1210.0803 (Ekaterina Shemyakova)

Invertible Darboux Transformations    [PDF]

Ekaterina Shemyakova

Darboux transformations were originally introduced for univariate operators and for hyperbolic bivariate operators of second order. These transformations were then generalized for other kinds of operators. Darboux's original formulas written in terms of Wronskians have been generalized for those cases also. For some cases it has been proved that no other Darboux transformations, other than those generated by these Wronskian formulas, are possible. Darboux transformations generated by Darboux Wronskian formulas are not invertible, in the sense that the corresponding mappings of the operator kernels are not invertible. Until now, only Laplace transformations, which are special cases of Darboux transformations for hyperbolic bivariate operators of second order, are known to be different from Darboux transformations generated by Darboux Wronskian formulas and are invertible in the above sense. In the present paper we show that for a bivariate linear partial differential operator of arbitrary order there are many Darboux transformations that, first, cannot be represented by Darboux Wronskian formulas, and, secondly, are invertible transformations. Among other results, we find a criteria for such operators to have invertible Darboux transformations. We give conditions under which an operator has a Darboux transformation generated by a Darboux Wronskian formula.

View original: http://arxiv.org/abs/1210.0803