Dynamics with Infinitely Many Time Derivatives and Rolling Tachyons

Both in string field theory and in p-adic string theory the equations ofmotion involve infinite number of time derivatives. We argue that the initialvalue problem is qualitatively different from that obtained in the limit ofmany time derivatives in that the space of initial conditions becomes stronglyconstrained. We calculate the energy-momentum tensor and study in detail timedependent solutions representing tachyons rolling on the p-adic string theorypotentials. For even potentials we find surprising small oscillations at thetachyon vacuum. These are not conventional physical states but ratheranharmonic oscillations with a nontrivial frequency--amplitude relation. Whenthe potentials are not even, small oscillatory solutions around the bottom mustgrow in amplitude without a bound. Open string field theory resembles thislatter case, the tachyon rolls to the bottom and ever growing oscillationsensue. We discuss the significance of these results for the issues of emergingclosed strings and tachyon matter.Comment: 46 pages, 14 figures, LaTeX. Replaced version: Minor typos corrected, some figures edited for clarit