Darboux transformation

The Darboux transformation is a transformation of the solutions and coefficient functions of a ( partial differential equation ), which generates a new solution of a form-like differential equation (with a different coefficient function). It is used, for example, to find further solutions for nonlinear partial differential equations, especially in the theory of solitons , where the associated differential equations are, for example, the nonlinear Schrödinger equation or the Korteweg-de-Vries equation .

It is named after Gaston Darboux , but were already Th.-F. Moutard (1875, 1878) known.

The classic example is the second order differential equation of the Sturm-Liouville type :

${\displaystyle -f_{xx}(x)+u(x)f(x)=\lambda f(x)}$

with the coefficient function and a constant . The second derivative refers to the independent variable . The differential equation corresponds to a stationary Schrödinger equation with potential . Let it be another solution to this differential equation. A Darboux transformation is then given by: ${\displaystyle u(x)}$ ${\displaystyle \lambda }$ ${\displaystyle f_{xx}}$ ${\displaystyle x}$ ${\displaystyle u(x)}$ ${\displaystyle g(x)}$

${\displaystyle h(x)=f_{x}(x)+k(x)f(x)}$ With ${\displaystyle k(x)=-{\frac {g_{x}(x)}{g(x)}}}$

Because is a solution of the form-like differential equation: ${\displaystyle h(x)}$

${\displaystyle -h_{xx}(x)+v(x)h(x)=\lambda h(x)}$

with the new coefficient function ${\displaystyle v(x)=u(x)+2k(x)}$

The spectral properties of the classical Darboux transformations were further investigated by MM Crum in 1955 and they were used (as Crum transformations) in the theory of solitons by M. Wadati and colleagues, who also examined the connection with the Bäcklund transformation , which Vladimir Borissowitsch Matvejew Continued in the late 1970s.

literature

• C. Rogers, WK Schief: Backlund and Darboux Transformations, Cambridge University Press 2002, chapter 7
• VB Matveev, MA Salle: Darboux transformations and solitons, Springer, 1991

Web links

• SB Leble, Darboux Transformation, Encyclopedia of Mathematics , Springer

References and comments

1. ↑ Darboux uses them in his Leçons sur la théorie général des surfaces , 2nd edition, Gauthier-Villars 1912
2. ^ Moutard, J. Ecole Polytechnique, 45, 1878, 1-11
3. ^ MM Crum, Associated Sturm-Liouville systems, QJ Math. Oxford, Vol. 6, 1955, pp. 121-127
4. ↑ M. Wadati, H. Sanuki, K. Konno, Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws, Progr. Theor. Phys., Vol. 53, 1975, pp. 419-436