- with David Hobby,
Classification of Darboux transformations: first order and iterated type .
Abstract. We analyze Darboux transformations (DT) for general multi-dimensional linear partial differential operators. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a complete description of all possible first order Darboux transformations. We also introduce two new types of Darboux transformations of higher order: a class of invertible transformations generalizing previously introduced "Type I" transformations (which are a generalization of classical Laplace transformations for 2D Schroedinger operators); and a class of "weak Wronskian type" transformations.
- with Simon Li (my student) and Theodore Voronov,
Differential operators on the superline, Berezinians, and Darboux transformations.
Abstract. We consider differential operators on a supermanifold of dimension $1|1$. We distinguish non-degenerate operators as those with an invertible top coefficient in the expansion in the superderivative $D$. They are remarkably similar with ordinary differential operators. We show that every non-degenerate operator can be written in terms of 'super Wronskians' (which are certain Berezinians). We apply this to Darboux transformations (DTs), proving that every DT of an arbitrary non-degenerate operator is the composition of elementary first order transformations. Hence every DT corresponds to an invariant subspace of the source operator and upon a choice of a basis in this subspace is expressed by a super-Wronskian formula. We consider also dressing transformations, i.e., the effect of a DT on the coefficients of an operator. We calculate them in examples and make some general statements.
- Factorization of Darboux Transformations of Arbitrary Order for Two-dimensional Schroedinger
operator, (submitted).
Informal abstract. We give a proof of Darboux's conjecture that every Darboux transformation of arbitrary order of a 2D Schroedinger type operator can be factorized into Darboux transformations of order one. The proof is constructive.
Besides the main result, the paper contains an algebraic definition of a Darboux transformations for an arbitrary operator, which naturally implies additional properties that are usually had to be added artificially.
2015
- with Sean Hill (my student) and Theodore Voronov, Darboux transformations for differential operators on the superline ,
Russian Mathematical Surveys, (6), pp. 207-208, 70, 2015.
Informal abstract. We show that an arbitrary Darboux transformation of a differential operator on the superline is the composition of "elementary" Darboux transformations of order one. This is an analog of the similar statement for operators on the line. One should note that examples of super Darboux transformations including the elementary transformations and their products for particular operators were considered before.
The novelty of our work is in the full description of Darboux transformations for arbitrary operators on the superline. This may be regarded as the first step in the algebraic theory of super Darboux transformations. As follows from the results of this paper, the case of the superline is close to the ordinary one-dimensional case. On the other hand, it is known that Darboux transformations in 2D differ substantially from the case of 1D. New possibilities that may open in the super version seem very attractive and we hope to consider them elsewhere. - Invertible Darboux Transformations of Type I ,
accepted to Programming and Computer Software, special issue Computer Algebra, (Russian Academy of Science, eds. S. Abramov, S.Tsarev), 2015.
Informal abstract. For operators of arbitrary dimension and arbitrary order we singled out Darboux transformations of certain form and call them transformation of type I (precisely M generates a Darboux transformation for L of type I if the remainder of the division of L by M is just a function).
Transformations of type I are always invertible, and amazingly we have simple coordinate-free formulas for all operators involved into the definition of Darboux transformation and its inverse in this case. All Darboux transformations of order one are of type I. Laplace transformations are of type I too.
Using those we obtained analogs of chains Darboux transformations for operators of third order . - An algorithm for construction of orbits of Darboux transformations of type I for third order hyperbolic operators of two variables,
accepted to Programming and Computer Software, special issue Computer Algebra, (Russian Academy of Science, eds. S. Abramov, S.Tsarev), 2015.
2014
- Orbits of Darboux groupoid for hyperbolic operators of order three , Geometric Methods in Physics,
P. Kielanowski, S. T. Ali, A.Odesski, A. Odzijewicz, M. Schlichenmaier, Th. Voronov (editors),
Trends in Mathematics. Springer, Basel, 2014.
Informal abstract. Darboux transformations are viewed as morphisms in a Darboux category. Darboux transformations of type I which we defined previously, make an important subgroupoid consists of Darboux transformations of type I. We describe the orbits of this subgroupoid for hyperbolic operators of order three. We consider the algebras of differential invariants for our operators. In particular, we show that the Darboux transformations of this class can be lifted to transformations of differential invariants (which we calculate explicitly).
- Darboux transformations for factorizable operators, Programming and Computer Software, special issue Computer Algebra,
(Russian Academy of Science, eds. S. Abramov, S.Tsarev), number 2, 2014.
