Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 016, 19 pages      arXiv:1302.6000
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

A Generalization of the Hopf-Cole Transformation

Paulius Miškinis
Department of Physics, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saulėtekio Ave 11, LT-10223, Vilnius-40, Lithuania

Received June 04, 2012, in final form February 17, 2013; Published online February 25, 2013

A generalization of the Hopf-Cole transformation and its relation to the Burgers equation of integer order and the diffusion equation with quadratic nonlinearity are discussed. The explicit form of a particular analytical solution is presented. The existence of the travelling wave solution and the interaction of nonlocal perturbation are considered. The nonlocal generalizations of the one-dimensional diffusion equation with quadratic nonlinearity and of the Burgers equation are analyzed.

Key words: nonlocality; nonlinearity; diffusion equation; Burgers equation.

pdf (390 kb)   tex (27 kb)


  1. Abraham-Shrauner B., Guo A., Hidden and nonlocal symmetries of nonlinear differential equations, in Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics (Acireale, 1992), Editors N.H. Ibragimov, M. Torrisi, A. Valenti, Kluwer Acad. Publ., Dordrecht, 1993, 1-5.
  2. Agarwal R.P., O'Regan D. (Editors), Integral and integrodifferential equations. Theory, methods and applications, Series in Mathematical Analysis and Applications, Vol. 2, Gordon and Breach Science Publishers, Amsterdam, 2000.
  3. Bardou F., Bouchaud J.-P., Aspect A., Cohen-Tannoudji C., Lévy statistics and laser cooling. How rare events bring atoms to rest, Cambridge University Press, Cambridge, 2002.
  4. Bateman H., Some recent researches in the motion of fluids, Monthly Weather Rev. 43 (1915), 163-167.
  5. Bertotti G., Hysteresis in magnetism: for physicists, materials scientists, and engineers, Academic Press, New York, 1998.
  6. Biler P., Funaki T., Woyczynski W.A., Fractal Burgers equations, J. Differential Equations 148 (1998), 9-46.
  7. Biler P., Karch G., Woyczyński W.A., Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 613-637.
  8. Burgers J.M., The nonlinear diffusion equation. Asymptotic solutions and statistical problems, D. Reidel Publishing Company, Dordrecht, 1974.
  9. Caputo M., Linear model of dissipation whose Q is almost frequency independent. II, Geophys. J. R. Astronom. Soc. 13 (1967), 529-539.
  10. Carpinteri A., Mainardi F. (Editors), Fractals and fractional calculus in continuum mechanics, CISM Courses and Lectures, Vol. 378, Springer-Verlag, Vienna, 1997.
  11. Clarkson P.A., Mansfield E.L., Symmetries of the nonlinear heat equation, in Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics (Acireale, 1992), Editors N.H. Ibragimov, M. Torrisi, A. Valenti, Kluwer Acad. Publ., Dordrecht, 1993, 155-171.
  12. Cole J.D., On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225-236.
  13. Fisher D.S., Huse D.A., Directed paths in a random potential, Phys. Rev. B 43 (1991), 10728-10742.
  14. Fokas A.S., Zakharov V.E. (Editors), Important developments in soliton theory, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1993.
  15. Galaktionov V.A., Posashkov S.A., New exact solutions of parabolic equations with quadratic nonlinearities, U.S.S.R. Comput. Math. and Math. Phys. 29 (1989), 112-119.
  16. Gerasimov A.N., A generalization of linear laws of deformation and its application to problems of internal friction, Prikl. Mat. Meh. 12 (1948), 251-260.
  17. Gurbatov S.N., Malakhov A.N., Saichev A.I., Nonlinear random waves and turbulence in nondispersive media: waves, rays, particles, Nonlinear Science: Theory and Applications, Manchester University Press, Manchester, 1991.
