zbMATH — the first resource for mathematics

Invertible Darboux transformations of type I. (English. Russian original) Zbl 1406.44003
Program. Comput. Softw. 41, No. 2, 119-125 (2015); translation from Programmirovanie 41, No. 2 (2015).
Summary: In the paper, a general category-based approach to invertible Darboux transformations is suggested. For operators of arbitrary dimension and arbitrary order, it is suggested to consider Darboux transformations of certain type, namely, of type I. Explicit compact formulas are obtained. In particular, any invertible Darboux transformations of first order and, for example, Laplace transformations are type I transformations. For operators of third order in the plane, analogues of the Darboux transformation chains are obtained.

44A15 Special integral transforms (Legendre, Hilbert, etc.)
35A30 Geometric theory, characteristics, transformations in context of PDEs
Full Text: DOI
[1] Grinevich, PG; Mironov, AE; Novikov, SP, 2D-Schrödinger operator, (2+1) evolution systems and new reductions, 2D-Burgers hierarchy and inverse problem data, Russ. Math. Surv., 65, 580-582, (2010) · Zbl 1231.35199
[2] Li, CX; Nimmo, JJC, Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2120, 2471-2493, (2010) · Zbl 1194.35374
[3] Grinevich, P.G. and Novikov, S.P., Discrete SL2 connections and self-adjoint difference operators on the triangulated 2-manifold, ArXiv preprint arXiv:1207. 1729, 2013.
[4] Tsarev, SP, Generalized Laplace transformations and integration of hyperbolic systems of linear partial differential equations, 325-331, (2005), Beijing · Zbl 1360.35116
[5] Tsarev, SP, On factorization and solution of multidimensional linear partial differential equations, (2007)
[6] Ganzha, EI, On Laplace and dini transformations for multidimensional equations with a decomposable principal symbol, Programming Comput. Software, 38, 150-155, (2012) · Zbl 1254.35135
[7] Ganzha, E.I., Intertwining Laplace transformations of linear partial differential equations, arXiv preprint arXiv: 1306.1113, 2013.
[8] Shemyakova, ES, Invertible Darboux transformations, (2013) · Zbl 1384.37097
[9] Shemyakova, ES, Proof of the completeness of Darboux Wronskian formulas for order two, Canadian J. Math., 65, 655-674, (2013) · Zbl 1275.53088
[10] Darboux, G., Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal, vol. 2, Paris: Gauthier-Villars, 1915. · JFM 45.0881.05
[11] Tsarev, SP, On Darboux integrable nonlinear partial differential equations, Proc. Steklov Inst. Math., 225, 372-381, (1999) · Zbl 0990.35008
[12] Shemyakova, ES, Invariant properties of third-order non-hyperbolic linear partial differential operators, Lecture Notes Computer Sci., 5625, 154-169, (2009) · Zbl 1247.68328
[13] Shemyakova, ES, A package to work with linear partial differential operators, Programming Comput. Software, 39, 212-220, (2013) · Zbl 1311.65180
[14] Shemyakova, E; Winkler, F, Obstacles to factorization of partial differential operators into several factors, Programming Comput. Software, 33, 67-73, (2007) · Zbl 1130.35007
[15] Shemyakova, E, Invariants for Darboux transformations of arbitrary order for \(D\)_{x}\(D\)_{y} + ad_{x} + bd_{y} + \(c\), 155-162, (2013), Basel
[16] Shemyakova, E, A full system of invariants for thirdorder linear partial differential operators, Lecture Notes Comput. Sci., 4120, 360-369, (2006)
[17] Shemyakova, E; Winkler, F, A full system of invariants for third-order linear partial differential operators in general form, Lecture Notes Comput. Sci., 4770, 360-369, (2007) · Zbl 1141.68711
[18] Shemyakova, ES; Mansfield, EL, Moving frames for Laplace invariants, 295-302, (2008), New York
[19] Shemyakova, E., Factorization of Darboux transformations of arbitrary order for two-dimensional Schroedinger operator, arXiv preprint arXiv:1304.7063, 2013. · Zbl 1305.70038
[20] Tsarev, S.P., Factorization in categories of systems of linear partial differential equations, Arxiv preprint arXiv:0801.1341, 2008.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.