Informal abstract. He we address a "non-general" case, which was skipped in the paper above, and which occurs when initial operator is factorizable. Darboux transformations for such operators have a number of additional interesting properties, while many usual facts which are true for a general Darboux transformation are not true for this one. Initially the case was skipped due to the two reasons: 1) it is kind of very degenerate case, because once operator is factorizable, we can write down its complete solution; 2) the methods that have been employed for the general case completely fail here. However, since Darboux transformations of an operator are often used not for the solution of the corresponding linear PDE, but for some integrable system based on it, we cannot completely neglect this case.
Darboux transformations for factorizable operator has lot of structure. In particular, there is a correspondence between Darboux transformations of factorizable second-order operators and Darboux transformation of first order operators. However, the correspondence is not one-to-one, and lifting back process is not immediate.
2013
- Invertible Darboux Transformations ,
SIGMA 9 (2013), 002,
special issue “Symmetries of Differential Equations: Frames, Invariants and Applications” in
honor of the 60th Birthday of Peter Olver. Eds: N.Kamran, G.M.Beffa, W. Miller, G. Sapiro, bibtex
Informal abstract. This paper is the first step towards the study of invertible Darboux transformations. If Darboux transformation is invertible then the corresponding mappings of the operator kernels is an isomorphism. For the classical case of operators of order two on the plane, there are only two invertible Darboux transformations, which are two famous particular cases of Darboux transformations, Laplace transformations.Here we find a criterion for an arbitrary operator on the plane to have an invertible Darboux transformation.
We also study Wronkian-type formulas which allow to construct Darboux transformations for many different types of operators. These are non-invertible by construction. From time to time new works appear showing that Wronskian-type formulas generate Darboux transformations for some new type of operators. However, up until this work there has been no explicit example of an operator for which Wronkiant-type formulas do NOT work. We give an explicit example in this paper, and find sufficient conditions for Wronkians to work.
- Invariants for Darboux transformations of Arbitrary Order for
$D_x D_y +aD_x + bD_y +c$ , Geometric Methods in Physics. XXXI Workshop, Bialowieza, Poland, June 30 to July 6, 2013,
P. Kielanowski, S. T. Ali, A.Odesski, A. Odzijewicz, M. Schlichenmaier, Th. Voronov (editors),
Trends in Mathematics. Springer, Basel, 2013. bibtex .
Informal abstract. One of my goals is to develop an invariant approach to Darboux transformations. A Darboux transformation of arbitrary order for an operator of the form $L=D_x D_y +aD_x + bD_y +c$ is generated by an arbitrary partial differential operator $M$. The fact that operator $M$ is arbitrary makes it hard to find compact constructive and explicit formulas for the joint differential invariants.
Such formulas is the achievement of the present paper. The formulas are given in terms of complete Bell polynomials. - Proof of the Completeness of Darboux Wronskian Formulae for Order Two,
Canadian Journal of Mathematics 65, no.3, 655-674, 2013.
Informal abstract. The paper contains a constructive proof that every Darboux transformation for operators of the form $D_x D_y +aD_x + bD_y +c$ can be factored into elementary Darboux transformations of order one. For such operators there are only two types of elementary Darboux transformations: Wronskian-type and Laplace transformations (this was proved in earlier paper Laplace Transformations as the Only Degenerate Darboux Transformations of First Order).
The conjecture was reduced to the solution of a complicated system of non-linear PDEs. Using moving frames method (of differential geometry, due to P.Olver et al.) I have found a generating set of joint differential invariants for the involved operators. Then after re-writing in terms of invariants, the initial system split into two much simpler systems, one of which was a system of linear PDEs.
2012
- Package LPDO for MAPLE, Programming and Computer Software, special issue Computer Algebra,
(Russian Academy of Science, eds. S. Abramov, S.Tsarev), number 4, 2013.
Informal abstract. Help file for my LPDO package . - Journal: Laplace Transformations as the Only Degenerate Darboux Transformations of First Order,
Programming and Computer Software, special issue Computer Algebra,
(Russian Academy of Science, eds. S. Abramov, S.Tsarev), volume 38, number 2, 2012, bibtex .
Informal abstract. Wronskian formulas allow one to construct Darboux Transformations (DTs). Laplace transformations (DT of order one) cannot be represented in this way. Here: among DTs of total order 1 - NO exceptions, other than Laplace transformations.2011
- X- and Y-invariants of partial differential operators in the plane , Programming and Computer Software, special issue devoted to Computer Algebra 2011, (Russian Academy of Science, eds. S. Abramov, S.Tsarev), (37), no.4, pp.192-196, 2011, bibtex.
