The paper describes some probabilistic and combinatorial aspects of the nonlinear Fourier transform associated with AKNS-ZS problems. One two main results shows that volumes a family polytopes appear in power expansion are distributed according to beta probability distribution. We establish this result by studying an Euler-type discretisation transform. This approach provides another result. di...

Using bidifferential calculus, we derive a vectorial binary Darboux transformation for the first member of "negative" part AKNS hierarchy. A reduction leads to "negative flow" NLS hierarchy, which in turn is rather simple nonlinear complex PDE two dimensions, with leading mixed third derivative. This may be regarded as describing geometric dynamics scalar field one dimension, since it invariant...

In this note we prove that the maximally defined operator associated with the Dirac-type differential expression M(Q) = i ( d dx Im −Q −Q − d dx Im ) , where Q represents a symmetric m × m matrix (i.e., Q(x) = Q(x) a.e.) with entries in L loc (R), is J -self-adjoint, where J is the antilinear conjugation defined by J = σ1C, σ1 = ( 0 Im Im 0 ) and C(a1, . . . , am, b1, . . . , bm) = (a1, . . . ,...