Laplace transformations as the only degenerate Darboux transformations of first order

Abstract

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.

Intertwining Laplace Transformations of Linear Partial Differential Equations

“…Definition 1 We will say that the differential operators L and L 1 defined above by the formulas (11) and (12) Proof. Since ord ω = ord X 2 − 1, the principal symbols of the operators M = X 2 and M 1 = X 2 + ω coincide.…”

“…In the past decade a number of publications [4,5,12,13,14,18,19,21,22] were devoted to application of various differential substitutions to construction of algorithms for closed-form solution of linear partial differential equations or systems of such equations. The obvious drawback was just the vast diversity of such differential substitutions, often considered as absolutely different in properties and necessary tools for their study.…”

Invertible Darboux Transformations

“…Proof . In [10] it has been proved that a Darboux transformation generated by M = D x + m or by M = D y + m is either a Laplace transformation, that is generated by M = D x + b or M = D y + a, or generated by M in the form D x − ψ 1,x ψ −1 1 , or D y − ψ 1,y ψ −1 1 , where ψ 1 ∈ ker L. In the latter case, ψ 1 ∈ (ker L ∩ ker M), and, therefore, the mapping ker L → ker L 1 is not invertible.…”

Proof of the Completeness of Darboux Wronskian Formulae for Order Two

“…After that work it was still unclear whether there are some exceptional transformations, that is such that cannot be described by Darboux Wronskian formulas among Darboux transformations of orders higher than one. This problem is reducing to solution of a system of two large non-linear PDEs, for which methods of the previous work Shemyakova (2012) were hard to apply. We, however, succeeded in proving that Darboux Wronskian formulas complete for transformations of order two in a different and rather elegant fashion, and present this proof in this paper.…”

“…Even for the simplest case of M of total degree 1, the problem of describing all Darboux transformations is not easy Shemyakova (2012). For the case of M of total degree 2, which is considered here the problem becomes very difficult.…”