Soliton-generating τ-functions revisited

Abstract

In the inverse-scattering formalism and the Hirota algorithm, soliton solutions of evolution equations are images of τ-functions, which are, often, unbounded in space and time. Hence, whereas solitons are bounded, localized structures, they are generated from unbounded, spatially extended entities that lack physically intuitive significance. The goal of this paper is to provide a representation of soliton solutions as images of bounded and localized sources. To this end, appropriate equivalent τ-functions, which are different in form from the traditionally used ones, are generated. In terms of the latter, the single-soliton solutions of the Korteweg-deVries (KdV) and Kadomtsev-Petviashvili II (KP II) equations are images of single solitons (NOT solutions of theses equations), whereas multi-soliton solutions are images of positive humps that are localized in the soliton interaction region. The structure of the equivalent τ-functions has another desirable attribute: It elucidates in a simple manner the role of the arbitrary shifts in the positions of soliton trajectories in controlling the behavior of multi-soliton solutions: Whether an N-soliton solution is reduced to a solution with (N − 1) or (N − 2) solitons when two wave numbers are made to coincide. Examples are discussed in the cases of the KdV and KP II equations. The modified KdV equation provides a unique observation. Depending on the shifts in the positions of soliton trajectories, when two wave numbers are made to coincide, the two-soliton solution may tend either to a single soliton or to a δ-function in the x-t plane.