N=2 supersymmetric unconstrained matrix GNLS hierarchies are consistent

We develop a pseudo-differential approach to the N=2 supersymmetric unconstrained matrix (k|n,m)-Generalized Nonlinear Schroedinger hierarchies and prove consistency of the corresponding Lax-pair representation (nlin.SI/0201026). Furthermore, we establish their equivalence to the integrable hierarchies derived in the super-algebraic approach of the homogeneously-graded loop superalgebra sl(2k+n|2k+m)\otimes C[{lambda},{lambda}^{-1}] (nlin.SI/0206037). We introduce an unconventional definition of N=2 supersymmetric strictly pseudo-differential operators so as to close their algebra among themselves.

💡 Research Summary

The paper presents a comprehensive pseudo‑differential formulation of the N=2 supersymmetric unconstrained matrix (k|n,m) Generalized Nonlinear Schrödinger (GNLS) hierarchies and rigorously establishes the consistency of the associated Lax‑pair representation. The authors begin by recalling that most previous work on N=2 supersymmetric GNLS systems has been limited to constrained superfields, which satisfy chirality or other differential constraints. In contrast, the present study deals with completely unconstrained matrix superfields, i.e., superfields that depend arbitrarily on the N=2 superspace coordinates (z, θ, \barθ). This generality demands a careful treatment of the underlying operator algebra because the usual pseudo‑differential operators, which involve both positive and negative powers of the ordinary derivative ∂, do not close under multiplication when supersymmetric covariant derivatives D and \bar D are present.

To overcome this obstacle the authors introduce an unconventional definition of “strictly pseudo‑differential operators”. In their construction the only allowed negative powers are those of the bosonic derivative ∂⁻¹; any term where ∂⁻¹ is multiplied by the Grassmann coordinates θ or \bar θ is explicitly excluded. Moreover, the covariant derivatives D and \bar D are required to commute with ∂⁻¹ (i.e., D∂⁻¹ = ∂⁻¹D, \bar D∂⁻¹ = ∂⁻¹\bar D). With these rules the set of strictly pseudo‑differential operators forms a closed algebra, which is essential for defining a well‑behaved Lax operator.

The Lax operator is taken in the standard form
L = ∂ + ∑{i≥0} U_i ∂^{-i},
where each coefficient U_i is an (k|n,m) matrix whose entries are unconstrained N=2 superfields. The positive‑degree part of L, denoted B_n, generates the nth flow through the Lax equation ∂{t_n}L =