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

TL;DR

This paper studies the solutions of nonlinear pseudodifferential equations modeling p-adic string dynamics, proving convergence for certain p-values and discussing existence and discontinuity of solutions.

Contribution

It introduces a method to analyze boundary value problems for p-adic strings, establishing convergence results and exploring solution existence for specific p-values.

Findings

Convergence of the method for p=4n+1 with known closed string solutions.

Existence and nonexistence results for p=2.

Indication of possible discontinuous solutions.

Abstract

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.