Skip to main content
SpringerLink
Log in

Moving boundary problems for an extended Dym equation. Reciprocal connection

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

Stefan-type moving boundary problems are investigated for an extended Dym equation originally introduced in work of Camassa and Holm. Reduction is made to an associated class of moving boundary value problems for the canonical integrable Dym equation and exact solution obtained in terms of Yablonski–Vorob’ev polynomials via a Painlevé II reduction.

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

Notes

  1. Hereafter denoted by \(P_{\mathrm {II}}\)

References

  1. Rubinstein LI (1971) The Stefan problem, American Mathematical Society Translations, vol 27. American Mathematical Society, Providence

    Google Scholar 

  2. Friedman A (1982) Variational principles and free boundary problems. Wiley, New York

    MATH  Google Scholar 

  3. Elliot CM, Ockendon JR (1982) Weak and variational methods for moving boundary problems, research notes in mathematics 59. Pitman, New York

    Google Scholar 

  4. Crank J (1984) Free and moving boundary value problems. Clarendon Press, Oxford

    MATH  Google Scholar 

  5. Alexides V, Solomon AD (1996) Mathematical modelling of melting and freezing processes. Hemisphere Publishing Corporation, Taylor and Francis, Washington

    Google Scholar 

  6. Tarzia DA (2000) A bibliography on moving-free boundary problems for heat diffusion equation. Stefan Probl Mat Ser A2:1–297

    Google Scholar 

  7. Rogers C, Schief WK (2002) Bäcklund and Darboux transformations. geometry and modern applications in soliton theory, cambridge texts in applied mathematics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  8. Chou K-S, Qu C (2002) Integrable equations arising from motions of plane curves. Physica D 162:9–33

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Vasconcelos GL, Kadanoff LP (1991) Stationary solutions for the Saffman–Taylor problem with surface tension. Phys Rev A 44:6490–6495

    Article  ADS  Google Scholar 

  10. Kruskal M (1975) Nonlinear wave equations. In: Moser Jürgen J (ed) Dynamical systems, theory and applications, lecture notes in physics. Springer, Heidelberg, pp 310–354

    Chapter  Google Scholar 

  11. Vassiliou PJ (2001) Harry Dym equation, In: Encyclopaedia of mathematics, Springer

  12. Tanveer S (1993) Evolution of Hele–Shaw interface for small surface tension. Phil Trans R Soc Lond A 343:155–204

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Fokas AS, Tanveer S (1998) A Hele–Shaw problem and the second Painlevé transcendent. Math Proc Camb Phil Soc 124:169–191

    Article  MATH  Google Scholar 

  14. Camassa R, Holm DD (1993) An integrable shallow water equation with peaked solitons. Phys Rev Lett 71:1661–1664

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Schief WK, Rogers C (1999) Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces. Proc R Soc Lond A 455:3163–3188

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Da Rios LS (1906) Sul moto d’un liquido indefinito con un filetto vorticoso. Rend Circ Mat Palermo 22:117–135

    Article  MATH  Google Scholar 

  17. Ricca RL (1991) Rediscovery of Da Rios equations. Nature 352:561–562

    Article  ADS  Google Scholar 

  18. Schief WK, Rogers C (2005) The Da Rios system under a geometric constraint: the Gilbarg problem. J Geom Phys 54:286–300

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Hasimoto H (1972) A soliton on a vortex filament. J Fluid Mech 51:477–485

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Rogers C, Schief WK (1998) Intrinsic geometry of the NLS equation and its auto-Bäcklund transformation. Stud Appl Math 101:267–287

    Article  MATH  MathSciNet  Google Scholar 

  21. Fokas A (1980) A symmetry approach to exactly solvable evolution equations. J Math Phys 21:1318–1325

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Calogero F, Degasperis A (1985) A modified modified Korteweg-de Vries equation. Inverse Probl 1:57–66

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. Ma WX (2010) An extended Harry Dym hierarchy. J Phys A Math Theor 43:165202

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Dai HH, Pavlov M (1998) Transformations for the Camassa–Holm equation, its high frequency limit and the sinh–Gordon equation. J Phys Soc Jpn 67:3655–3657

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. Lukashevich NA (1971) The second Painlevé equation. Diff Eq 7:853–854

    MATH  Google Scholar 

  26. Gromak VI (1999) Bäcklund transformations of Painlevé equations and their applications. In: Conte R (ed) The Painlevë property: one century later. Springer-Verlag, New York, pp 687–734

    Chapter  Google Scholar 

  27. Yablonskii AI (1959) On rational solutions of the second Painlevé equation. Vesti Akad Nauk B55R ser Fiz Tkh Nauk 3:30–35

    Google Scholar 

  28. Vorob’ev AP (1965) On the rational solutions of the second Painlevé equation. Diff Eq 1:79–81

    MATH  Google Scholar 

  29. Rogers C, Bassom A, Schief WK (1999) On a Painlevé model in steady electrolysis: application of a Bäcklund transformation. J Math Anal Appl 240:367–381

    Article  MATH  MathSciNet  Google Scholar 

  30. Bass LK, Nimmo J, Rogers C, Schief WK (2010) Electrical structures of interfaces. A Painlevé II model. Proc R Soc Lond A 466:2117–2136

    Article  ADS  MATH  MathSciNet  Google Scholar 

  31. Bracken AJ, Bass LK, Rogers C (2012) Bäcklund flux-quantization in a model of electrodiffusion based on Painlevé II. J Phys A Math Theor 45:105204

