Vladimirov, V. S. Nonlinear equations for \(p\)-adic open, closed, and open-closed strings. (English) Zbl 1177.81118 Theor. Math. Phys. 149, No. 3, 1604-1616 (2006); translation from Teor. Mat. Fiz. 149, No. 3, 354-367 (2006). Summary: We investigate the structure of solutions of boundary value problems for a one-dimensional nonlinear system of pseudodifferential equations describing the dynamics (rolling) of \(p\)-adic open, closed, and open-closed strings for a scalar tachyon field using the method of successive approximations. For an open-closed string, we prove that the method converges for odd values of \(p\) of the form \(p = 4n+1\) under the condition that the solution for the closed string is known. For \(p = 2\), we discuss the questions of the existence and the nonexistence of solutions of boundary value problems and indicate the possibility of discontinuous solutions appearing. Cited in 30 Documents MSC: 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 35S05 Pseudodifferential operators as generalizations of partial differential operators 45G10 Other nonlinear integral equations Keywords:string; tachyon PDFBibTeX XMLCite \textit{V. S. Vladimirov}, Theor. Math. Phys. 149, No. 3, 1604--1616 (2006; Zbl 1177.81118); translation from Teor. Mat. Fiz. 149, No. 3, 354--367 (2006) Full Text: DOI arXiv References: [1] L. Brekke and P. G. O. Freund, Phys. Rep., 233, 1 (1993). [2] N. Moeller and M. Schnabl, JHEP, 0401, 011 (2004). [3] I. M. Gelfand and G. E. Shilov, Generalized Functions and Operations on Them [in Russian], Vol. 2, Spaces of Fundamental and Generalized Functions, Fizmatlit., Moscow (1958); English transl.: Generalized Functions, Vol. 2, Spaces of Fundamental and Generalized Functions, Acad. Press, New York (1968). [4] V. S. Vladimirov and Ya. I. Volovich, Theor. Math. Phys., 138, 297 (2004); math-ph/0306018 (2003). · Zbl 1178.81174 [5] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Vols. 1, 2, Cambridge Univ. Press, Cambridge (1987, 1988). [6] L. Brekke, P. G. O. Freund, M. Olson, and E. Witten, Nucl. Phys. B, 302, 365 (1988); 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, Singapore (1994); A. Sen, JHEP, 0204, 048 (2002); hep-th/0203211 (2002); D. Ghoshal and A. Sen, Nucl. Phys. B, 584, 300 (2000); I. V. Volovich, Class. Q. Grav., 4, L83 (1987); J. A. Minahan, JHEP, 0103, 028 (2001); N. Barnaby, JHEP, 0407, 025 (2004); hep-th/0406120 (2004); E. Coletti, I. Sigalov, and W. Taylor, JHEP, 0508, 104 (2005); hep-th/0505031 (2005). [7] P. H. Frampton and Y. Okada, Phys. Rev. D, 37, 3077 (1988). [8] N. Moeller and B. Zwiebach, JHEP, 0210, 034 (2002); hep-th/0207107 (2002). [9] I. Ya. Aref’eva, L. V. Joukovskaja, and A. S. Koshelev, JHEP, 0309, 012 (2003); hep-th/0301137 (2003). [10] Ya. I. Volovich, J. Phys. A, 36, 8685 (2003); math-ph/0301028 (2003). · Zbl 1040.35088 [11] V. S. Vladimirov, Izv. Math., 69, 487 (2005); math-ph/0507018 (2005). · Zbl 1086.81073 [12] L. V. Joukovskaja, Theor. Math. Phys., 146, 335 (2006). · Zbl 1177.81113 [13] G. Calcagni, JHEP, 0605, 012 (2006); hep-th/0512259 (2005). [14] V. S. Vladimirov, Russ. Math. Surveys, 60, 1077 (2005). · Zbl 1138.45006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.