Abstract. Darboux Wronskian formulas allow to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one cannot be represented this way. It has been a long standing problem on what are other exceptions. In our previous work we proved that among transformations of total order one there are no other exceptions. Here we prove that for transformations of total order two there are no exceptions at all. We also obtain a simple explicit invariant description of all possible Darboux Transformations of total order two.
Abstract. We consider differential operators on a supermanifold of dimension 1|1. We define non-degenerate operators as those with an invertible top coefficient in the expansion in the "superderivative" D (which is the square root of the shift generator, the partial derivative in an even variable, with the help of an odd indeterminate). They are remarkably similar to ordinary differential operators. We show that every non-degenerate operator can be written in terms of 'super Wronskians' (which are certain Berezinians). We apply this to Darboux transformations (DTs), proving that every DT of an arbitrary non-degenerate operator is the composition of elementary first order transformations. Hence every DT corresponds to an invariant subspace of the source operator and, upon a choice of basis in this subspace, is expressed by a super-Wronskian formula. We consider also dressing transformations, i.e., the effect of a DT on the coefficients of the non-degenerate operator. We calculate these transformations in examples and make some general statements.
The paper is devoted to the Darboux transformations, an effective algorithm for finding analytical solutions of partial differential equations. It is proved that Wronskian like formulas suggested by G. Darboux for the second order linear operators on the plane describe all possible differential transformations with ᏹ of the form D x + m(x, y) and D y + m(x, y), except for the Laplace transformations.
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