Acta Applicandae Mathematicae

, Volume 100, Issue 2, pp 113–185 | Cite as

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

  • Roman O. Popovych
  • Michael Kunzinger
  • Nataliya M. Ivanova
Article

Abstract

We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.

Keywords

Lie symmetry of differential equations Group classification problem Potential symmetry of differential equations Conservation laws Fokker–Planck equations Kolmogorov equations Darboux transformation Equivalence group Admissible transformation Normalized class of differential equations Generalized potential symmetry of differential equations 

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References

  1. 1.
    Akhatov, I.S., Gazizov, R.K., Ibragimov, N.K.: Group classification of equation of nonlinear filtration. Dokl. AN SSSR 293, 1033–1035 (1987) MathSciNetGoogle Scholar
  2. 2.
    Akhatov, I.S., Gazizov, R.K., Ibragimov, N.K.: Nonlocal symmetries. A heuristic approach. In: Itogi Nauki i Tekhniki, Current Problems in Mathematics. Newest Results, vol. 34, pp. 3–83. VINITI, Moscow (1989). (in Russian, English translation in J. Sov. Math. 55, 1401–1450 (1991)) Google Scholar
  3. 3.
    Anco, S.C., Bluman, G.: Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations. J. Math. Phys. 38, 3508–3532 (1997) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Anco, S.C., Bluman, G.: Direct construction method for conservation laws of partial differential equations I. Examples of conservation law classifications. Eur. J. Appl. Math. 13(5), 545–566 (2002); arXiv:math-ph/0108023 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Anco, S.C., Bluman, G.: Direct construction method for conservation laws of partial differential equations II. General treatment. Eur. J. Appl. Math. 13(5), 567–585 (2002); arXiv:math-ph/0108024 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Anco, S.C., The, D.: Symmetries, conservation laws, and cohomology of Maxwell’s equations using potentials. Acta Appl. Math. 89, 1–52 (2005); arXiv:math-ph/0501052 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Atherton, R.W., Homsy, G.M.: On the existence and formulation of variational principles for nonlinear differential equations. Stud. Appl. Math. 54, 31–60 (1975) MathSciNetGoogle Scholar
  8. 8.
    Basarab-Horwath, P., Lahno, V., Zhdanov, R.: The structure of Lie algebras and the classification problem for partial differential equations. Acta Appl. Math. 69, 43–94 (2001) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bluman, G.W.: On the transformation of diffusion processes into the Wiener process. SIAM J. Appl. Math. 39, 238–247 (1980) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Bluman, G.W.: Simplifying the form of Lie groups admitted by a given differential equation. J. Math. Anal. Appl. 145, 52–62 (1990) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bluman, G.: Connections between symmetries and conservation laws. SIGMA 1, paper 011, 16 pp. (2005) Google Scholar
  12. 12.
    Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989) MATHGoogle Scholar
  13. 13.
    Bluman, G.W., Reid, G.J., Kumei, S.: New classes of symmetries for partial differential equations. J. Math. Phys. 29, 806–811 (1988) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bluman, G., Shtelen, V.: Nonlocal transformations of Kolmogorov equations into the backward heat equation. J. Math. Anal. Appl. 291, 419–437 (2004) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bluman, G., Temuerchaolu, Anco, S.C.: New conservation laws obtained directly from symmetry action on a known conservation law. J. Math. Anal. Appl. 322, 233–250 (2006) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Bocharov, A.V., Chetverikov, V.N., Duzhin, S.V., Khor’kova, N.G., Krasil’shchik, I.S., Samokhin, A.V., Torkhov, Y.N., Verbovetsky, A.M., Vinogradov, A.M.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Faktorial, Moscow (1997) Google Scholar
