### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 010, 20 pages      arXiv:1605.04362      https://doi.org/10.3842/SIGMA.2017.010

### Classification of Multidimensional Darboux Transformations: First Order and Continued Type

David Hobby and Ekaterina Shemyakova
1 Hawk dr., Department of Mathematics, State University of New York at New Paltz, USA

Received October 10, 2016, in final form February 14, 2017; Published online February 24, 2017

Abstract
We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. 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 introduce a large class of invertible Darboux transformations of higher order, which we call Darboux transformations of continued Type I. This generalizes the class of Darboux transformations of Type I, which was previously introduced. There is also a modification of this type of Darboux transformations, continued Wronskian type, which generalize Wronskian type Darboux transformations.

Key words: Darboux transformations; Laplace transformations; linear partial differential operators; continued Darboux transformations.

pdf (411 kb)   tex (33 kb)

References

1. Adler V.E., Marikhin V.G., Shabat A.B., Lagrangian lattices and canonical Bäcklund transformations, Theoret. and Math. Phys. 129 (2001), 1448-1465.
2. Bagrov V.G., Samsonov B.F., Darboux transformation, factorization and supersymmetry in one-dimensional quantum mechanics, Theoret. and Math. Phys. 104 (1995), 1051-1060.
3. Bagrov V.G., Samsonov B.F., Darboux transformation of the Schrödinger equation, Phys. Part. Nuclei 28 (1997), 374-397.
4. Berest Yu., Veselov A., Singularities of the potentials of exactly solvable Schrödinger equations, and the Hadamard problem, Russian Math. Surveys 53 (1998), no. 1, 208-209.
5. Berest Yu., Veselov A., On the structure of singularities of integrable Schrödinger operators, Lett. Math. Phys. 52 (2000), 103-111.
6. Blumberg H., Über algebraische Eigenschaften von linearen homogenen Differentialausdrücken, Ph.D. Thesis, Göttingen, 1912.
7. Cannata F., Ioffe M., Junker G., Nishnianidze D., Intertwining relations of non-stationary Schrödinger operators, J. Phys. A: Math. Gen. 32 (1999), 3583-3598, quant-ph/9810033.
8. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, hep-th/9405029.
9. Crum M.M., Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. (2) 6 (1955), 121-127.
10. Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. II, Gauthier-Villars, Paris, 1989.
11. Doktorov E.V., Leble S.B., A dressing method in mathematical physics, Mathematical Physics Studies, Vol. 28, Springer, Dordrecht, 2007.
12. Etingof P., Gelfand I., Retakh V., Factorization of differential operators, quasideterminants, and nonabelian Toda field equations, Math. Res. Lett. 4 (1997), 413-425, q-alg/9701008.
13. Fels M., Olver P.J., Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213.
14. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
15. Ganzha E.I., Intertwining Laplace transformations of linear partial differential equations, in Algebraic and Algorithmic Aspects of Differential and Integral Operators, Lecture Notes in Comput. Sci., Vol. 8372, Springer, Heidelberg, 2014, 96-115.
16. Hill S., Shemyakova E., Voronov Th., Darboux transforms for differential operators on the superline, Russian Math. Surveys 70 (2015), 1173-1175, arXiv:1505.05194.
17. Infeld L., Hull T.E., The factorization method, Rev. Modern Phys. 23 (1951), 21-68.
18. Ioffe M.V., Supersymmetrical separation of variables in two-dimensional quantum mechanics, SIGMA 6 (2010), 075, 10 pages, arXiv:1009.4764.
19. Li C.X., Nimmo J.J.C., Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010), 2471-2493, arXiv:1009.4764.
20. Li S., Shemyakova E., Voronov Th., Differential operators on the superline, Berezinians, and Darboux transformations, Lett. Math. Phys., to appear, arXiv:1605.07286.
21. Liu Q.P., Darboux transformations for supersymmetric Korteweg-de Vries equations, Lett. Math. Phys. 35 (1995), 115-122, hep-th/9409008.
22. Liu Q.P., Mañas M., Crum transformation and Wronskian type solutions for supersymmetric KdV equation, Phys. Lett. B 396 (1997), 133-140, solv-int/9701005.
23. Liu Q.P., Mañas M., Darboux transformation for the Manin-Radul supersymmetric KdV equation, Phys. Lett. B 394 (1997), 337-342, solv-int/9701007.
24. Matveev V.B., Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation, depending on functional parameters, Lett. Math. Phys. 3 (1979), 213-216.
25. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.
26. Novikov S.P., Dynnikov I.A., Discrete spectral symmetries of small-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds, Russian Math. Surveys 52 (1997), 1057-1116, math-ph/0003009.
27. Samsonov B.F., On the $N$-th order Darboux transformation, Russian Math. (Iz. VUZ) (1999), no. 6, 62-65.
28. Schrödinger E., Further studies on solving eigenvalue problems by factorization, Proc. Roy. Irish Acad. Sect. A. 46 (1940), 183-206.
29. Shabat A.B., The infinite-dimensional dressing dynamical system, Inverse Problems 8 (1992), 303-308.
30. Shabat A.B., On the theory of Laplace-Darboux transformations, Theoret. and Math. Phys. 103 (1995), 482-485.
31. Shemyakova E., Factorization of Darboux transformations of arbitrary order for 2D Schrödinger operator, arXiv:1304.7063.
32. Shemyakova E., Invertible Darboux transformations, SIGMA 9 (2013), 002, 10 pages, arXiv:1210.0803.
33. Shemyakova E., Proof of the completeness of Darboux Wronskian formulae for order two, Canad. J. Math. 65 (2013), 655-674, arXiv:1111.1338.
34. Shemyakova E., Orbits of Darboux groupoid for hyperbolic operators of order three, in Geometric Methods in Physics (Białowieża, Poland, 2014), Editors P. Kielanowski, A. Bieliavsky, A. Odzijewicz, M. Schlichenmaier, Th. Voronov, Birkhäuser, Basel, 2015, 303-312, arXiv:1411.6491.
35. Shemyakova E., Mansfield E.L., Moving frames for Laplace invariants, in ISSAC 2008, ACM, New York, 2008, 295-302.
36. Tsarev S.P., Factorization of linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations, Theoret. and Math. Phys. 122 (2000), 121-133.
37. Tsarev S.P., Factorization of linear differential operators and systems, in Algebraic Theory of Differential Equations, London Math. Soc. Lecture Note Ser., Vol. 357, Cambridge University Press, Cambridge, 2009, 111-131.
38. Tsarev S.P., Shemyakova E., Differential transformations of second-order parabolic operators in the plane, Proc. Steklov Inst. Math. 266 (2009), 219-227, arXiv:0811.1492.
39. Veselov A.P., Shabat A.B., Dressing chains and the spectral theory of the Schrödinger operator, Func. Anal. Appl. 27 (1993), 81-96.
40. Witten E., Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692.