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


SIGMA 3 (2007), 005, 16 pages      math-ph/0701012      https://doi.org/10.3842/SIGMA.2007.005
Contribution to the Vadim Kuznetsov Memorial Issue

Symmetry Operators for the Fokker-Plank-Kolmogorov Equation with Nonlocal Quadratic Nonlinearity

Alexander V. Shapovalov a, Roman O. Rezaev b and Andrey Yu. Trifonov b
a) Theoretical Physics Department, Tomsk State University, 36 Lenin Ave., 660050, Tomsk, Russia
b) Laboratory of Mathematical Physics, Mathematical Physics Department, Tomsk Polytechnical University, 30 Lenin Ave., 660034, Tomsk, Russia

Received October 11, 2006, in final form December 09, 2006; Published online January 05, 2007

Abstract
The Cauchy problem for the Fokker-Plank-Kolmogorov equation with a nonlocal nonlinear drift term is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Fokker-Plank-Kolmogorov equations is considered. Illustrative examples of the one-dimensional symmetry operators are presented.

Key words: symmetry operators; Fokker-Plank-Kolmogorov equation; nonlinear partial differential equations.

pdf (271 kb)   ps (165 kb)   tex (18 kb)

References

  1. Miller W., Jr., Symmetry and separation of variables, Addison-Wesley, London - Amsterdam - Ontario - Tokio - New York, 1977.
  2. Malkin M.A., Manko V.I., Dynamic symmetries and coherent states of quantum systems, Nauka, Moscow, 1979 (in Russian).
  3. Ovsjannikov L.V., Group analysis of differential equations, Nauka, Moscow, 1978 (English transl.: Academic Press, New York, 1982).
  4. Anderson R.L., Ibragimov N.H., Lie-Bäcklund transformations in applications, Philadelphia, SIAM, 1979.
  5. Olver P.J., Application of Lie groups to differential equations, Springer, New York, 1986.
  6. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of equations of nonlinear mathematical physics, Kluwer, Dordrecht, 1993.
  7. Fushchych W.I., Nikitin A.G., Symmetries of equations of quantum mechanics, Allerton Press Inc., New York, 1994.
  8. Krasilshchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of jet spaces and nonlinear partial differential equations, Advanced Studies in Contemporary Mathematics, Vol. 1, Gordon and Breach Science Publishers, New York, 1986.
  9. Gaeta G., Nonlinear symmetries and nonlinear equations, Mathematics and its Applications, Vol. 299, Kluwer Academic Publishers Group, Dordrecht, 1994.
  10. Frank T.D., Nonlinear Fokker-Planck equations, Springer, Berlin, 2004.
  11. Bellucci S., Trifonov A.Yu., Semiclassically-concentrated solutions for the one-dimensional Fokker-Planck equation with a nonlocal nonlinearity, J. Phys. A: Math. Gen. 38 (2005), L103-L114.
  12. Maslov V.P., The complex WKB method for nonlinear equations. I. Linear theory, Birkhauser Verlag, Basel - Boston - Berlin, 1994.
  13. Belov V.V., Dobrokhotov S.Yu., Semiclassical Maslov asymptotics with complex phases. I. General approach, Teor. Mat. Fiz. 92 (1992), 215-254 (English transl.: Theoret. and Math. Phys. 92 (1992), 843-868).
  14. Dobrokhotov S.Yu., Martinez Olive V., Localized asymptotic solutions of the magneto dynamo equation in ABC-fields, Mat. Zametki 54 (1993), no. 4, 45-68 (English transl.: Math. Notes 54 (1993), 1010-1025).
  15. Albeverio S., Dobrokhotov S.Yu., Poteryakhin M., On quasimodes of small diffusion operators corresponding to stable invariant tori with nonregular neighborhoods, Asymptot. Anal. 43 (2005), no. 3, 171-203.