  17. 17.
    Caviglia, G.: Conservation laws for the Navier–Stokes equations. Int. J. Eng. Sci. 24, 1295–1302 (1986) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Cherkasov, I.D.: On the transformation of the diffusion process to a Wiener process. Theory Probab. Appl. 2, 373–377 (1957) CrossRefGoogle Scholar
  19. 19.
    Cicogna, G., Vitali, D.: Classification of the extended symmetries of Fokker–Planck equations. J. Phys. A 23, L85–L88 (1990) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Crum, M.M.: Associated Sturm–Liouville systems. Q. J. Math. Oxf. Ser. (2) 6, 121–127 (1955) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Dorodnitsyn, V.A., Svirshchevskii, S.R.: On Lie–Bäcklund groups admitted by the heat equation with a source. Preprint No. 101, Keldysh Institute of Applied Mathematics of Academy of Sciences USSR, Moscow (1983) Google Scholar
  22. 22.
    Edelen, D.G.B.: Isovector Methods for Equations of Balance. Sijthoff & Noordhoff, Alphen aan den Rijn (1980) MATHGoogle Scholar
  23. 23.
    Feller, W.: Diffusion processes in genetics. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley (1951) Google Scholar
  24. 24.
    Fels, M., Olver, P.: Moving coframes I. A practical algorithm. Acta Appl. Math. 51, 161–213 (1998) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Fels, M., Olver, P.: Moving coframes II. Regularization and theoretical foundations. Acta Appl. Math. 55, 127–208 (1999) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Finlayson, B.A.: Existence of variational principles for the Navier–Stokes equation. Phys. Fluids 15, 963–967 (1972) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Fokker, A.D.: Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Ann. Phys. 43, 810–820 (1914) CrossRefGoogle Scholar
  28. 28.
    Fushchych, W.I., Nikitin, A.G.: Symmetries of Equations of Quantum Mechanics. Allerton, New York (1994) Google Scholar
  29. 29.
    Fushchych, W.I., Shtelen, W.M., Serov, M.I., Popovych, R.O.: Q-conditional symmetry of the linear heat equation. Proc. Acad. Sci. Ukr. 12, 28–33 (1992) MathSciNetGoogle Scholar
  30. 30.
    Gardiner, C.W.: Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences, 2nd edn. Springer Series in Synergetics, vol. 13. Springer, Berlin (1985) Google Scholar
  31. 31.
    Gihman, I.I., Skorohod, A.V.: The Theory of Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 210. Springer, Berlin/New York (1980) MATHGoogle Scholar
  32. 32.
    Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Mathematics and Its Applications (Soviet Series). Reidel, Dordrecht (1985) MATHGoogle Scholar
  33. 33.
    Ibragimov, N.H.: Laplace type invariants for parabolic equations. Nonlinear Dyn. 28, 125–133 (2002) MATHCrossRefGoogle Scholar
  34. 34.
    Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333, 311–328 (2007) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Ibragimov, N.H., Kolsrud, T.: Lagrangian approach to evolution equations: symmetries and conservation laws. Nonlinear Dyn. 36, 29–40 (2004) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Ivanova, N.M.: Local and nonlocal conservation laws of diffusion–convection equations. Collect. Works Inst. Math. (Kyiv) 3(2), 148–158 (2006) Google Scholar
  37. 37.
    Ivanova, N.M.: Conservation laws of multidimensional diffusion–convection equations. Nonlinear Dyn. 49, 71–81 (2007) CrossRefMathSciNetGoogle Scholar
  38. 38.
    Ivanova, N.M., Popovych, R.O.: Equivalence of conservation laws and equivalence of potential systems. Int. J. Theor. Phys. 46, 2658–2668 (2007). Preprint No. 1885, ESI for Mathematical Physics; arXiv:math-ph/0611032 MATHCrossRefGoogle Scholar
  39. 39.
    Johnpillai, I.K. Mahomed, F.M.: Singular invariant equation for the (1+1) Fokker–Planck equation. J. Phys. A 34, 11033–11051 (2001) MATHMathSciNetGoogle Scholar
