## Classification of Darboux transformations for operators of the form $$\partial_x\partial_y+a\partial_x+b\partial_y+c$$.(English)Zbl 1445.35014

Summary: Darboux transformations are nongroup-type symmetries of linear differential operators. One can define Darboux transformations algebraically by the intertwining relation $$ML=L_1M$$ or the intertwining relation $$ML=L_1N$$ in the cases when the former is too restrictive.
Here we show that Darboux transformations for operators of the form $$L=\partial_x\partial_y+a\partial_x+b\partial_y+c$$ (sometimes referred to as 2D Schrödinger operators or Laplace operators) are always compositions of atomic Darboux transformations of two different well-studied types of Darboux transformations, provided that the chain of Laplace transformations for the original operator is long enough.

### MSC:

 35A22 Transform methods (e.g., integral transforms) applied to PDEs 35F05 Linear first-order PDEs 35B06 Symmetries, invariants, etc. in context of PDEs 35Q51 Soliton equations 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry

### Keywords:

nongroup-type symmetries
Full Text:

### References:

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