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Uspekhi Mat. Nauk, 2005, Volume 60, Issue 6(366), Pages 73–88 (Mi umn1677)  

This article is cited in 11 scientific papers (total in 11 papers)

On the non-linear equation of a $p$-adic open string for a scalar field

V. S. Vladimirov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: A survey is given of results concerning the structure of solutions of a one-dimensional non-linear pseudodifferential equation describing the dynamics (rolling) of a $p$-adic open string for the scalar tachyon field. It is indicated that discontinuous solutions can occur for $p=2$. The method of expanding the solutions in series of Hermite polynomials and the method of reduction to a non-linear boundary-value problem for the heat equation are used. Some unsolved problems are given, in particular, the existence problem for a solution of the boundary-value problem for $p=2$.

DOI: https://doi.org/10.4213/rm1677

Full text: PDF file (319 kB)
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English version:
Russian Mathematical Surveys, 2005, 60:6, 1077–1092

Bibliographic databases:

Document Type: Article
UDC: 510.6+519.21
MSC: Primary 45G10, 35K05; Secondary 35S15, 33C45, 81T30
Received: 17.08.2005

Citation: V. S. Vladimirov, “On the non-linear equation of a $p$-adic open string for a scalar field”, Uspekhi Mat. Nauk, 60:6(366) (2005), 73–88; Russian Math. Surveys, 60:6 (2005), 1077–1092

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. S. Vladimirov, “Nonlinear equations for $p$-adic open, closed, and open-closed strings”, Theoret. and Math. Phys., 149:3 (2006), 1604–1616  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Vladimirov V.S., “The equation of the p-adic closed strings for the scalar tachyon field”, Sci. China, Ser. A, 51:4 (2008), 754–764  crossref  mathscinet  zmath  isi  elib
    3. Dragovich B., Khrennikov A.Yu., Kozyrev S.V., Volovich I.V., “On p-adic mathematical physics”, P-Adic Num. Ultrametr. Anal. Appl., 1:1 (2009), 1–17  crossref  mathscinet  zmath
    4. Vasily S. Vladimirov, “Mathematical aspects of nonlinear pseudodifferential equations of p-adic strings”, P-Adic Num Ultrametr Anal Appl, 3:3 (2011), 236  crossref  mathscinet  zmath
    5. V. S. Vladimirov, “Matematicheskie voprosy teorii nelineinykh psevdodifferentsialnykh uravnenii $p$-adicheskikh strun”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(22) (2011), 34–41  mathnet  crossref  elib
    6. V. S. Vladimirov, “On some exact solutions in p-adic open-closed string theory”, P-Adic Num Ultrametr Anal Appl, 4:1 (2012), 57  crossref  mathscinet  zmath
    7. S. N. Askhabov, “Periodic solutions of convolution type equations with monotone nonlinearity”, Ufa Math. J., 8:1 (2016), 20–34  mathnet  crossref  isi  elib
    8. Zuniga-Galindo W.A., “Non-Archimedean White Noise, Pseudodifferential Stochastic Equations, and Massive Euclidean Fields”, J. Fourier Anal. Appl., 23:2 (2017), 288–323  crossref  mathscinet  zmath  isi  scopus
    9. Vourdas A., “A Quantum System With Positions in the Profinite Group Z(P)”: Vourdas, A, Finite and Profinite Quantum Systems, Quantum Science and Technology-Series, Springer-Verlag Berlin, 2017, 161–180  crossref  mathscinet  isi
    10. Dragovich B., Khrennikov A.Yu., Kozyrev S.V., Volovich I.V., Zelenov E.I., “P-Adic Mathematical Physics: the First 30 Years”, P-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121  crossref  mathscinet  zmath  isi
    11. Kh. A. Khachatryan, “On the solvability of a boundary value problem in $ p$-adic string theory”, Trans. Moscow Math. Soc., 2018, 101–115  mathnet  crossref  elib
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