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.

##### MSC:

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 |

##### Keywords:

Darboux transformation; inverse transformation; gauge transformation; third-order hyperbolic operator; operator orbit; directed graph##### References:

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