  40. 40.
    Khamitova, R.S.: The structure of a group and the basis of conservation laws. Teor. Mat. Fiz. 52(2), 244–251 (1982) MATHMathSciNetGoogle Scholar
  41. 41.
    Kingston, J.G., Sophocleous, C.: On form-preserving point transformations of partial differential equations. J. Phys. A 31, 1597–1619 (1998) MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Kolmogorov, A.N.: On analytic methods in probability. Uspekhi Mat. Nauk 5, 5–41 (1938) MathSciNetGoogle Scholar
  43. 43.
    Lahno, V.I., Spichak, S.V., Stognii, V.I.: Symmetry Analysis of Evolution Type Equations. RCD, Moscow/Izhevsk (2004) Google Scholar
  44. 44.
    Liboff, R.L.: Introduction to the Theory of Kinetic Equations. Wiley, New York (1969) MATHGoogle Scholar
  45. 45.
    Lie, S.: Über die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung. Arch. Math. 6(3), 328–368 (1881) (English translation by N.H. Ibragimov: Lie, S. On integration of a class of linear partial differential equations by means of definite integrals. CRC Handbook of Lie Group Analysis of Differential Equations, vol. 2, pp. 473–508 (1994)) Google Scholar
  46. 46.
    Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991) MATHGoogle Scholar
  47. 47.
    Morozov, O.I.: Contact equivalence problem for linear parabolic equations. arXiv:math-ph/0304045, 19 pp. Google Scholar
  48. 48.
    Morse, P.M., Feshbach, H.: Methods of Theoretical Physics, vol. 1. McGraw-Hill, New York (1953) Google Scholar
  49. 49.
    Olver, P.: Applications of Lie Groups to Differential Equations. Springer, New York (1986) MATHGoogle Scholar
  50. 50.
    Olver, P.: Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995) MATHGoogle Scholar
  51. 51.
    Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982) MATHGoogle Scholar
  52. 52.
    Planck, M.: Sitzungsber. Preuss. Akad. Wiss. Phys. Math. K1 (1917) Google Scholar
  53. 53.
    Popovych, R.O.: On the symmetry and exact solutions of a transport equation. Ukr. Math. J. 47, 142–148 (1995) CrossRefMathSciNetGoogle Scholar
  54. 54.
    Popovych, R.O.: Normalized classes of nonlinear Schroedinger equations. Bulg. J. Phys. 33(s2), 211–222 (2006). (Proceedings of the VI International Workshop “Lie Theory and Its Application to Physics”, 15–21 August 2005, Varna, Bulgaria) MATHGoogle Scholar
  55. 55.
    Popovych, R.O.: No-go theorem on reduction operators of linear second-order parabolic equations. Collect. Works Inst. Math. (Kyiv) 3(2), 231–238 (2006) Google Scholar
  56. 56.
    Popovych, R.O.: Classification of admissible transformations of differential equations. Collect. Works Inst. Math. (Kyiv) 3(2), 239–254 (2006) Google Scholar
  57. 57.
    Popovych, R.O., Eshraghi, H.: Admissible point transformations of nonlinear Schrodinger equations. In: Proceedings of the 10th International Conference in Modern Group Analysis (MOGRAN X), Larnaca, Cyprus, 2004, pp. 168–176 (2005) Google Scholar
  58. 58.
    Popovych, R.O., Ivanova, N.M.: New results on group classification of nonlinear diffusion–convection equations. J. Phys. A 37, 7547–7565 (2004); arXiv:math-ph/0306035 MATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Popovych, R.O., Ivanova, N.M.: Hierarchy of conservation laws of diffusion–convection equations. J. Math. Phys. 46, 043502 (2005); arXiv:math-ph/0407008 CrossRefMathSciNetGoogle Scholar
  60. 60.
    Popovych, R.O., Ivanova, N.M.: Potential equivalence transformations for nonlinear diffusion–convection equations. J. Phys. A 38, 3145–3155 (2005); arXiv:math-ph/0402066 MATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    Popovych, R.O., Kunzinger, M., Eshraghi, H.: Admissible point transformations of nonlinear Schrödinger equations. arXiv:math-ph/0611061, 35 pp. Google Scholar
  62. 62.
    Prokhorova, M.: The structure of the category of parabolic equations. arXiv:math.AP/0512094, 24 pp. Google Scholar
  63. 63.
