Hamiltonian formulation of the supersymmetric KdV equation

TL;DR

This paper develops a Hamiltonian formulation for the supersymmetric KdV equation, revealing unique constraints and a nonlocal Hamiltonian component, and confirms the consistency of superspace and component descriptions.

Contribution

It provides the first detailed constrained Hamiltonian analysis of the supersymmetric KdV equation, including a nonlocal Hamiltonian density and superspace formulation.

Findings

Derived primary and secondary constraints for the system.

Identified a nonlocal contribution to the Hamiltonian density.

Confirmed the Hamiltonian equations reproduce the supersymmetric KdV system.

Abstract

We studied the constrained Hamiltonian formulation of a supersymmetric Korteweg-de Vries (KdV) equation, which is observed to be a constrained system similar to its classical version. We found a nontrivial Lagrangian description, where we select $a=2$ for the free parameter $a$ in the supersymmetric extension. The corresponding degenerate Lagrangian requires an exclusive consideration and the utilization of the Dirac-Bergmann algorithm. We explicitly determined the full set of primary and secondary constraints and constructed the total Hamiltonian governing the dynamics of the system. In this analysis, in addition to a nontrivial constraint involving the fermionic fields, the consistency conditions give rise to a nonlocal contribution to the Hamiltonian density. This highlights a distinctive feature of this supersymmetric extension. We showed that the resulting Hamilton equations of motion reproduce the supersymmetric KdV system in the component form. Finally, we derived a compact superspace representation of the Hamiltonian and demonstrated its consistency with the component-level formulation.