In mathematics, the Darboux transformation, named after Gaston Darboux (1842–1917), is a method of generating a new equation and its solution from the known ones.
It is widely used in inverse scattering theory, in the theory of orthogonal polynomials,[1][2] and as a way of constructing soliton solutions of the KdV hierarchy.
[3] From the operator-theoretic point of view, this method corresponds to the factorization of the initial second order differential operator into a product of first order differential expressions and subsequent exchange of these factors, and is thus sometimes called the single commutation method in mathematics literature.
[4] The Darboux transformation has applications in supersymmetric quantum mechanics.
[5][6] The idea goes back to Carl Gustav Jacob Jacobi.
be a fixed strictly positive solution of the same equation for some
and its general solution can be found by d’Alembert's method: where
Darboux transformation modifies not only the differential equation but also the boundary conditions.
[8][9][10][11] On the other hand, it also allows one to convert inverse square singularities to Dirichlet boundary conditions and vice versa.
[12][13] Thus Darboux transformations relate eigenparameter-dependent boundary conditions with inverse square singularities.