  18. Hernández Heredero R., Levi D., Winternitz P., Symmetry preserving discretization of the Burgers equation, in SIDE III - Symmetries and Integrability of Difference Equations (Sabaudia, 1998), CRM Proc. Lecture Notes, Vol. 25, Amer. Math. Soc., Providence, RI, 2000, 197-208.
  19. Hilfer R. (Editor), Applications of fractional calculus in physics, World Scientific Publishing Co. Inc., River Edge, NJ, 2000.
  20. Hopf E., The partial differential equation ut+uuxuxx, Comm. Pure Appl. Math. 3 (1950), 201-230.
  21. Hwa T., Nonequilibrium dynamics of driven line liquids, Phys. Rev. Lett. 69 (1992), 1552-1555, cond-mat/9206008.
  22. Hwa T., Kardar M., Avalanches, hydrodynamics, and discharge events in models of sandpiles, Phys. Rev. A 45 (1992), 7002-7023.
  23. Kardar M., Parisi G., Zhang Y.-C., Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892.
  24. Kilbas A.A., Generalized fractional differential and difference equations, in Fractional Integral Equations, Proceedinds AMAD-2003, Belarusian State University, Minsk, 2003, 5-13.
  25. Klafter J., Shlesinger M.F., Zumofen G., Beyond Brownian motion, Phys. Today 49 (1996), no. 2, 33-39.
  26. Lakes R.S., Viscoelastic solids, CRC Mechanical Engineering Series, CRC Press, New York, 1998.
  27. Miller K.S., Ross B., An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1993.
  28. Miškinis P., Nonlinear and nonlocal integrable models, Technika, Vilnius, 2003.
  29. Miškinis P., The nonlinear heterogeneous diffusion equation and distribution of matter in the early Universe, Astrophys. J. Lett. 543 (2000), L95-L98.
  30. Nigmatullin R.R., To the theoretical explanation of the "universal response", Phys. Stat. Sol. B 123 (1984), 739-745.
  31. Novikov S., Manakov S.V., Pitaevskiĭ L.P., Zakharov V.E., Theory of solitons. The inverse scattering method, Contemporary Soviet Mathematics, Plenum, New York, 1984.
  32. Oldham K.B., Spanier J., The fractional calculus. Theory and applications of differentiation and integration to arbitrary order, Mathematics in Science and Engineering, Vol. 111, Academic Press, New York - London, 1974.
  33. Podlubny I., Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Vol. 198, Academic Press, San Diego, CA, 1999.
  34. Polyanin A.D., Manzhirov A.V., Handbook of integral equations, CRC Press, Boca Raton, FL, 1998.
  35. Riemann B., Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abh. Königl. Ges. Wiss. Göttingen 8 (1860), 43-65.
  36. Sakhnovich L.A., Integral equations with difference kernels on finite intervals, Operator Theory: Advances and Applications, Vol. 84, Birkhäuser Verlag, Basel, 1996.
  37. Samko S.G., Kilbas A.A., Marichev O.I., Fractional integrals and derivatives. Theory and applications, Gordon and Breach Science Publishers, Yverdon, 1993.
  38. Shandarin S.F., Zel'dovich Ya.B., The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium, Rev. Modern Phys. 61 (1989), 185-220.
  39. Shlesinger M.F., Zaslavsky G.M., Frisch U. (Editors), Lévy flights and related topics in physics, Lecture Notes in Physics, Vol. 450, Springer-Verlag, Berlin, 1995.
  40. Volterra V., Lecons sur la théorie mathématique de la lutte pour la vie, Gauthier-Villars, Paris, 1931.
  41. Woyczyński W.A., Burgers-KPZ turbulence. Göttingen lectures, Lecture Notes in Mathematics, Vol. 1700, Springer-Verlag, Berlin,
  42. Zaslavsky G.M., Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371 (2002), 461-580.
  43. Zel'dovich Ya.B., Gravitational instability: an approximate theory for large density perturbations, Astronom. Astrophys. 5 (1970), 84-95.

Previous article  Next article   Contents of Volume 9 (2013)