Skip to main content
SpringerLink
Log in

Moutard transforms for the conductivity equation

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We construct Darboux–Moutard-type transforms for the two-dimensional conductivity equation. This result continues our recent studies of Darboux–Moutard-type transforms for generalized analytic functions. In addition, at least, some of the Darboux–Moutard-type transforms of the present work admit direct extension to the conductivity equation in multidimensions. Relations to the Schrödinger equation at zero energy are also shown.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Allaire, G., Malige, F.: Analyse asymptotique spectrale d’un problme de diffusion neutronique. C. R. Acad. Sci. Paris. Série I 324, 939–944 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Brown, R.M., Uhlmann, G.A.: Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions. Commun. Partial Differ. Equ. 22(5–6), 1009–1027 (1997). https://doi.org/10.1080/03605309708821292

    Article  MathSciNet  MATH  Google Scholar 

  3. Chaabi, S., Rigat, S., Wielonsky, F.: A boundary value problem for conjugate conductivity equations. Stud. Appl. Math. 137(3), 328–355 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Grinevich, P.G., Novikov, R.G.: Moutard transform for the generalized analytic functions. J. Geom. Anal. 26(4), 2984–2995 (2016). https://doi.org/10.1007/s12220-015-9657-8

    Article  MathSciNet  MATH  Google Scholar 

  5. Grinevich, P.G., Novikov, R.G.: Generalized analytic functions, Moutard-type transforms, and holomorphic maps. Funct. Anal. Appl. 50(2), 150–152 (2016). https://doi.org/10.1007/s10688-016-0140-5

    Article  MathSciNet  MATH  Google Scholar 

  6. Grinevich, P.G., Novikov, R.G.: Moutard transform approach to generalized analytic functions with contour poles. Bull. des Sci. Math. 140(6), 638–656 (2016). https://doi.org/10.1016/j.bulsci.2016.01.003

    Article  MathSciNet  MATH  Google Scholar 

  7. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, vol. 6, 2nd edn. Butterworth-Heinemann, Oxford (1987)

    Google Scholar 

  8. Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, vol. 7, 3rd edn. Butterworth-Heinemann, Oxford (1986)

    MATH  Google Scholar 

  9. Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P.: Electrodynamics of Continuous Media, vol. 8, 2nd edn. Butterworth-Heinemann, Oxford (1984)

    Google Scholar 

  10. Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  11. Matuev, R.M., Taimanov, I.A.: The Moutard transformation of two-dimensional Dirac operators and the conformal geometry of surfaces in four-space. Math. Notes 100(6), 835–846 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Moutard, T.F.: Sur la construction des équations de la forme \(\frac{1}{z}\frac{\partial ^2 z}{\partial x\partial y}= \lambda (x, y)\) qui admettenent une intégrale générale explicite. J. École Polytech. 45, 1–11 (1878)

    MATH  Google Scholar 

  13. Nimmo, J.J.C., Schief, W.K.: Superposition principles associated with the Moutard transformation: an integrable discretization of a 2+1-dimensional sine-Gordon system. Proc. R. Soc. Lond. A 453, 255–279 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Novikov, R.G., Taimanov, I.A., Tsarev, S.P.: Two-dimensional von Neumann–Wigner potentials with a multiple positive eigenvalue. Funct. Anal. Appl. 48(4), 295–297 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Novikov, R.G., Taimanov, I.A.: Moutard type transformation for matrix generalized analytic functions and gauge transformations. Russ. Math. Surv. 71(5), 970–972 (2016). https://doi.org/10.1070/RM9741

    Article  MATH  Google Scholar 

  16. Novikov, R.G., Taimanov, I.A.: Darboux-Moutard transformations and Poincaré–Steklov operators. Proc. Steklov Inst. Math. 302, 315–324 (2018)

    Article  Google Scholar 

  17. Taimanov, I.A.: Blowing up solutions of the modified Novikov–Veselov equation and minimal surfaces. Theor. Math. Phys. 182(2), 173–181 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Taimanov, I.A.: The Moutard transformation of two-dimensional Dirac operators and Möbius geometry. Math. Notes 97(1), 124–135 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Taimanov, I.A., Tsarev, S.P.: On the Moutard transformation and its applications to spectral theory and Soliton equations. J. Math. Sci. 170(3), 371–387 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Vekua, I.N.: Generalized Analytic Functions. Pergamon Press Ltd, Oxford (1962)

    MATH  Google Scholar 

  21. Vanninathan, M.: Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. (Math. Sci.) 90(3), 239–271 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yu, D., Liu, Q.P., Wang, S.: Darboux transformation for the modified Veselov–Novikov equation. J. Phys. A Math. Gen. 35(16), 3779–3786 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Grégoire Allaire for drawing our attention to the articles [1, 21], which use a reduction in Eq. (47) to (49) via (48) and (50).

Author information

Authors and Affiliations

  1. L.D. Landau Institute for Theoretical Physics, pr. Akademika Semenova 1a, Chernogolovka, Russia, 142432

    P. G. Grinevich

  2. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, GSP-1, Leninskiye Gory 1, Main Building, Moscow, Russia, 119991

    P. G. Grinevich

  3. CNRS (UMR 7641), Centre de Mathématiques Appliquées, École Polytechnique, 91128, Palaiseau, France

    R. G. Novikov

  4. IEPT RAS, Moscow, Russia, 117997

    R. G. Novikov

Authors
  1. P. G. Grinevich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. G. Novikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. G. Grinevich.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The main part of the work was fulfilled during the visit of the first author to the Institut des Hautes Études Scientifiques in November 2017 and to the Centre de Mathématiques Appliquées of École Polytechnique in September–October 2018. The first author was partially supported by the Russian Foundation for Basic Research, Grant 17-51-150001 “Quasilinear equations, inverse problems, and their applications”. The second author was partially supported by PRC 1545 CNRS/RFBR: “Équations quasi-linéaires, problèmes inverses et leurs applications”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grinevich, P.G., Novikov, R.G. Moutard transforms for the conductivity equation. Lett Math Phys 109, 2209–2222 (2019). https://doi.org/10.1007/s11005-019-01183-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-019-01183-x

Keywords

Mathematics Subject Classification

Search

Navigation