Geodesic
On the non-linear equation of a $p$-adic open string for a~scalar field
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1077-1092

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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 $p=2$. 
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     author = {V. S. Vladimirov},
     title = {On the non-linear equation of a $p$-adic open string for a~scalar field},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {1077--1092},
     publisher = {mathdoc},
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     number = {6},
     year = {2005},
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V. S. Vladimirov. On the non-linear equation of a $p$-adic open string for a~scalar field. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1077-1092. http://geodesic.mathdoc.fr/item/RM_2005_60_6_a5/