Darboux transformation

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.^[7]

Let ${\displaystyle y=y(x)}$ be a solution of the equation

${\displaystyle -y''(x)+q(x)y(x)=\lambda y(x)}$

and ${\displaystyle y=z(x)}$ be a fixed strictly positive solution of the same equation for some ${\displaystyle \lambda =\lambda _{0}}$. Then for ${\displaystyle \lambda \neq \lambda _{0}}$,

${\displaystyle Y(x):=y'(x)-{\frac {z'(x)}{z(x)}}y(x)=z(x)\left({\frac {y(x)}{z(x)}}\right)'}$

is a solution of the equation

${\displaystyle -Y''(x)+Q(x)Y(x)=\lambda Y(x),}$

where ${\displaystyle Q(x)=q(x)-2\left({\frac {z'(x)}{z(x)}}\right)'.}$ Also, for ${\displaystyle \lambda =\lambda _{0}}$, one solution of the latter differential equation is ${\displaystyle 1/z(x)}$ and its general solution can be found by d’Alembert's method:

${\displaystyle Y(x)={\frac {1}{z(x)}}\left(C_{1}+C_{2}\int ^{x}z^{2}(x)dx\right),}$

where ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ are arbitrary constants.

Darboux transformation modifies not only the differential equation but also the boundary conditions. This transformation makes it possible to reduce eigenparameter-dependent boundary conditions to boundary conditions independent of the eigenvalue parameter – one of the Dirichlet, Neumann or Robin 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.^[14]