Advertisement

Theoretical and Mathematical Physics

, Volume 138, Issue 3, pp 297–309 | Cite as

Nonlinear Dynamics Equation in p-Adic String Theory

  • V. S. Vladimirov
  • Ya. I. Volovich
Article

Abstract

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation is of interest in mathematical physics and its applications, in particular, in string theory and cosmology. We undertake a systematic mathematical investigation of the properties of these equations and prove the main uniqueness theorem for the solution in an algebra of generalized functions. We discuss boundary problems for bounded solutions and prove the existence theorem for spatially homogeneous solutions for odd p. For even p, we prove the absence of a continuous nonnegative solution interpolating between two vacuums and indicate the possible existence of discontinuous solutions. We also consider the multidimensional equation and discuss soliton and q-brane solutions.

p-adic string pseudodifferential operator nonlinear equations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    L. Brekke, P. G. O. Freund, M. Olson, and E. Witten, Nucl. Phys. B, 302, 365 (1988).Google Scholar
  2. 2.
    P. H. Frampton and Y. Okada, Phys. Rev. D, 37, 3077–3079 (1988).Google Scholar
  3. 3.
    V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics [in Russian], Nauka, Moscow (1994); English transl., World Scientific, River Edge, N. J. (1994).Google Scholar
  4. 4.
    L. Brekke and P. G. O. Freund, Phys. Rep. (Rev. Sct. Phys. Lett.), 233, No. 1, 1–66 (1993).Google Scholar
  5. 5.
    A. Sen, JHEP, 0204, 048 (2002); hep-th/0203211 (2002).Google Scholar
  6. 6.
    N. Moeller and B. Zwiebach, JHEP, 0210, 034 (2002); hep-th/0207107 (2002).Google Scholar
  7. 7.
    I. Ya. Aref'eva, L. V. Joukovskaya, and A. S. Koshelev, JHEP, 0309, 012 (2003); hep-th/0301137 (2003).Google Scholar
  8. 8.
    A. Yu. Khrennikov, p-Adic Valued Distributions in Mathematical Physics, Kluwer, Dordrecht (1994).Google Scholar
  9. 9.
    M. B. Green, J.-H. Schwarz, and E. Witten, Superstring Theory, Cambridge Univ. Press, Cambridge (1987).Google Scholar
  10. 10.
    Ya. Volovich, J. Phys. A, 36, 8685–8702 (2003); math-ph/0301028 (2003).Google Scholar
  11. 11.
    V. S. Vladimirov, Lett. Math. Phys., 28, 123–131 (1993).Google Scholar
  12. 12.
    I. M. Gelfand and G. E. Shilov, Generalized Functions: Issue 2. Spaces of Fundamental and Generalized Functions [in Russian], Fizmatlit, Moscow (1958); English transl., Acad. Press, New York (1968).Google Scholar
  13. 13.
    V. S. Vladimirov, Methods in the Theory of Functions of Several ComplexV ariables [in Russian], Nauka, Moscow (1964).Google Scholar
  14. 14.
    D. Ghoshal and A. Sen, Nucl. Phys. B, 584, 300–312 (2000).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • V. S. Vladimirov
    • 1
  • Ya. I. Volovich
    • 2
  1. 1.Steklov Mathematical InstituteRASMoscowRussia
  2. 2.Moscow State UniversityMoscowRussia

Personalised recommendations