On multidimensional Darboux transformations

TLDR: In contrast to known results on one-dimensional Darboux transformations, results on multidimensional transformations do not seem convenient for disentangling the sets of solutions in global inverse problems as discussed by the authors.

Abstract: In contrast to known results on one-dimensional Darboux transformations, results on multidimensional Darboux transformations do not seem convenient for disentangling the sets of solutions in global inverse problems. In the present paper, first, we give a broad generalization of vectorial Darboux transformations and a more limited generalization of binary transformations. Second, the reduction of multidimensional transformations in special examples is described to show the limits of their use in inverse theory, to generate one-dimensional transformations, and to give a better understanding of how the latter ones work.