
SIGMA 9 (2013), 002, 10 pages arXiv:1210.0803
http://dx.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)
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References
 Bagrov V.G., Samsonov B.F., Darboux transformation of the Schrödinger
equation, Phys. Part. Nuclei 28 (1997), 374397.
 Darboux G., Leçons sur la théorie générale des surfaces et les
applications géométriques du calcul infinitésimal. II, GauthierVillars, Paris, 1889.
 Ganzha E.I., On Laplace and Dini transformations for multidimensional equations
with a decomposable principal symbol, Program. Comput. Softw.
38 (2012), 150155.
 Grinevich P.G., Novikov S.P., Discrete SL_{2} connections and selfadjoint
difference operators on the triangulated 2manifold, arXiv:1207.1729.
 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), 24712493,
arXiv:0911.1413.
 Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series
in Nonlinear Dynamics, SpringerVerlag, Berlin, 1991.
 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, 119138.
 Novikov S.P., Veselov A.P., Exactly solvable twodimensional 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, 109132, mathph/0003008.
 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.
 Shemyakova E., Laplace transformations as the only degenerate Darboux
transformations of first order, Program. Comput. Softw. 38
(2012), 105108.
 Shemyakova E., Proof of the completeness of Darboux Wronskian formulas for
order two, Canad. J. Math., to appear, arXiv:1111.1338.
 Tsarev S.P., Factorization of linear partial differential operators and the
Darboux method for integrating nonlinear partial differential equations,
Theoret. Math. Phys. 122 (2000), 121133.
 Tsarev S.P., Shemyakova E., Differential transformations of secondorder
parabolic operators in the plane, Proc. Steklov Inst. Math.
266 (2009), 219227, arXiv:0811.1492.

