Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 002, 10 pages      arXiv:1210.0803      https://doi.org/10.3842/SIGMA.2013.002
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Invertible Darboux Transformations

Ekaterina Shemyakova
Department of Mathematics, SUNY at New Paltz, 1 Hawk Dr. New Paltz, NY 12561, USA

Received October 01, 2012, in final form January 01, 2013; Published online January 04, 2013

Abstract
For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding mappings of the operator kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases of Darboux transformations for hyperbolic bivariate operators of order 2. In the present paper we find a criteria for a bivariate linear partial differential operator of an arbitrary order d to have an invertible Darboux transformation. We show that Wronkian formulae may fail in some cases, and find sufficient conditions for such formulae to work.

Key words: Darboux transformations; Laplace transformations; 2D Schrödinger operator; invertible Darboux transformations.

pdf (319 kb)   tex (15 kb)

References

  1. Bagrov V.G., Samsonov B.F., Darboux transformation of the Schrödinger equation, Phys. Part. Nuclei 28 (1997), 374-397.
  2. Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. II, Gauthier-Villars, Paris, 1889.
  3. Ganzha E.I., On Laplace and Dini transformations for multidimensional equations with a decomposable principal symbol, Program. Comput. Softw. 38 (2012), 150-155.
  4. Grinevich P.G., Novikov S.P., Discrete SL2 connections and self-adjoint difference operators on the triangulated 2-manifold, arXiv:1207.1729.
  5. Li C.X., Nimmo J.J.C., Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010), 2471-2493, arXiv:0911.1413.
  6. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.
  7. Novikov S.P., Four lectures: discretization and integrability. Discrete spectral symmetries, in Integrability, Lecture Notes in Physics, Vol. 767, Editor A.V. Mikhailov, Springer, Berlin, 2009, 119-138.
  8. Novikov S.P., Veselov A.P., Exactly solvable two-dimensional Schrödinger operators and Laplace transformations, in Solitons, geometry, and topology: on the crossroad, Amer. Math. Soc. Transl. Ser. 2, Vol. 179, Amer. Math. Soc., Providence, RI, 1997, 109-132, math-ph/0003008.
  9. Rogers C., Schief W.K., Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
  10. Shemyakova E., Laplace transformations as the only degenerate Darboux transformations of first order, Program. Comput. Softw. 38 (2012), 105-108.
  11. Shemyakova E., Proof of the completeness of Darboux Wronskian formulas for order two, Canad. J. Math., to appear, arXiv:1111.1338.
  12. Tsarev S.P., Factorization of linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations, Theoret. Math. Phys. 122 (2000), 121-133.
  13. Tsarev S.P., Shemyakova E., Differential transformations of second-order parabolic operators in the plane, Proc. Steklov Inst. Math. 266 (2009), 219-227, arXiv:0811.1492.

Previous article  Next article   Contents of Volume 9 (2013)