On equivalence between solution structures of the anti-self-dual Yang--Mills equations

We establish an exact correspondence between the quasi-Wronskian and quasi-Grammian constructions for the anti-self-dual Yang–Mills (ASDYM) equations with complex $2\times 2$ matrix-valued fields. Starting from vacuum wave functions associated with $2M$ pairwise distinct nonzero constant spectral parameters, we restrict attention to a nonempty local domain on which the corresponding interpolation matrix is invertible. We show that both constructions are governed by the same uniquely solvable matrix-polynomial interpolation problem. In particular, the quasi-Wronskian Darboux polynomial $T(\lambda )$ and the quasi-Grammian rational dressing matrix $W(\lambda )$ satisfy $$T(\lambda )=q_{\xi }(\lambda )W(\lambda ),$$ where $q_{\xi }(\lambda )$ is the monic scalar polynomial whose zeros are the prescribed poles of $W(\lambda )$. This identity yields explicit relations between the corresponding Yang and potential matrices and shows that the two constructions generate identical ASDYM gauge potentials in the same gauge.