Remarks on matrix hierarchies, d bar dressing, and the spectral transform

Abstract

MatrixHierarchies are construced within the framework of D bar dressingála Carroll-Konopelchenko [ 19 [ with particular emphasis on DS, Akns, and Kdv. For Akns and Ds a new a1 gebraic way of looking at certain aspects o f the inversespectralitransform (IST) emerges where suitable D bar data R_o ~ x = 1 + σ₁xi^⁁ (4 = xexp(-) is the matrix wave function ispaired with xl , and xi s explicitly computed from XI forsuitable xl . The DS case is in the general form with boundary ltnkagea_i, which appear i n xl. We also utilizerelations with tau functions via the Hirota bilinear identity obtained in the D bar context i n [ 119.[ CalculationsforKdV can be made vi a twisted or untwisted A, and one makes contact with a number of the standard algebraic and geometrical structures . These results can be viewed as a development i n a general program of 1inking scattering - dressing (analytic ) data tohierarchy ( algebraicj data, where the asymptotics of x leading t o the Hirotabilinear identity generate the a1 gebraic context