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

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 $p=2$.