Darboux transformation
In mathematics, the Darboux transformation, named after Gaston Darboux, 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, and as a way of constructing soliton solutions of the KdV hierarchy. 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. The Darboux transformation has applications in supersymmetric quantum mechanics.
History
The idea goes back to Carl Gustav Jacob Jacobi.Method
Let be a solution of the equationand be a fixed strictly positive solution of the same equation for some. Then for,
is a solution of the equation
where
Also, for,
one solution of the latter differential equation is and its general solution can be found by d'Alembert's method:
where and are arbitrary constants.