My Refereed Publications (published/accepted - 23, submitted - 1)

Research interests: integrable systems, in particular algebraic theory of Darboux transformations for integrable systems, factorization of linear partial differential operators.

2014

1. Journal: Factorization of Darboux Transformations of Arbitrary Order for Two-dimensional Schroedinger operator, 2014.
2. Journal: Darboux transformations for factorizable operators, accepted to Programming and Computer Software, special issue Computer Algebra, (Russian Academy of Science, eds. S. Abramov, S.Tsarev), number 2, 2014.

2013

3. Journal: 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, 2013, bibtex
Informal abstract. A Darboux transformation is invertible if 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 criteria 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.

4. Refereed conference procs: 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.
5. Journal: Proof of the Completeness of Darboux Wronskian Formulae for Order Two, Canadian Journal of Mathematics 65, no.3, 655-674, 2013. Get it here or at arxiv
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

6. Journal: 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 .
7. 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

8. Journal: 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.
9. 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

10. Journal: 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 .
11. Journal: 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

12. Refereed conference procs: : Multiple factorizations of bivariate linear partial differential operators, Lecture Notes in Computer Science, vol. 5743, pp. 299--309, 2009, bibtex .
13. Refereed conference procs: : 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 .
14. Journal: 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

15. Refereed conference procs: : 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)

16. Refereed conference procs: : 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 .
17. Refereed conference procs: : 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 .
18. Journal: 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 .
19. Journal: 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 .
20. Refereed conference procs: : with F.Winkler, Symbolic and Algebraic Methods for Linear Partial Differential Operators , Lecture Notes in Computer Science, vol.4770, 2007, bibtex .

2006

21. Refereed conference procs: : with F.Winkler, Obstacle to Factorization of LPDOs, in Proc. Transgressive Computing 2006 (J.-G. Dumas, ed.), pp.435--441, 2006.
22. Refereed conference procs: : 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)

23. Journal: Involutive divisions. Graphs., Programming and Computer Software, vol.30, no.2, pp.68--74, 2004.

2003 (while 4th year undergraduate student)

24. Journal: Involutive divisions for effective involutive algorithms, Fundam. Prikl. Mat., vol.9, no.3, pp.237--253, 2003.

Minor Refereed Publications

1. Abstract of the PhD thesis, M.Giesbrecht (eds.), ACM Communications in Computer Algebra, 41, N.3, issue 161, 2007.
2. 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.
3. 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

1. with J. Middeke, F. Winkler, Proceedings of DEAM (Workshop for Differential Equations by Algebraic Methods), 2009, RISC Report Series, University of Linz, Austria.