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

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 trans...

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Bibliographic Details
Published in:Mathematics (Basel) 2021-05, Vol.9 (9), p.926
Main Authors: Albares, Paz, Estévez, Pilar Garcia
Format: Article
Language:English
Subjects:
conservation laws
Differential equations
Mathematical analysis
Mathematics
Partial differential equations
symmetry groups
Transformations
Variables
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math9090926