Some Properties of Dynamical Equations in $p$-Adic String and SFT Models

We consider nonlinear pseudodifferential equations with an infinite number of derivatives. These equations form a new class of equations in mathematical physics, which first appeared in $p$-adic string theory. The investigation of these equations is of much interest for mathematical physics and its applications, in particular, in string field theory and cosmology. In this paper, we study the existence and the properties of spatially homogeneous solutions of these equations.