneric" class, [Formula: see text] but breaks down beyond lowest order in the expansion for potentials belonging to the "generic" class, R(0) = −1. Re-examination of the generic problem in the vicinity of k = 0 leads to a "renormalization" in each order of the expansion, which enables the generic case to be correctly solved. Unlike the nongeneric case, the generic solution is not found to be asymptotically valid.