Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation

A Kac-Moody algebra construction for the integrable hierarchy containing the Gardner equation is proposed. Solutions are systematically constructed employing the dressing method and deformed vertex operators which takes into account the nonvanishing boundary value problem for the mKdV hierarchy. Explicit examples are given and besides usual KdV like solitons, our solutions contemplate the large amplitude table-top solitons, kinks, dark solitons, breathers and wobbles.

💡 Research Summary

The paper presents a unified algebraic framework for integrable hierarchies that incorporates the Gardner equation as a member, using the affine Kac‑Moody algebra (\widehat{\mathfrak{sl}}{2}). Starting from the well‑known KdV and mKdV hierarchies, the authors first recast them in terms of a zero‑curvature condition based on the even‑odd grading of (\widehat{\mathfrak{sl}}{2}). They demonstrate that the Miura transformation (u=v^{2}+v_{x}) is not merely a map between two isolated equations but a homomorphism between the whole KdV hierarchy and the whole mKdV hierarchy. This is proved by constructing the recursion operators (R) for KdV and (R_{0}) for mKdV and showing that the Miura map factorises the higher‑order flows as well.

To treat non‑vanishing boundary conditions, a constant parameter (\mu) is introduced, leading to a deformed zero‑curvature equation (23) where the semi‑simple element (E(0){\alpha}+E(1){-\alpha}) is shifted by (\mu H(0)). This simple translation (v\rightarrow v+\mu) changes the vacuum of the Lax pair and yields, for the lowest odd grade (n=3), the Gardner equation \