Modified KdV hierarchy : Lax pair representation and bi-Hamiltonian structure
Amitava Choudhuri, B. Talukdar, U. Das
TL;DR
This paper explores the modified KdV hierarchy, deriving Lax pairs and bi-Hamiltonian structures using the Miura transformation, and provides a Lagrangian formulation for the complex mKdV equation with semi-analytical solutions.
Contribution
It introduces a Lagrangian-based approach to study the bi-Hamiltonian structure of the mKdV hierarchy and constructs semi-analytical solutions for the complex mKdV equation.
Findings
Derived Lax pair expressions using Miura transformation
Established Lagrangian representation for cmKdV equation
Constructed semi-analytical solutions for cmKdV
Abstract
We consider equations in the modified KdV (mKdV) hierarchy and make use of the Miura transformation to construct expressions for their Lax pair. We derive a Lagrangian-based approach to study the bi-Hamiltonian structure of the mKdV equations. We also show that the complex modified KdV (cmKdV) equation follows from the action principle to have a Lagrangian representation. This representation not only provides a basis to write the cmKdV equation in the canonical form endowed with an appropriate Poisson structure but also help us construct a semianalytical solution of it. The solution obtained by us may serve as a useful guide for purely numerical routines which are currently being used to solve the cmKdV eqution.
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