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

Paz Albares and Pilar Garcia Estévez
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Paz Albares: Departamento de Física Fundamental, Universidad de Salamanca, 37008 Salamanca, Spain
Pilar Garcia Estévez: Departamento de Física Fundamental, Universidad de Salamanca, 37008 Salamanca, Spain

Mathematics, 2021, vol. 9, issue 9, 1-11

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.

Keywords: symmetry groups; conservation laws; partial differential equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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