- Book chapter: with F. Winkler, Linear Partial Differential Equations and Linear Partial Differential Operators in Computer Algebra,
in Monographs in Symbolic Computation, Springer (book chapter),
editors: P. Paule et al., vol. Progress and Prospects in Numerical and Symbolic Scientific Computing, 2011.2010
- Refinement of Two-Factor Factorizations of a Linear Partial Differential Operator of Arbitrary Order and Dimension, Mathematics in Computer Science, (4), no.2-3, pp. 223-230, 2010, bibtex .
- with S.I.Khashin, and D.J.Jeffrey,
Conjecture concerning
a completely monotonic function,
Computers & Mathematics with Applications, vol. 60, issue 5, pp.1360-1363, 2010.
ScienceDirect
, bibtex .
2009
- : Multiple factorizations of bivariate linear partial differential operators, Lecture Notes in Computer Science, vol. 5743, pp. 299--309, 2009, bibtex .
- : On the invariant properties of non-hyperbolic third-order linear partial differential operators, Conferences on Intelligent Computer Mathematics, vol.5625, pp.154--169, 2009, bibtex .
- with S.Tsarev
Differential transformations of
parabolic second-order operators in the plane,
Proceedings Steklov Inst. Math. (Moscow), vol.266, pp.219--227, 2009, bibtex .
2008
- : with E.Mansfield, Moving frames for Laplace
invariants, Proceedings of ISSAC 2008 (The International Symposium on Symbolic and
Algebraic Computation), pp.295--302, 2008, bibtex .
2007 (PhD degree obtained in 2007)
- : with F.Winkler, On the invariant properties of hyperbolic bivariate third-order linear partial differential operators, Lecture Notes in Artificial Intelligence, vol.5081, pp.199--212, 2007, bibtex .
- : with F.Winkler, A full system of invariants for third-order linear partial differential operators in general form, Lecture Notes in Comput. Sci., vol.4770, pp.360--369, 2007, bibtex .
- The Parametric Factorizations of Second-, Third- and Fourth-Order Linear Partial Differential Operators on the Plane , Mathematics in Computer Science, vol.1, no.2, pp.225--237, 2007, bibtex .
- with F.Winkler, Obstacles to the Factorization of Linear Partial Differential Operators into Several Factors , Programming and Computer Software, vol.33, no.2, pp.67--73, 2007, bibtex .
- : with F.Winkler, Symbolic and Algebraic Methods for Linear Partial Differential Operators ,
Lecture Notes in Computer Science, vol.4770, 2007, bibtex .
2006
- : with F.Winkler, Obstacle to Factorization of LPDOs, in Proc. Transgressive Computing 2006 (J.-G. Dumas, ed.), pp.435--441, 2006.
- : A full system of invariants for third-order linear
partial
differential operators, Lecture Notes in Computer Science, vol.4120,
pp.360--369, 2006, bibtex .
2004 (while 5th year undergraduate student)
- Involutive divisions. Graphs., Programming
and
Computer Software, vol.30, no.2, pp.68--74, 2004.
2003 (while 4th year undergraduate student)
- Involutive divisions for effective involutive algorithms, Fundam. Prikl. Mat., vol.9, no.3, pp.237--253, 2003.
Research interests:
Symmetries, Integrable Systems, Differential Geometry, Algebraic theory of Darboux transformations, Computational Pure Mathematics.My Refereed Publications (published/accepted - 27, submitted - 3)
2016
Minor Refereed Publications
- Abstract of the PhD thesis, M.Giesbrecht (eds.), ACM Communications in Computer Algebra, 41, N.3, issue 161, 2007.
- with F. Winkler. Extended abstract: Algebraic Methods for Linear Partial Differential Operators. M.Giesbrecht, I.Kotsireas, A.Lobo (eds.), ACM Communications in Computer Algebra, 41, N.2, issue 160, 2007.
- with F. Winkler. Extended abstract: Approximate Factorization of Linear Partial Differential Operators. Full System of Invariants for Order Three. Zh. Wan, A.Lobo(eds.), ACM Communications in Computer Algebra, 40, N.2, issue 156, 2006.
Technical Reports
- with J. Middeke, F. Winkler, Proceedings of DEAM (Workshop for Differential Equations by Algebraic Methods), 2009, RISC Report Series, University of Linz, Austria.