Miura transformation for the "good'' Boussinesq equation

TL;DR

This paper extends the concept of the Miura transformation to the 'good' Boussinesq equation, establishing a new link between solutions of second- and fourth-order equations via Riemann-Hilbert problems.

Contribution

It introduces a novel Miura-type transformation for the 'good' Boussinesq equation, connecting solutions through Riemann-Hilbert problem techniques.

Findings

Established a Miura-type transformation for the 'good' Boussinesq equation.

Connected solutions of second- and fourth-order equations via Riemann-Hilbert problems.

Provided a new method to generate solutions for the Boussinesq equation.

Abstract

It is well-known that each solution of the mKdV equation gives rise, via the Miura transformation, to a solution of the KdV equation. In this work, we show that a similar Miura-type transformation exists also for the ``good'' Boussinesq equation. This transformation maps solutions of a second-order equation to solutions of the fourth-order Boussinesq equation. Just like in the case of mKdV and KdV, the correspondence exists also at the level of the underlying Riemann--Hilbert problems and this is in fact how we construct the new transformation.