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Discrete mKdV Equation via Darboux Transformation


We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.


  1. 1.

    Bianchi, L.: Sulla trasformazione di Bäcklund per le superficie pseudosferiche. Rend. Lincei 5(1), 3–12 (1892)

    MATH  Google Scholar 

  2. 2.

    Bobenko, A.I., Pinkall, U.: Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996). MR1396732

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Burstall, F.E., Hertrich-Jeromin, U., Müller, C, Rossman, W.: Semi-discrete isothermic surfaces. Geom. Dedicata. 183, 43– 58 (2016). MR3523116

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cho, J., Rossman, W., Seno, T.: Infinitesimal Darboux transformation and semi-discrete mKdV equation. arXiv:2010.07846 (2020)

  5. 5.

    Goldstein, R.E., Petrich, D.M.: The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane. Phys. Rev. Lett. 67(23), 3203–3206 (1991). MR1135964

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Hirota, R.: Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation. J. Phys. Soc. Jpn. 43(4), 1424–1433 (1977). MR0460934

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Hirota, R.: Discretization of the potential modified KdV equation. J. Phys. Soc. Jpn. 67(7), 2234–2236 (1998). MR1647153

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Inoguchi, J.-I., Kajiwara, K., Matsuura, N., Ohta, Y.: Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves. J. Phys. A 45(4), 045206,16 (2012). MR2874242

    MathSciNet  Article  Google Scholar 

  9. 9.

    Inoguchi, J.-I., Kajiwara, K., Matsuura, N., Ohta, Y.: Motion and Bäcklund transformations of discrete plane curves. Kyushu J. Math. 66(2), 303–324 (2012). MR3051339

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kaji, S., Kajiwara, K., Park, H.: Linkage mechanisms governed by integrable deformations of discrete space curves. In: Euler, N., Nucci, M.C. (eds.) Nonlinear systems and their remarkable mathematical structures, vol. 2, pp 356–381. Chapman and Hall/CRC, New York (2019)

  11. 11.

    Lamb Jr., G.L.: Solitons and the motion of helical curves. Phys. Rev. Lett. 37(5), 235–237 (1976). MR473584

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Matsuura, N.: Discrete KdV and discrete modified KdV equations arising from motions of planar discrete curves. Int. Math. Res. Not. IMRN 8, 1681–1698 (2012). MR2920827

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Müller, C., Wallner, J.: Semi-discrete isothermic surfaces. Results Math. 63(3-4), 1395–1407 (2013). MR3057376

    MathSciNet  Article  Google Scholar 

  14. 14.

    Rogers, C., Schief, W.K.: Bäcklund and Darboux Transformations, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002). MR1908706

    Book  Google Scholar 

  15. 15.

    Wadati, M.: Bäcklund transformation for solutions of the modified Korteweg-de Vries equation. J. Phys. Soc. Jpn 36(5), 1498 (1974)

    ADS  Article  Google Scholar 

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The authors would like to thank Professor Udo Hertrich-Jeromin and the referees for valuable comments. The authors gratefully acknowledge the support from the JSPS/FWF Bilateral Joint Project I3809-N32 “Geometric shape generation” and JSPS Grants-in-Aid for: JSPS Fellows 19J10679, Scientific Research (C) 15K04845, (C) 20K03585 and (S) 17H06127 (P.I.: M.-H. Saito).


Open access funding provided by TU Wien (TUW).

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Correspondence to Joseph Cho.

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Communicated by:Kenji Kajiwara

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Cho, J., Rossman, W. & Seno, T. Discrete mKdV Equation via Darboux Transformation. Math Phys Anal Geom 24, 25 (2021).

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  • mKdV equation
  • Discrete mKdV equation
  • Darboux transformations
  • Permutability
  • Transformation theory
  • Integrable systems

Mathematics Subject Classification (2020)

  • Primary 53A70
  • Secondary 35Q53
  • 53A04