Nonlinearization of the KdV-type and mKdV-type bilinear equations

TL;DR

This paper introduces a general method using Bell polynomials to nonlinearize bilinear equations, transforming integrable KdV and mKdV equations into nonlinear forms, and also discusses non-integrable cases.

Contribution

It presents a novel procedure for nonlinearizing bilinear equations, expanding the understanding of KdV and mKdV types with new nonlinear forms.

Findings

Derived nonlinear forms of KdV and mKdV bilinear equations

Identified non-integrable bilinear equations of KdV type

Provided examples demonstrating the nonlinearization process

Abstract

In this paper, we show a general procedure to nonlinearize bilinear equations by using the Bell polynomials. As applications, we obtain nonlinear forms of some integrable bilinear equations (in the sense having 3-soliton solutions) of the KdV type and mKdV type that were found by Jarmo Hietarinta in 1980s. Examples of non-integrable bilinear equations of the KdV type are also given.