Spectral problem for the complex mKdV equation: singular manifold method and Lie symmetries

TL;DR

This paper investigates the complex mKdV equation using the singular manifold method to derive spectral problems and transformations, and identifies Lie symmetries to find reductions and integrals for associated ODEs.

Contribution

It introduces a combined approach of the singular manifold method and Lie symmetry analysis for the complex mKdV equation, providing new spectral and reduction results.

Findings

Derived the spectral problem and Darboux transformations for the complex mKdV.

Identified Lie symmetries and obtained similarity reductions.

Established first integrals for reduced ordinary differential equations.

Abstract

This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary Darboux transformations and $\tau$-functions. The second part concerns the identification of the classical Lie symmetries for the spectral problem. The similarity reductions associated to these symmetries allow us to derive the reduced spectral problems and first integrals for the ordinary differential equations arising from such reductions.