Ekaterina Shemyakova

Moving frames for laplace invariants

The development of symbolic methods for the factorization and integration of linear PDEs, many of the methods being generalizations of the Laplace transformations method, requires the finding of complete generating sets of invariants for the corresponding linear operators and their systems with respect to the gauge transformations L -> g(x,y)-1 O L O g(x,y). Within the theory of Laplace-like methods, there is no uniform approach to this problem, though some individual invariants for hyperbolic bivariate operators, and complete generating sets of invariants for second- and third-order hyperbolic bivariate ones have been obtained. Here we demonstrate a systematic and much more efficient approach to the same problem by application of moving-frame methods. We give explicit formulae for complete generating sets of invariants for second- and third-order bivariate linear operators, hyperbolic and non-hyperbolic, and also demonstrate the approach for pairs of operators appearing in Darboux transformations.

Differential transformations of parabolic second-order operators in the plane

Darboux’s classical results about transformations of second-order hyperbolic equations by means of differential substitutions are extended to the case of parabolic equations of the form Lu = (Dx2 + a(x, y)Dx + b(x, y)Dy + c(x, y))u = 0. We prove a general theorem that provides a way to determine admissible differential substitutions for such parabolic equations. It turns out that higher order transforming operators can always be represented as a composition of first-order operators that define a series of consecutive transformations. The existence of inverse transformations imposes some differential constrains on the coefficients of the initial operator. We show that these constraints may imply famous integrable equations, in particular, the Boussinesq equation.

A full system of invariants for third-order linear partial differential operators in general form

We find a full system of invariants with respect to gauge transformations L → g-1Lg for third-order hyperbolic linear partial differential operators on the plane. The operators are considered in a normalized form, in which they have the symbol SymL = (pX + qY)XY for some non-zero bivariate functions p and q. For this normalized form, explicit formulae are given. The paper generalizes a previous result for the special, but important, case p = q = 1.

Invertible Darboux Transformations

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 opera- tor 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 trans- formation. We show that Wronkian formulae may fail in some cases, and find sufficient conditions for such formulae to work.

Proof of the Completeness of Darboux Wronskian Formulae for Order Two

Darboux Wronskian formulas allow to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one cannot be represented this way. It has been a long standing problem on what are other exceptions. In our previous work we proved that among transformations of total order one there are no other exceptions. Here we prove that for transformations of total order two there are no exceptions at all. We also obtain a simple explicit invariant description of all possible Darboux Transformations of total order two.

Linear Partial Differential Equations and Linear Partial Differential Operators in Computer Algebra

In this survey paper we describe our recent contributions to symbolic algorithmic problems in the theory of Linear Partial Differential Operators (LPDOs). Such operators are derived from Linear Partial Differential Equations in the usual way. The theory of LPDOs has a long history, dealing with problems such as the determination of differential invariants, factorization, and exact methods of integration. The study of constructive factorization have led us to the notion of obstacles to factorization, to the construction of a full generating set of invariants for bivariate LPDOs of order 3, to necessary and sufficient conditions for the existence of a factorization in terms of generating invariants, and a result concerning multiple factorizations of LPDOs. We give links to our further work on generalizations of these results to n-variate LPDOs of arbitrary order.

Parametric Factorizations of Second-, Third- and Fourth-Order Linear Partial Differential Operators with a Completely Factorable Symbol on the Plane

Abstract.Parametric factorizations of linear partial operators on the plane are considered for operators of orders two, three and four. The operators are assumed to have a completely factorable symbol. It is proved that “irreducible” parametric factorizations may exist only for a few certain types of factorizations. Examples are given of the parametric families for each of the possible types. For the operators of orders two and three, it is shown that any factorization family is parameterized by a single univariate function (which can be a constant function).

A full system of invariants for third-order linear partial differential operators

A full system of invariants for a third-order bivariate hyperbolic linear partial differential operator L is found under the gauge transformation g(x1,x2)−−1Lg(x1,x2). That is, all other invariants can be obtained from this full system, and two operators are equivalent with respect to the gauge transformations if and only if their full systems of invariants are equal. To obtain the invariants, we generalize the notion of Laplace invariants from the case of order two to that of arbitrary order. This is done through the notion of common obstacles to factorizations into first-order factors. Explicit formulae for the invariants of a general operator are given in terms of the coefficients of the operator. The majority of the results were obtained using Maple 9.5.

Laplace transformations as the only degenerate Darboux transformations of first order

The paper is devoted to the Darboux transformations, an effective algorithm for finding analytical solutions of partial differential equations. It is proved that Wronskian-like formulas suggested by G. Darboux for the second-order linear operators on the plane describe all possible differential transformations with M of the form Dx + m(x, y) and Dy + m(x, y), except for the Laplace transformations.

Refinement of Two-Factor Factorizations of a Linear Partial Differential Operator of Arbitrary Order and Dimension

Given a right factor and a left factor of a Linear Partial Differential Operator (LPDO), under which conditions we can refine these two-factor factorizations into one three-factor factorization? This problem is solved for LPDOs of arbitrary order and number of variables. A more general result for the incomplete factorizations of LPDOs is proved as well.