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) 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. To Anatolii Alekseevich Logunov on his 80th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 354–367, December, 2006.