On the equations for p-adic closed and open strings

Research Articles

Abstract

A survey of several pure mathematical results concerning the boundary-value problems for nonlinear pseudo-differential equation for closed and open strings in d-dimensional flat spacetime is presented. We obtained some results on existence or nonexistence of solutions. In particular, the absence of almost-periodic solutions was shown. We consider also some numerical approaches to the problems.

Key words

p-adic strings tachyons 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. H. Frampton and Y. Okada, “Effective scalar field theory of p-adic string,” Phys. Rev. D 37(10), 3077–3079 (1989).CrossRefMathSciNetGoogle Scholar
  2. 2.
    L. Brekke and P. G. O. Freund, “p-Adic numbers in physics,” Phys. Rep. 233(1), 1–66 (1993).CrossRefMathSciNetGoogle Scholar
  3. 3.
    N. Moeller and M. Schnabl, “Tachyon condensation in open-closed p-adic string theory,” JHEP 0401, 011 (2004).CrossRefMathSciNetGoogle Scholar
  4. 4.
    M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory Vols. 1, 2, (Cambridge Univ. Press, Cambridge, 1987, 1988).MATHGoogle Scholar
  5. 5.
    E. Witten, “Noncommutative geometry and string field theory” Nucl. Phys. B 268, 253 (1986).CrossRefMathSciNetGoogle Scholar
  6. 6.
    T. G. Erler and D. J. Gross, “Locality, causality, and an initial value formulation for open string field theory,” arXiv:hep-th/0406199.Google Scholar
  7. 7.
    I. V. Volovich, “p-Adic string,” Class. Quantum Grav. 4, L83–L87 (1987).CrossRefMathSciNetGoogle Scholar
  8. 8.
    L. Brekke, P. G. O. Freund, M. Olson and E. Witten, “Non-archimedian string dynamics,” Nucl. Phys. B 302, 365 (1988).CrossRefMathSciNetGoogle Scholar
  9. 9.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).Google Scholar
  10. 10.
    I. Ya. Aref’eva, L. V. Joukovskaya and A. S. Koshelev, “Time evolution in superstring field theory on non-BPS brane. I. Rolling tachyon and energy-momentum conservation,” JHEP 0309, 012 (2003); arXiv: hep-th/0301137.CrossRefMathSciNetGoogle Scholar
  11. 11.
    I. Ya. Aref’eva and A. S. Koshelev, “Cosmic acceleration and crossing of w = −1 barrier in non-local cubic superstring field theory model,” JHEP 041 (2007).Google Scholar
  12. 12.
    G. Calcagni, “Cosmological tachyon from cubic string field theory,” JHEP 05 012 (2006); arXiv:hep-th/0512259.Google Scholar
  13. 13.
    I. Ya. Aref’eva, “Nonlocal string tachyon as a model for cosmological dark energy,” AIP Conf. Proc. 826, 301–311 (2006); arXiv:astro-ph/0410443.CrossRefMathSciNetGoogle Scholar
  14. 14.
    N. Barnaby, T. Biswas and J.M. Cline, “p-Adic inflation,” JHEP 0704, 056 (2007); arXiv:hep-th/0612230.CrossRefMathSciNetGoogle Scholar
  15. 15.
    I. Ya. Aref’eva, L. V. Joukovskaya and S. Yu. Vernov, “Bouncing and accelerating solutions in nonlocal stringy models,” JHEP 0707, 087 (2007); arXiv:hep-th/0701184v4.CrossRefMathSciNetGoogle Scholar
  16. 16.
    L. Joukovskaya, “Dynamics in nonlocal cosmological models derived from string field theory,” Phys. Rev. D 76, 105007 (2007); arXiv:0707.1545v2[hep-th].Google Scholar
  17. 17.
