Miura-reciprocal transformation and symmetries for the spectral problems of KdV and mKdV

TL;DR

This paper introduces reciprocal transformations for the spectral problems of KdV and mKdV equations, deriving new equations and symmetries, and analyzing their Lax pairs and similarity reductions.

Contribution

It presents novel reciprocal transformations connecting KdV and mKdV spectral problems, and derives their Lax pairs and symmetry reductions.

Findings

Derived reciprocal KdV and mKdV equations from original spectral problems.

Identified classical Lie symmetries for the new Lax pairs.

Obtained non-autonomous ODEs through similarity reductions.

Abstract

We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax pairs for RKdV and RmKdV are straightforwardly obtained by means of the aforementioned reciprocal transformations. We have also identified the classical Lie symmetries for the Lax pairs of RKdV and RmKdV. Non-trivial similarity reductions are computed and they yield non-autonomous ordinary differential equations (ODEs), whose Lax pairs are obtained as a consequence of the reductions.