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Shemyakova, Ekaterina

Classification of Darboux transformations for operators of the form \(\partial_x\partial_y+a\partial_x+b\partial_y+c\). (English) Zbl 1445.35014

Ill. J. Math. 64, No. 1, 71-92 (2020).
Summary: Darboux transformations are nongroup-type symmetries of linear differential operators. One can define Darboux transformations algebraically by the intertwining relation \(ML=L_1M\) or the intertwining relation \(ML=L_1N\) in the cases when the former is too restrictive.
Here we show that Darboux transformations for operators of the form \(L=\partial_x\partial_y+a\partial_x+b\partial_y+c\) (sometimes referred to as 2D Schrödinger operators or Laplace operators) are always compositions of atomic Darboux transformations of two different well-studied types of Darboux transformations, provided that the chain of Laplace transformations for the original operator is long enough.

Cited in 1 Document

MSC:

35A22 Transform methods (e.g., integral transforms) applied to PDEs
35F05 Linear first-order PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
35Q51 Soliton equations
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry

Keywords:

nongroup-type symmetries
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\textit{E. Shemyakova}, Ill. J. Math. 64, No. 1, 71--92 (2020; Zbl 1445.35014)
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References:

[1] V. È. Adler, V. G. Marikhin, and A. B. Shabat, Lagrangian lattices and canonical Bäcklund transformations, Theoret. and Math. Phys. 129 (2001), no. 2, 1448-1465. · Zbl 1029.37042
[2] C. Athorne, Laplace maps and constraints for a class of third-order partial differential operators, J. Phys. A 51 (2018), no. 8, 085205. · Zbl 1391.35012
[3] V. G. Bagrov and B. F. Samsonov, Darboux transformation, factorization and supersymmetry in one-dimensional quantum mechanics, Theoret. and Math. Phys. 104 (1995), no. 2, 1051-1060. · Zbl 0857.34070
[4] V. G. Bagrov and B. F. Samsonov, Darboux transformation of the Schrödinger equation, Phys. Part. Nucl. 28 (1997), no. 4, 374-397.
[5] Y. Berest and A. Veselov, On the structure of singularities of integrable Schrödinger operators, Lett. Math. Phys. 52 (2000), no. 2, 103-111. · Zbl 0969.47018
[6] Y. Y. Berest and A. P. Veselov, Singularities of the potentials of exactly solvable Schrödinger equations, and the Hadamard problem, Russian Math. Surveys 53 (1997), no. 1, 208-209. · Zbl 0918.35041
[7] F. Cannata, M. Ioffe, G. Junker, and D. Nishnianidze, Intertwining relations of non-stationary Schrödinger operators, J. Phys. A 32 (1999), no. 19, 3583. · Zbl 0962.81003
[8] F. Cooper, A. Khare, and U. Sukhatme, Supersymmetry and quantum mechanics, Physics Rep. 251 (1995), nos. 5-6, 267-385. · Zbl 0988.81001
[9] M. M. Crum, Associated Sturm-Liouville systems, Q. J. Math. 6 (1955), no. 1, 121-127. · Zbl 0065.31901
[10] G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, Vol. 2, Gauthier-Villars, Paris, 1889. · JFM 53.0659.02
[11] E. Doktorov and S. Leble, A Dressing Method in Mathematical Physics, Math. Phys. Stud. 28, Springer Dordrecht, 2007. · Zbl 1142.35002
[12] P. Etingof, I. Gelfand, and V. Retakh, Factorization of differential operators, quasideterminants, and nonabelian Toda field equations, Math. Res. Lett. 4 (1997), nos. 2-3, 413-425. · Zbl 0959.37054
[13] M. Fels and P. J. Olver, Moving coframes, I: A practical algorithm, Acta Appl. Math. 51 (1998), no. 2, 161-213. · Zbl 0937.53012
[14] M. Fels and P. J. Olver, Moving coframes, II: Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), no. 2, 127-208. · Zbl 0937.53013
[15] E. I. Ganzha, “Intertwining Laplace transformations of linear partial differential equations” in Algebraic and Algorithmic Aspects of Differential and Integral Operators: 5th International Meeting (Sofia, Bulgaria), Selected and Invited Papers, Springer, Berlin, 2014, 96-115. · Zbl 1375.35010
[16] S. Hill, E. Shemyakova, and T. Voronov, Darboux transformations for differential operators on the superline, Russian Math. Surveys 70 (2015), no. 6, 207-208. · Zbl 1343.37073
[17] D. Hobby and E. Shemyakova, Classification of multidimensional Darboux transformations: First order and continued type, SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), 010, 20 pp. · Zbl 1359.16021
