Miura transformation for the “good” Boussinesq equation

Abstract

AbstractIt is well known that each solution of the modified Korteveg–de Vries (mKdV) equation gives rise, via the Miura transformation, to a solution of the Korteveg–de Vries (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.