On Nonlinear Equations of $p$-adic Strings for Scalar Tachyon Fields

V. S. Vladimirov

Informatics and Automation, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 254-272.

Résumé

We consider boundary value problems for open and closed $p$-adic strings for scalar tachyon fields. Estimates for solutions to these problems and possible ways of constructing these solutions are obtained by reducing the problems to linear parabolic equations with nonlinear boundary conditions. We give an application of Gauss-type quadrature formulas to the numerical solution of the boundary value problems, and discuss the possibility of using these methods in multidimensional problems ($d=2$).

Bibliographie

[1] Frampton P. H., Okada Y., “Effective scalar field theory of $p$-adic string”, Phys. Rev. D, 37:10 (1988), 3077–3079 | DOI | MR

[2] Brekke L., Freund P. G. O., “$p$-Adic numbers in physics”, Phys. Rep., 233:1 (1993), 1–66 | DOI | MR

[3] Moeller N., Schnabl M., “Tachyon condensation in open-closed $p$-adic string theory”, J. High Energy Phys., 2004, no. 01, Pap. 011 | MR

[4] Vladimirov V. S., “O nelineinykh uravneniyakh $p$-adicheskikh otkrytykh, zamknutykh i otkryto-zamknutykh strun”, TMF, 149:3 (2006), 354–367 | DOI | MR | Zbl

[5] Grin M., Shvarts Dzh., Vitten E., Teoriya superstrun, T. 1, 2, Mir, M., 1990

[6] Witten E., “Non-commutative geometry and string field theory”, Nucl. Phys. B, 268 (1986), 253–294 | DOI | MR

[7] Erler T. G., Gross D. J., Locality, causality, and an initial value formulation for open string field theory, E-print , 2004 arXiv: hep-th/0406199

[8] Volovich I. V., “$p$-Adic string”, Class. and Quantum Grav., 4 (1987), L83–L87 | DOI | MR

[9] Brekke L., Freund P. G. O., Olson M., Witten E., “Non-archimedean string dynamics”, Nucl. Phys. B, 302 (1988), 365–402 | DOI | MR

[10] Vladimirov V. S., Volovich I. V., Zelenov E. I., $p$-Adicheskii analiz i matematicheskaya fizika, Nauka, M., 1994 | MR

[11] Aref'eva I. Ya., Koshelev A. S., Joukovskaya L. V., “Time evolution in superstring field theory on non-BPS brane. I: Rolling tachyon and energy-momentum conservation”, J. High Energy Phys., 2003, no. 09, Pap. 012 ; arXiv: hep-th/0301137 | MR

[12] Aref'eva I. Ya., Koshelev A. S., “Cosmic acceleration and crossing of $w=-1$ barrier in non-local cubic superstring field theory model”, J. High Energy Phys., 2007, no. 02, Pap. 041 | MR

[13] Calcagni G., “Cosmological tachyon from cubic string field theory”, J. High Energy Phys., 2006, no. 05, Pap. 012 ; arXiv: hep-th/0512259 | MR

[14] Aref'eva I. Ya., “Nonlocal string tachyon as a model for cosmological dark energy”, $p$-Adic mathematical physics, AIP Conf. Proc., 826, Amer. Inst. Phys., Melville, 2006, 301–311 ; arXiv: astro-ph/0410443 | DOI | MR | Zbl

[15] Volovich Ya., “Numerical study of nonlinear equations with an infinite number of derivatives”, J. Phys. A, 36:32 (2003), 8685–8701 ; arXiv: math-ph/0301028 | DOI | MR | Zbl

[16] Zhukovskaya L. V., “Iteratsionnyi metod resheniya nelineinykh integralnykh uravnenii, opisyvayuschikh rollingovye resheniya v teorii strun”, TMF, 146:3 (2006), 402–409 | DOI | MR

[17] Barnaby N., Biswas T., Cline J. M., “$p$-Adic inflation”, J. High Energy Phys., 2007, no. 04, Pap. 056 ; arXiv: hep-th/0612230 | MR

[18] Aref'eva I. Ya., Joukovskaya L. V., Vernov S. Yu., “Bouncing and accelerating solutions in nonlocal stringy models”, J. High Energy Phys., 2007, no. 07, Pap. 087 ; arXiv: hep-th/0701184v4 | MR

[19] Joukovskaya L., “Dynamics in nonlocal cosmological models derived from string field theory”, Phys. Rev. D, 76:10 (2007), Pap. 105007 ; arXiv: 0707.1545v2[hep-th] | DOI | MR

[20] Vladimirov V. S., Volovich Ya. I., “O nelineinom uravnenii dinamiki v teorii $p$-adicheskoi struny”, TMF, 138:3 (2004), 355–368 ; arXiv: math-ph/0306018 | DOI | MR | Zbl

[21] Vladimirov V. S., “Ob uravnenii $p$-adicheskoi otkrytoi struny dlya skalyarnogo polya takhionov”, Izv. RAN. Ser. mat., 69:3 (2005), 55–80 ; arXiv: math-ph/0507018 | DOI | MR | Zbl

[22] Moeller N., Zwiebach B., “Dynamics with infinitely many time derivatives and rolling tachyons”, J. High Energy Phys., 2002, no. 10, Pap. 034 ; arXiv: hep-th/0207107 | MR

[23] Prokhorenko D. V., On some nonlinear integral equation in the (super)string theory, E-print , 2006 arXiv: math-ph/0611068

[24] Vladimirov V. S., “The equation of the $p$-adic closed strings for the scalar tachyon field”, Sci. China. A: Math., 51:4 (2008), 754–764 | DOI | MR | Zbl

[25] Barnaby N., Kamran N., Dynamics with infinitely many derivatives: The initial value problem, E-print , 2007 arXiv: 0709.3968v2[hep-th]

[26] Sen A., “Rolling tachyon”, J. High Energy Phys., 2002, no. 04, Pap. 048 ; arXiv: hep-th/0203211 | MR

[27] Ghoshal D., Sen A., “Tachyon condensation and brane descent relations in $p$-adic string theory”, Nucl. Phys. B, 584 (2000), 300–312 | DOI | MR | Zbl

[28] Barnaby N., “Caustic formation in tachyon effective field theories”, J. High Energy Phys., 2004, no. 07, Pap. 025 ; arXiv: hep-th/0406120 | MR

[29] Coletti E., Sigalov I., Taylor W., “Taming the tachyon in cubic string field theory”, J. High Energy Phys., 2005, no. 08, Pap. 104 ; arXiv: hep-th/0505031 | MR

[30] Minahan J. A., Mode interactions of the tachyon condensate in $p$-adic string theory, E-print , 2001 arXiv: hep-th/0102071v1 | MR

[31] Lib E., Loss M., Analiz, Nauch. kn., Novosibirsk, 1998

[32] Nikiforov A. F., Uvarov V. B., Spetsialnye funktsii matematicheskoi fiziki, Nauka, M., 1978 | MR

[33] Natanson I. P., Konstruktivnaya teoriya funktsii, Gostekhizdat, M.–L., 1949

[34] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, 4-e izd., Fizmatgiz, M., 1963 | MR

[35] Levitan B. M., Pochti periodicheskie funktsii, Gostekhizdat, M., 1953 | Zbl