Abstract:
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.