Nonlinear equations for $p$-adic open, closed, and open-closed strings

We investigate the structure of solutions of boundary value problems for a one-dimensional nonlinear system of pseudodifferential equations describing the dynamics {(}rolling{\rm)} 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.