Advertisement

Moutard transforms for the conductivity equation

  • P. G. GrinevichEmail author
  • R. G. Novikov
Article
  • 24 Downloads

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.

Keywords

Darboux–Moutard transforms Conductivity equation Integrability Generalized analytic functions 

Mathematics Subject Classification

35Q79 35Q60 35J15 35C05 30G20 

Notes

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).

References

  1. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chaabi, S., Rigat, S., Wielonsky, F.: A boundary value problem for conjugate conductivity equations. Stud. Appl. Math. 137(3), 328–355 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, vol. 6, 2nd edn. Butterworth-Heinemann, Oxford (1987)Google Scholar
  8. 8.
    Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, vol. 7, 3rd edn. Butterworth-Heinemann, Oxford (1986)zbMATHGoogle Scholar
  9. 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. 10.
    Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)zbMATHGoogle Scholar
  13. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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 CrossRefzbMATHGoogle Scholar
  16. 16.
    Novikov, R.G., Taimanov, I.A.: Darboux-Moutard transformations and Poincaré–Steklov operators. Proc. Steklov Inst. Math. 302, 315–324 (2018)CrossRefGoogle Scholar
  17. 17.
    Taimanov, I.A.: Blowing up solutions of the modified Novikov–Veselov equation and minimal surfaces. Theor. Math. Phys. 182(2), 173–181 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Taimanov, I.A.: The Moutard transformation of two-dimensional Dirac operators and Möbius geometry. Math. Notes 97(1), 124–135 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vekua, I.N.: Generalized Analytic Functions. Pergamon Press Ltd, Oxford (1962)zbMATHGoogle Scholar
  21. 21.
    Vanninathan, M.: Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. (Math. Sci.) 90(3), 239–271 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.L.D. Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Faculty of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia
  3. 3.CNRS (UMR 7641), Centre de Mathématiques Appliquées, École PolytechniquePalaiseauFrance
  4. 4.IEPT RASMoscowRussia

Personalised recommendations