Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV

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.

Lie group approach for constructing all reciprocal transformations: The two‐dimensional stationary gas dynamics equations

“…For example, the nonlinear heat equation $$$$ {u}_t={\left({u}^{-2}{u}_x\right)}_x $$$$ is mapped into the classical linear heat equation $$$ {u}_t={u}_{xx} $$$ by the transformation 12,13 $$$$ U={u}^{-1}, dX=u\kern0.1em dx+{u}^{-2}{u}_x\kern0.1em dt, dT= dt. $$$$ Applications of reciprocal transformations to the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations and advantages of the reciprocal transformations can be found in Albares and Estévez 14 …”

“…Applications of reciprocal transformations to the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations and advantages of the reciprocal transformations can be found in Albares and Estévez. 14 The present paper deals with invariant reciprocal transformations, where a nondegenerate change (1) maps a system of differential equations of a given class into a system of the same class (only arbitrary elements can be changed). In the group analysis method, such these types of transformations are called equivalence transformations.…”

Integrability, rational solitons and symmetries for nonlinear systems in Biology and Materials Physics

“…In this context, transformations among families of integrable equations, such as Bäcklund and Miura transformations [361,362], hodograph transformations [97,158] or reciprocal transformations [30,217,243,358,359], emerge as a useful tool when it comes to identifying their integrability. These techniques have been successfully applied in literature, highlighting the work carried out by Estévez and collaborators [27,30,139,140,158,162], two of which are contributions of the author of this thesis. The Painlevé Property, as defined previously, might turn out to be an overly restrictive imposition to characterize the integrability of a differential equation, since it exclusively considers ordinary poles for the singularities of the solutions.…”

“…Furthermore, Lie symmetries for Lax pairs yield much more information than just the symmetries of a set of PDEs. This procedure allows us not only to obtain the symmetry transformations of the fields, but also provides how the eigenfunctions and the spectral parameter are transformed under the action of the symmetry group [24,25,27,28,144,155].…”

“…A promising line of research in this regard requires a detailed consideration of the different existing transformations among families of integrable systems, such as Bäcklund transformations, Miura transformations, hodograph or reciprocal transformations. The author of this manuscript has carried out further research on reciprocal transformations and the composition of Miura-reciprocal transformations [27,30], not included in this dissertation. Reciprocal transformations have proved to be a remarkably fruitful technique to identify the integrability of PDEs, derive Lax pairs and may have an impact on the obtention of solutions of interest for those systems.…”