[18] G. Hovhannisyan, O. Ruff, and Z. Zhang, Higher dimensional Darboux transformations, J. Math. Anal. Appl. 464 (2018), no. 1, 776-805. · Zbl 1394.35007
[19] G. Hovhannisyan and O. Ruff, Darboux transformations on a space scale, J. Math. Anal. Appl. 434 (2016), no. 2, 1690-1718. · Zbl 1329.35014
[20] L. Infeld and T. E. Hull, The factorization method, Rev. Mod. Phys. 23 (1951), no. 1, 21-68. · Zbl 0043.38602
[21] M. V. Ioffe, Supersymmetrical separation of variables in two-dimensional quantum mechanics, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), 075, 10 pp. · Zbl 1217.81099
[22] C. X. Li and J. J. C. Nimmo, Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation, Proc. Roy. Soc. A Math. Phys. Eng. Sci. 466 (2010), no. 2120, 2471-2493. · Zbl 1194.35374
[23] S. Li, E. Shemyakova, and T. Voronov, Differential operators on the superline, Berezinians, and Darboux transformations, Lett. Math. Phys. 107 (2017), no. 9, 1689-1714. · Zbl 1394.16027
[24] Q. P. Liu, Darboux transformations for supersymmetric Korteweg-de Vries equations, Lett. Math. Phys. 35 (1995), no. 2, 115-122. · Zbl 0834.35110
[25] Q. P. Liu and M. Mañas, Crum transformation and Wronskian type solutions for supersymmetric KdV equation, Phys. Lett. B 396 (1997), nos. 1-4, 133-140.
[26] Q. P. Liu and M. Mañas, Darboux transformation for the Manin-Radul supersymmetric KdV equation, Phys. Lett. B 394 (1997), nos. 3-4, 337-342.
[27] V. B. Matveev, Darboux transformation and explicit solutions of the Kadomtsev-Petviashvili equation, depending on functional parameters, Lett. Math. Phys. 3 (1979), no. 3, 213-216. · Zbl 0418.35005
[28] V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer Ser. Nonlinear Dyn., Springer-Verlag, Berlin, 1991. · Zbl 0744.35045
[29] S. P. Novikov and I. A. Dynnikov, Discrete spectral symmetries of small-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds, Russian Math. Surveys 52 (1997), no. 5, 1057-1116. · Zbl 0928.35107
[30] P. J. Olver and J. Pohjanpelto, Differential invariant algebras of Lie pseudo-groups, Adv. Math. 222 (2009), no. 5, 1746-1792. · Zbl 1194.58018
[31] B. F. Samsonov, On the \(N\)-th order Darboux transformation, Russian Math. (Iz. VUZ) 43 (1999), no. 6, 62-65.
[32] E. Schrödinger, Further studies of eigenvalue problems by factorization, Proc. Roy. Ir. Acad. A 46 (1940),183-206. · Zbl 0063.06818
[33] A. Shabat, The infinite-dimensional dressing dynamical system, Inverse Problems 8 (1992), no. 2, 303-308. · Zbl 0762.35098
[34] A. B. Shabat, On the theory of Laplace-Darboux transformations, Theoret. Math. Phys. 103 (1995), no. 1, 482-485. · Zbl 0855.35004
[35] E. Shemyakova, Invertible Darboux transformations, SIGMA Symmetry, Integrability Geom. Meth. Appl. 9 (2013), 002, 10 pp.
[36] E. Shemyakova, Proof of the completeness of Darboux Wronskian formulae for order two, Canad. J. Math. 65 (2013), no. 3, 655-674. · Zbl 1275.53088
[37] E. Shemyakova, “Orbits of Darboux groupoid for hyperbolic operators of order three” in Geometric Methods in Physics, Trends Math., Springer, Basel, 2015. · Zbl 1366.70022
[38] E. Shemyakova and E. L. Mansfield, “Moving frames for Laplace invariants” in Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2008, 295-302.
[39] E. S. Shemyakova, Darboux transformations for factorizable Laplace operators, Program. Comput. Softw. 40 (2014), 151-157. Translated from Programmirovanie 40 (2014), no. 3. · Zbl 1311.35004
[40] S. V. Smirnov, Factorization of Darboux-Laplace transformations for discrete hyperbolic operators, Theoret. Math. Phys. 119 (2019), no. 2, 621-636. · Zbl 1439.44003
[41] S. P. Tsarev, Factorization of linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations, Theoret. Math. Phys. 122 (2000), no. 1, 121-133.
[42] S. P. Tsarev and E. Shemyakova, “Differential transformations of parabolic second-order operators in the plane” in Proceedings of the Steklov Institute of Mathematics 266 (2009), no. 1, 219-227. · Zbl 1188.35008
[43] A. P. Veselov and A. B. Shabat, A dressing chain and the spectral theory of the Schrödinger operator, Funct. Anal. Appl. 27 (1993), no. 2, 81-96. · Zbl 0813.35099
[44] E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), no. 4, 661-692. · Zbl 0499.53056
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