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An algorithm for constructing Darboux transformations of type I for third-order hyperbolic operators of two variables. (English. Russian original) Zbl 1344.65119
Program. Comput. Softw. 42, No. 2, 112-119 (2016); translation from Programmirovanie 42, No. 2 (2016).
Summary: Darboux transformations of type I are invertible Darboux transformations with explicit short formulas for inverse transformations. These transformations are invariant with respect to gauge transformations, and, for gauge transformations acting on third-order hyperbolic operators of two variables, a general-form system of generating differential invariants is known. In the paper, first-order Darboux transformations of type I for this class of operators are considered. The corresponding operator orbits are directed graphs with at most three edges originating from each vertex. In the paper, an algorithm for constructing such orbits is suggested. We have derived criteria for existence of first-order Darboux transformations of type I in terms of the generating invariants, formulas for transforming invariants, and the so-called “triangle rule” property of orbits. The corresponding implementation in the LPDO package is described. The orbits are constructed in two different forms, one of which outputs the graph in the format of the well-known built-in Maple package Graph Theory.
65P99 Numerical problems in dynamical systems
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37D05 Dynamical systems with hyperbolic orbits and sets
GraphTheory; LPDO; Maple
Full Text: DOI
[1] Grinevich, P.G.; Mironov, A.E.; Novikov, S.P., The two-dimensional Schrödinger operator: evolution (2+1)-systems and new reductions, the two-dimensional Burgers hierarchy and inverse problem data, Usp. Mat. Nauk, 65, 195-196, (2010) · Zbl 1231.35199
[2] 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, 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, 2013. ArXiv preprint arXiv: 1207.1729. · Zbl 1282.39006
[4] Voronov, F.F.; Gill, S.P.; Shemyakova, E.S., Darboux transformations for differential operators on a superstraight line, (2015)
[5] Tsarev, S.P., Generalized Laplace transformations and integration of hyperbolic systems of linear partial differential equations, 325-331, (2005) · Zbl 1360.35116
[6] Tsarev, S.P., On factorization and solution of multidimensional linear partial differential equations, (2007)
[7] Ganzha, E.I., On Laplace and dini transformations for multidimensional equations with a decomposable principal symbol, Program. Comput. Software, 38, 150-155, (2012) · Zbl 1254.35135
[8] Ganzha, E.I., Intertwining Laplace transformations of linear partial differential equations, 2013. arXiv preprint arXiv:1306.1113. · Zbl 1375.35010
[9] Shemyakova, E., Invertible Darboux transformations of type I, Program. Comput. Software, 41, 119-125, (2015) · Zbl 1406.44003
[10] Shemyakova, E., Invertible Darboux transformations, Proc. of Conf. “Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 2013, no. 9(002). http://arxivorg/abs/1210.0803. · Zbl 1384.37097
[11] Shemyakova, E.S.; Kielanowski, P. (ed.); Ali, S.T. (ed.); Odesski, A. (ed.); Odzijewicz, A. (ed.); Schlichenmaier, M. (ed.); Voronov, Th. (ed.), Orbits of Darboux groupoid for hyperbolic operators of order three, (2014), Basel
[12] Tsarev, S.P., On Darboux integrable nonlinear partial differential equations, Proc. Steklov Inst. Math., 225, 372-381, (1999) · Zbl 0990.35008
[13] Shemyakova, E., Proof of the completeness of Darboux Wronskian formulas for order two, Canadian J. Math., 65, 655-674, (2013) · Zbl 1275.53088
[14] Shemyakova, E.; Winkler, F., A full system of invariants for third-order linear partial differential operators in general form, Lect. Notes Comput. Sci., 4770, 360-369, (2007) · Zbl 1141.68711
[15] Shemyakova, E.S., A package to work with linear partial differential operators, Program. Comput. Software, 39, 212-220, (2013) · Zbl 1311.65180
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