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Proof of the Completeness of Darboux Wronskian Formulae for Order Two

Published online by Cambridge University Press:  20 November 2018

E. Shemyakova
E. Shemyakova*
Affiliation:
Mathematics Department, University of Western Ontario, LondonON
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Abstract

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Darboux Wronskian formulas allow us 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 to discover what other exceptions exist. 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.

Keywords

53Z0535Q99completeness of Darboux Wronskian formulascompleteness of Darboux determinantsDarboux transformationsinvariants for solution of PDEs
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 3 , 01 June 2013 , pp. 655 - 674
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Cassidy, P. and Singer, M. F., A Jordan-Hälder theorem for differential algebraic groups. J. Algebra 328(2011), 190–217.Google Scholar
[2] Darboux, G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. Deuxième partie, Chelsea Publishing Co., Bronx, N.Y., 1972.Google Scholar
[3] Fels, M.and Olver, P. J., Moving coframes. I. A practical algorithm. Acta Appl. Math. 51(1998), no. 2, 161–213.Google Scholar
[4] Gomez-Ullate, D., Kamran, N., and Milson, R., Two-step Darboux transformations and exceptional Laguerre polynomials J. Math. Anal. Appl. 387(2012), no. 1, 410–418.Google Scholar
[5] Krichever, I. M., Algebraic-geometric n-orthogonal curvilinear coordinate systems and the solution of associativity equations. Funct. Anal. Appl. 31(1997), no. 1, 25–39. http://dx.doi.org/10.1007/BF02466001 Google Scholar
[6] Mansfield, E. L., A practical guide to the invariant calculus. Cambridge Monographs on Applied and Computational Mathematics, 26 Cambridge University Press, Cambridge, 2010.Google Scholar
[7] Matveev, V. B. and Salle, M. A., Darboux transformations and solitons. Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.Google Scholar
[8] Olver, P. J., Differential invariant algebras. Symmetries and related topics in differential and difference equations, Contemp. Math. 549, American Mathematical Society, Providence, RI, pp. 95–121.Google Scholar
[9] Olver, P. J. and Pohjanpelto, J.,Maurer-Cartan forms and the structure of Lie pseudo-groups. Selecta Math. 11(2005), no. 1, 99–126.Google Scholar
[10] Shemyakova, E., Laplace transformations as the only degenerate Darboux transformations of first order. Programming and Computer Software 38(2012), no. 2, 105–108. http://dx.doi.org/10.1134/S0361768812020053 Google Scholar
[11] S. P.Tsarev, , Factorization in categories of systems of linear partial differential equations. arxiv:0801.1341. Google Scholar
[12] S. P.Tsarev, , Generalized Laplace transformations and integration of hyperbolic systems of linear partial differential equations. In: ISSAC ‘05: Proceedings of the 2005 international symposium on Symbolic and algebraic computation, ACM, New York, 2005, pp. 325–331.Google Scholar
[13] Vassiliou, P. J., Method for solving the multidimensional n-wave resonant equations and geometry of generalized Darboux-Manakov-Zakharov systems. Stud. Appl. Math. 126(2011), no. 3, 203–243.Google Scholar