Soliton-Generating ${\tau}$-Functions Revisited

TL;DR

This paper revisits soliton-generating { au}-functions within the inverse-scattering and Hirota frameworks, revealing their non-uniqueness and constructing bounded, localized solutions that extend traditional soliton families.

Contribution

It uncovers the non-uniqueness of { au}-functions, enabling the construction of bounded, localized solutions and extending the traditional soliton solution family.

Findings

{ au}-functions are not unique and can generate extended solution families.

Bounded { au}-functions lead to localized source images of solitons.

Traditional solutions are a subset within a broader parametric family.

Abstract

Within the framework of the Inverse-Scattering formalism and the Hirota algorithm, soliton solutions of evolution equations are images of {\tau}-functions. Typically, the latter are expressed in terms of exponentials, the arguments of which are linear in the coordinates. Consequently, often, {\tau}-functions are unbounded in space and time. However, they are not unique. Exploitation of their non-uniqueness uncovers physically interesting possibilities: 1) One can construct equivalent {\tau}-functions, which generate the same traditional (Inverse-Scattering/Hirota)) soliton solutions, yet allow for the extension of the family of soliton solutions to a wider, parametric family, in which the traditional solutions are a subset. The parameters are shifts in individual soliton trajectories. 2) When two wave numbers in a multi-soliton solution are made to coincide, the reduction of the solution to one with a lower number of solitons is qualitatively different for solutions that are within the traditional subset and those that are outside this subset. 3) One can construct {\tau}-functions that are bounded in space and time, in terms of which soliton solutions become images of localized sources.