  16. Maslov V.P., Global exponential asymptotic behavior of the solutions of tunnel-type equations and the problem of large deviations, Trudy Mat. Inst. Steklov. 193 (1984), 150-180.
  17. Belov V.V., Shapovalov A.V., Trifonov A.Yu., The trajectory-coherent approximation and the system of moments for the Hartree type equation, Int. J. Math. Math. Sci. 32 (2002), 325-370, math-ph/0012046.
  18. Belov V.V., Trifonov A.Yu., Shapovalov A.V., Semiclassical trajectory-coherent approximations to Hartree type equations, Teor. Mat. Fiz. 130 (2002), 460-492 (English transl.: Theoret. and Math. Phys. 130 (2002), 391-418).
  19. Lisok A.L., Trifonov A.Yu., Shapovalov A.V., The evolution operator of the Hartree-type equation with a quadratic potential, J. Phys. A: Math. Gen. 37 (2004), 4535-4556, math-ph/0312004.
  20. Shapovalov A.V., Trifonov A.Yu., Lisok A.L., Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential, SIGMA 1 (2005), 007, 14 pages, math-ph/0511010.
  21. Shvedov O.Yu., Semiclasical symmetries Ann. Phys. 296 (2002), 51-89.
  22. Trifonov A.Yu., Trifonova L.B., Fokker-Planck-Kolmogorov equation's with a nonlocal nonlinearity in a semiclassical approximation, Izv. Vyssh. Uchebn. Zaved. Fiz. 45 (2002), no. 2, 121-129 (English transl.: Russian Phys. J. 45 (2002), 118-128).
  23. Bezverbny A.V., Gogolev A.S., Trifonov A.Yu., Rezaev R.O., Nonlinear Fokker-Planck-Kolmogorov equation in the semiclassical coherent trajectory approximation Izv. Vyssh. Uchebn. Zaved. Fiz. 48 (2005), no. 6, 38-47 (English transl.: Russian Phys. J. 48 (2005), 592-604).
  24. Shiino M., Yoshida K., Chaos-nonchaos phase transitions induced by external noise in ensembles of nonlinearly coupled oscillators, Phys. Rev. E 63 (2001), 026210, 6 pages.
  25. Shiino M., Stability analysis of mean-field-type nonlinear Fokker-Planck equations associated with a generalized entropy and its application to the self-gravitating system, Phys. Rev. E 67 (2003), 056118, 5 pages.
  26. Drozdov A.N., Morillo M., Validity of basic concepts in nonlinear cooperative Fokker-Plank models, Phys. Rev. E 54 (1996), 3304-3313.
  27. Schuller F.P., Vogt P., Product structure of heat phase space and branching Brownian motion, Ann. Phys. 308 (2003), 528-554, math-ph/0209016.
  28. Frank T.D., Classical Langevin equations for the free electron gas and blackbody radiation, J. Phys. A: Math. Gen. 37 (2004), 3561-3567.
  29. Tatarskii V.I., The Wigner representation of quantum mechanics, Uspekhi Fiz. Nauk 139 (1983), 587-619 (English transl.: Soviet Phys. Uspekhi 139 (1983), 311-327).
  30. Popov M.M., Green functions for the Schrödinger equation with a quadric potential, in Problems of Mathematical Physics, Vol. 6, Leningrad, 1973, 119-125 (in Russian).
  31. Dodonov V.V., Malkin I.A., Man'ko V.I., Integrals of motion, Green functions and coherent states of dynamic systems, Internat. J. Theoret. Phys. 14 (1975), 37-54.
  32. Pukhnachov V.V., Transformations of equivalence and the hidden symmetry of evolution equations, Dokl. Akad. Nauk SSSR 294 (1987), 535-538 (English transl.: Sov. Math. Dokl. 34 (1987), 555-558).
  33. Lahno V.I., Spichak S.V., Stogniy V.I., Symmetry analysis of evolution type equations, Computer Research Institute, Moscow - Igevsk, 2004 (in Russian).


Previous article   Next article   Contents of Volume 3 (2007)