    Ya. I. Volovich, “Numerical study of nonlinear equations with infinite numbers of derivatives,” J. Phys. A: Math. Gen. 36(32), 8685–8701 (2003); arXiv:math-ph/0301028.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    V. S. Vladimirov and Ya. I. Volovich, “Nonlinear dynamics equation in p-adic string theory,” Theor. Math. Phys. 138(3), 297–309 (2004); arXiv:math-ph/0306018.CrossRefMathSciNetGoogle Scholar
  19. 19.
    L. V. Joukovskaja, “Iterative method for solving nonlinear integral equations describing rolling sollutions in string theory,” Theor. Math. Phys. 146(3), 335–342 (2006).CrossRefGoogle Scholar
  20. 20.
    D. V. Prokhorenko, “On some nonlinear integral equations in the (super)string theory,” arXiv:mathph/0611068.Google Scholar
  21. 21.
    V. S. Vladimirov, “The equation of the p-adic string for the scalar tachyon field,” Izvestiya: Mathematics 69 (3), 487–512 (2005); arXiv:math-ph/0507018.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    N. Moeller and B. Zwiebach, “Dynamics with infinitely many time derivatives and rolling tachyons,” JHEP 0210, 034 (2002); arXiv:hep-th/0207107.CrossRefMathSciNetGoogle Scholar
  23. 23.
    N. Barnaby and N. Kamran, “Dynamics with infinitely many derivatives: The initial value problem,“ JHEP 0802, 008 (2008); arXiv:0709.3968v2[hep-th].CrossRefMathSciNetGoogle Scholar
  24. 24.
    A. Sen, “Rolling tachyon,” JHEP 0204, 048 (2002); arXiv:hep-th/0203211.CrossRefGoogle Scholar
  25. 25.
    D. Ghoshal and A. Sen, “Tachyon condensation and brane descent relations in p-adic string theory,” Nucl. Phys. B 584, 300–312 (2002).CrossRefMathSciNetGoogle Scholar
  26. 26.
    N. Barnaby, “Caustic formation in tachyon effective field theories,” JHEP 0407, 025 (2004); arXiv:hepth/0406120.CrossRefMathSciNetGoogle Scholar
  27. 27.
    E. Coletti, I. Sigalov and W. Taylor, “Taming the tachyon in cubic string field theory,” JHEP 0508, 104 (2005); arXiv:hep-th/0505031.CrossRefMathSciNetGoogle Scholar
  28. 28.
    J. A. Minahan, “Mode interactions of the tachyon condensate in p-adic string theory,” JHEP 0103, 028 (2001); arXiv:hep-th/01020071v1.CrossRefMathSciNetGoogle Scholar
  29. 29.
    V. S. Vladimirov, “Nonlinear equations for p-adic open, closed, and open-closed strings,” Theor. Math. Phys. 149(3), 1604–1616 (2006); arXiv:0705.4600vl[math-ph].CrossRefGoogle Scholar
  30. 30.
    V. S. Vladimirov, “The equation of the p-adic closed strings for the scalar tachyons fields,” Science in China Series A: Mathematics, 51:4, 563–573 (2006).Google Scholar
  31. 31.
    V. S. Vladimirov, “On nonlinear equations of p-adic strings for scalar tachyon fields,” to appear in Proc. Third Intern. Conf. on p-AdicMath. Physics,Moscow, 2007, Proc.SteklovMath. Inst. 265 (2009).Google Scholar
  32. 32.
    E. H. Lieb and M. Loss, Analysis, AMS, Graduate Studies in Math. 14.Google Scholar
  33. 33.
    A. F. Nikiforov and V. B. Uvarov, Special Functions in Mathematical Physics (Nauka, Moscow, 1978) [in Russian].Google Scholar
  34. 34.
    I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Summs, Series and Products (Moscow, 1963) [in Russian].Google Scholar
  35. 35.
    I. P. Natanson, Constructive Theory of Functions (Gostekhizdat, Moscow, 1949) [in Russian].Google Scholar
  36. 36.
    B.M. Levitan, Almost Periodic Functions (Gostekhizdat, Moscow, 1953) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

Personalised recommendations