Comments on the negative grade KdV hierarchy

Adans, Y. F.; Gomes, Jose F.; Lobo, G. V.; Zimerman, A. H.

doi:10.21468/scipostphysproc.14.014

SciPost Phys. Proc.

2023

DOI: 10.21468/scipostphysproc.14.014

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Comments on the negative grade KdV hierarchy

Y. F. Adans,

Jose F. Gomes,

G. V. Lobo

et al.

Abstract: The construction of negative grade KdV hierarchy is proposed in terms of a Miura-gauge transformation. Such gauge transformation is employed within the zero curvature representation and maps the Lax operator of the mKdV into its couterpart within the KdV setting. Each odd negative KdV flow is obtained from an odd and its subsequent even negative mKdV flows. The negative KdV flows are shown to inherit the two different vacua structure that characterizes the associated mKdV flows.

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2024

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SKdV, SmKdV flows and their supersymmetric gauge-Miura transformations

Adans,

Aguirre,

Gomes

et al. 2024

Open Communications in Nonlinear Mathematical Physics

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The construction of Integrable Hierarchies in terms of zero curvature representation provides a systematic construction for a series of integrable non-linear evolution equations (flows) which shares a common affine Lie algebraic structure. The integrable hierarchies are then classified in terms of a decomposition of the underlying affine Lie algebra $\hat \lie $ into graded subspaces defined by a grading operator $Q$. In this paper we shall discuss explicitly the simplest case of the affine $\hat {sl}(2)$ Kac-Moody algebra within the principal gradation given rise to the KdV and mKdV hierarchies and extend to supersymmetric models. It is known that the positive mKdV sub-hierachy is associated to some positive odd graded abelian subalgebra with elements denoted by $E^{(2n+1)}$. Each of these elements in turn, defines a time evolution equation according to time $t=t_{2n+1}$. An interesting observation is that for negative grades, the zero curvature representation allows both, even or odd sub-hierarchies. In both cases, the flows are non-local leading to integro-differential equations. Whilst positive and negative odd sub-hierarchies admit zero vacuum solutions, the negative even admits strictly non-zero vacuum solutions. Soliton solutions can be constructed by gauge transforming the zero curvature from the vacuum into a non trivial configuration (dressing method). Inspired by the dressing transformation method, we have constructed a gauge-Miura transformation mapping mKdV into KdV flows. Interesting new results concerns the negative grade sector of the mKdV hierarchy in which a double degeneracy of flows (odd and its consecutive even) of mKdV are mapped into a single odd KdV flow. These results are extended to supersymmetric hierarchies based upon the affine $\hat {sl}(2,1)$ super-algebra.

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SKdV, SmKdV flows and their supersymmetric gauge-Miura transformations

Adans,

Aguirre,

Gomes

et al. 2024

Open Communications in Nonlinear Mathematical Physics

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Comments on the negative grade KdV hierarchy

Cited by 1 publication

References 7 publications

SKdV, SmKdV flows and their supersymmetric gauge-Miura transformations

SKdV, SmKdV flows and their supersymmetric gauge-Miura transformations

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