    Pucci, E., Saccomandi, G.: Potential symmetries and solutions by reduction of partial differential equations. J. Phys. A 26, 681–690 (1993) MATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Pucci, E., Saccomandi, G.: Potential symmetries of Fokker Plank equations. In: Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, Acireale, 1992, pp. 291–298. Kluwer, Dordrecht (1993) Google Scholar
  65. 65.
    Risken, H.: The Fokker–Planck Equation. Springer, Berlin (1989) MATHGoogle Scholar
  66. 66.
    Roussopoulos, P.: Métodes variationnelles en théorie des collisions. C. R. Acad. Sci. Paris 236, 1858–1860 (1953) MATHGoogle Scholar
  67. 67.
    Saccomandi, G.: Potential symmetries and direct reduction methods of order two. J. Phys. A 30, 2211–2217 (1997) MATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    Sastri, C.C.A., Dunn, K.A.: Lie symmetries of some equations of the Fokker–Planck type. J. Math. Phys. 26, 3042–3047 (1985) MATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    Shtelen, W.M., Stogny, V.I.: Symmetry properties of one- and two-dimensional Fokker–Planck equations. J. Phys. A 22, L539–L543 (1989) CrossRefMathSciNetGoogle Scholar
  70. 70.
    Sophocleous, C.: Potential symmetries of nonlinear diffusion–convection equations. J. Phys. A 29, 6951–6959 (1996) MATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Spichak, S., Stognii, V.: One-dimensional Fokker–Planck equation invariant under four- and six-parametrical group. In: Proceedings of the Third Int. Conf. on Symmetry in Nonlinear Mathematical Physics, Kyiv, 1999. Pr. Inst. Mat. Nat. Akad. Nauk Ukr. Mat. Zastos., vol. 30, pp. 204–209 (2000) Google Scholar
  72. 72.
    Spichak, S., Stognii, V.: Symmetry classification and exact solutions of the one-dimensional Fokker–Planck equation with arbitrary coefficients of drift and diffusion. J. Phys. A 32, 8341–8353 (1999) MATHCrossRefMathSciNetGoogle Scholar
  73. 73.
    Spichak, S.V., Stognii, V.I.: Symmetric classification of the one-dimensional Fokker–Planck–Kolmogorov equation with arbitrary drift and diffusion coefficients. Neliniĭni Kolyv. 2, 401–413 (1999) (in Russian) MATHMathSciNetGoogle Scholar
  74. 74.
    Stohny, V.: Symmetry properties and exact solutions of the Fokker–Planck equation. J. Nonlinear Math. Phys. 4, 132–136 (1997) MATHCrossRefMathSciNetGoogle Scholar
  75. 75.
    Tonti, E.: On the variational formulation for linear initial value problems. Ann. Mat. Pura Appl. (4) 95, 331–359 (1973) MATHCrossRefMathSciNetGoogle Scholar
  76. 76.
    Tsujishita, T.: On variation bicomplexes associated to differential equations. Osaka J. Math. 19, 311–363 (1982) MATHMathSciNetGoogle Scholar
  77. 77.
    Vinogradov, A.M.: The \(\mathcal{C}\) -spectral sequence, Lagrangian formalism, and conservation laws I. The linear theory. II. The nonlinear theory. J. Math. Anal. Appl. 100, 1–40 (1984); 41–129 MATHCrossRefMathSciNetGoogle Scholar
  78. 78.
    Zharinov, V.V.: Conservation laws of evolution systems. Teor. Mat. Fiz. 68(2), 163–171 (1986) MathSciNetGoogle Scholar
  79. 79.
    Wahlquist, H.D., Estabrook, F.B.: Prolongation structures of nonlinear evolution equations. J. Math. Phys. 16, 1–7 (1975) MATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    Wolf, T.: A comparison of four approaches to the calculation of conservation laws. Eur. J. Appl. Math. 13(5), 129–152 (2002) MATHCrossRefGoogle Scholar
  81. 81.
    Zhdanov, R., Lahno, V.: Group classification of the general evolution equation: local and quasilocal symmetries. SIGMA 1, paper 009, 7 pp. (2005); arXiv:nlin.SI/0510003 Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Roman O. Popovych
    • 1
    • 2
  • Michael Kunzinger
    • 2
  • Nataliya M. Ivanova
    • 1
  1. 1.Institute of Mathematics of NAS of UkraineKievUkraine
  2. 2.Fakultät für MathematikUniversität WienViennaAustria

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