    Article  ADS  MATH  Google Scholar 

  32. Bass LK, Bracken AJ (2014) Emergent behaviour in electrodiffusion: Planck’s other quanta. Rep Math Phys 73:65–75

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Rogers C (1968) Reciprocal relations in non-steady one-dimensional gasdynamics. Zeit Angew Math Phys 19:58–63

    Article  MATH  Google Scholar 

  34. Rogers C (1969) Invariant transformations in non-steady gasdynamics and magneto-gasdynamics. Zeit Angew Math Phys 20:370–382

    Article  MATH  Google Scholar 

  35. Rogers C, Wong P (1984) On reciprocal Bäcklund transformations of inverse scattering schemes. Phys Scr 30:10–14

    Article  ADS  MATH  Google Scholar 

  36. Rogers C (1987) The Harry Dym equation in 2+1-dimensions: a reciprocal link with the Kadomtsev-Petviashvili equation. Phys Lett 120A:15–18

    Article  ADS  MathSciNet  Google Scholar 

  37. Oevel W, Rogers C (1993) Gauge transformations and reciprocal links in 2+1-dimensions. Rev Math Phys 5:299–330

    Article  MATH  MathSciNet  Google Scholar 

  38. Hereman W, Banerjee PP, Chatterjee MR (1989) Derivation and explicit solution of the Harry Dym equation and its connections with the Korteweg-de-Vries equation. J Phys A 22:241–255

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Rogers C, Carillo S (1987) On reciprocal properties of the Caudrey–Dodd–Gibbon and Kaup–Kuperschmidt hierarchies. Phys Scr 36:865–869

    Article  ADS  MATH  Google Scholar 

  40. Fuchssteiner B, Schulze T, Carillo S (1992) Explicit solutions for the Harry Dym equation. J Phys A 25:223–230

    Article  ADS  MATH  MathSciNet  Google Scholar 

  41. Konopelchenko B, Rogers C (1992) Bäcklund and reciprocal transformations: gauge connections. In: Ames WF, Rogers C (eds) Nonlinear equations in applied science. Academic Press, New York, pp 317–362

    Chapter  Google Scholar 

  42. Dmitrieva LA (1993) Finite-gap solutions of the Harry Dym equation. Phys Lett A 182:65–70

    Article  ADS  MathSciNet  Google Scholar 

  43. Donato A, Ramgulam U, Rogers C (1992) The 3+1-dimensional Monge-Ampère equation in discontinuity wave theory: application of a reciprocal transformation. Meccanica 27:257–262

    Article  MATH  Google Scholar 

  44. Rogers C, Shadwick WE (1982) Bäcklund transformations and their applications. Academic Press, New York

    MATH  Google Scholar 

  45. Rogers C, Kingston JG, Shadwick WF (1980) On reciprocal-type invariant transformations in magnetogasdynamics. J Math Phys 21:395–397

    Article  ADS  MATH  Google Scholar 

  46. Rogers C (1985) Application of a reciprocal transformation to a two-phase Stefan problem. J Phys A Math Gen 18:L105–L109

    Article  ADS  MATH  MathSciNet  Google Scholar 

  47. Rogers C (1986) On a class of moving boundary problems in nonlinear heat conduction. Application of a Bäcklund transformation. Int J Nonlinear Mech 21:249–256

    Article  MATH  Google Scholar 

  48. Storm ML (1951) Heat conduction in simple metals. J Appl Phys 22:940–951

    Article  ADS  MATH  MathSciNet  Google Scholar 

  49. Rogers C, Guo BY (1988) A note on the onset of melting in a class of simple metals. Conditions on the applied boundary flux. Acta Math Sci 8:425–430

    MATH  MathSciNet  Google Scholar 

  50. Tarzia DA (1981) An inequality for the coefficient \(\sigma\) of the free boundary \(s(t)=\sigma \sqrt{t}\) of the Neumann problem for the two-phase Stefan problem. Quart Appl Math 39:491–497

    Article  MathSciNet  Google Scholar 

  51. Solomon AD, Wilson DG, Alexides V (1983) Explicit solution to phase problems. Quart Appl Math 41:237–243

    Article  MathSciNet  Google Scholar 

  52. Fokas AS, Rogers C, Schief WK (2005) Evolution of methacrylate distribution during wood saturation. A nonlinear moving boundary problem. Appl Math Lett 18:321–328

    Article  MATH  MathSciNet  Google Scholar 

  53. Fokas AS (1997) A unified transform method for solving linear and certain nonlinear PDEs. Proc R Soc Lond A 453:1411–1443

    Article  ADS  MATH  MathSciNet  Google Scholar 

  54. Rogers C (2015) On a class of reciprocal Stefan moving boundary problems. Zeit Ang Math Phys 66:2069–2079

    Article  MATH  MathSciNet  Google Scholar 

  55. Rogers C, Nucci MC (1986) On reciprocal Bäcklund transformations and the Korteweg-de Vries hierarchy. Phys Scr 33:289–292

    Article  ADS  MATH  Google Scholar 

  56. Rogers C (2015) Moving boundary problems for the Harry Dym equation and its reciprocal associates. Z Angew Math Phys 66:3025–3220

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics and Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems, The University of New South Wales, Sydney, NSW, 2052, Australia

    Colin Rogers

Authors
  1. Colin Rogers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Colin Rogers.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rogers, C. Moving boundary problems for an extended Dym equation. Reciprocal connection. Meccanica 52, 3531–3540 (2017). https://doi.org/10.1007/s11012-017-0662-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-017-0662-9

Keywords

Mathematics Subject Classification

Search

Navigation